# Wednesday, August 18: Sheaves ::: {.remark} We'll be covering Hartshorne, chapter 2: - Sections 1-5: Fundamental, sheaves, schemes, morphisms, constant sheaves. - Sections 6-9: Divisors, linear systems of differentials, nonsingular varieties. Note that most of the important material of this book is contained in the exercises! ::: ::: {.remark} Recall that a **topological space** $$X$$ is collection of *open* sets $${\mathcal{U}}= \left\{{U_i \subseteq X}\right\}$$ which is closed under arbitrary unions and finite intersections, where $$X, \emptyset\in {\mathcal{U}}$$. ::: ::: {.definition title="Presheaf"} A **presheaf of abelian groups** $${\mathcal{F}}$$ on $$X$$ a topological space is an assignment to every open $$U \subseteq X$$ an abelian group $${\mathcal{F}}(U)$$ and restriction morphisms $$\rho_{UV}: {\mathcal{F}}(U) \to {\mathcal{F}}(V)$$ for every inclusion $$V \subseteq U$$ satisfying - $${\mathcal{F}}(\emptyset) = 0$$ - $$\rho_{UU}: {\mathcal{F}}(U) \to {\mathcal{F}}(U)$$ is $$\operatorname{id}_{{\mathcal{F}}(U)}$$. - If $$W \subseteq V \subseteq U$$ are opens, then $\rho_{UW} = \rho_{VW} \circ \rho_{UV} .$ We'll write $${\mathcal{F}}(U)$$ to be the **sections of $${\mathcal{F}}$$ over $$U$$**, also notated $${\mathsf{\Gamma}\qty{U; {\mathcal{F}}} }$$ and write the restrictions as $${ \left.{{s}} \right|_{{v}} } = \rho_{UV}(s)$$ for $$V \subseteq U$$. ::: ::: {.example title="Presheaf of continuous functions"} Let $$X \coloneqq{\mathbb{R}}$$ with the standard topology and take $${\mathcal{F}}= C^0({-}; {\mathbb{R}})$$ (continuous real-valued functions) as the associated presheaf. So for $$U \subset {\mathbb{R}}$$ open, the sections are $${\mathcal{F}}(U) \coloneqq\left\{{f: U\to {\mathbb{R}}\text{ continuous}}\right\}$$. For restriction maps, given $$U \subseteq V$$ take the actual restriction of functions $$C^0(V; {\mathbb{R}}) \to C^0(U; {\mathbb{R}})$$. One needs to check the 3 conditions, but we can declare $$C^0(\emptyset; {\mathbb{R}}) = \left\{{0}\right\} = 0$$, and the others follow right away. ::: ::: {.example title="Constant presheaves"} The **constant presheaf** associated to $$A\in {\mathsf{Ab}}$$ on $$X\in {\mathsf{Top}}$$ is denote $$F = \underline{A}$$, where $\underline{A}(U) \coloneqq \begin{cases} A & U \neq \emptyset \\ 0 & U = \emptyset. \end{cases}$ and $\rho_{UV} \coloneqq \begin{cases} \operatorname{id}_A & V \neq \emptyset \\ 0 & V=\emptyset . \end{cases} .$ ::: ::: {.warnings} The constant sheaf is not the sheaf of constant functions! Instead these are *locally* constant functions. ::: ::: {.remark} Let $${\mathsf{Open}}_{/ {X}}$$ denote the category of open sets of $$X$$, defined - $${\operatorname{Ob}}({\mathsf{Open}}_{/ {X}} ) \coloneqq\left\{{U_i}\right\}$$, so each object is an open set. - $${\mathsf{Open}}_{/ {X}} (U, V)$$ is empty when $$V\not\subset U$$ and is the singleton inclusion $$\left\{{\iota: U\hookrightarrow V}\right\}$$ otherwise. ::: ::: {.example title="Of $\\Open\\slice{X}$ "} Take $$X\coloneqq\left\{{p, q}\right\}$$ with the discrete topology to obtain a category with 4 objects: {=tex} \begin{tikzcd} & {\left\{{p, q}\right\}} \\ {\left\{{p}\right\}} && {\left\{{q}\right\}} \\ & \emptyset \arrow[from=3-2, to=1-2] \arrow[from=3-2, to=2-1] \arrow[from=3-2, to=2-3] \arrow[from=2-1, to=1-2] \arrow[from=2-3, to=1-2] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMSwwLCJcXHRze3AsIHF9Il0sWzAsMSwiXFx0c3twfSJdLFsyLDEsIlxcdHN7cX0iXSxbMSwyLCJcXGVtcHR5c2V0Il0sWzMsMF0sWzMsMV0sWzMsMl0sWzEsMF0sWzIsMF1d) Similarly, the indiscrete topology yields $$\emptyset \to \left\{{p, q}\right\}$$, a category with two objects. ::: ::: {.remark} Then a presheaf is a contravariant functor $${\mathcal{F}}: {\mathsf{Open}}_{/ {X}} \to {\mathsf{Ab}}$$ which sends the cofinal/initial object $$\left\{{\emptyset}\right\} \in {\mathsf{Open}}_{/ {X}}$$ to the final/terminal object $$0 \in {\mathsf{Ab}}$$. More generally, we can replace $${\mathsf{Ab}}$$ with any category $$\mathsf{C}$$ admitting a final object: - $$\mathsf{C} \coloneqq\mathsf{CRing}$$ the category of commutative rings, which we'll use to define schemes. - $$\mathsf{C} = {\mathsf{Grp}}$$, the full category of (potentially nonabelian) groups. - $$\mathsf{C} \coloneqq{\mathsf{Top}}$$, arbitrary topological spaces. ::: ::: {.example title="of presheaves"} Let $$X\in {\mathsf{Var}}_{/ {k}}$$ a variety over $$k\in \mathsf{Field}$$ equipped with the Zariski topology, so the opens are complements of vanishing loci. Given $$U \subseteq X$$, define a presheaf of regular functions $${\mathcal{F}}\coloneqq{\mathcal{O}}$$ where - $${\mathcal{O}}(U)$$ are the regular functions $$f:U\to k$$, i.e. functions on $$U$$ which are locally expressible as a ratio $$f = g/h$$ with $$g, h\in k[x_1, \cdots, x_{n}]$$. - Restrictions are restrictions of functions. Taking $$X = {\mathbb{A}}^1_{/ {k}}$$, the Zariski topology is the cofinite topology, so every open $$U$$ is the complement of a finite set and $$U = \left\{{t_1, \cdots, t_m}\right\}^c$$. Then $${\mathcal{O}}(U) = \left\{{\phi: U\to k}\right\}$$ which is locally a fraction, and it turns out that these are all globally fractions and thus ${\mathcal{O}}(U) = \left\{{ {f \over g} {~\mathrel{\Big|}~}f,g\in k[t], g(t) \neq 0 \,\,\, \forall t\in U}\right\} = \left\{{{ f \over \prod (t-t_i)^{m_i}} {~\mathrel{\Big|}~}f\in k[t] }\right\} = k[t] \left[ { \scriptstyle { {S}^{-1}} } \right] ,$ where $$S = \left\langle{\prod t-t_i}\right\rangle$$ is the multiplicative set generated by the factors. This gives an abelian group since we can take least common denominators, and we have restrictions. ::: ::: {.warnings} Note that there are two similar notations for localization which mean different things! For a multiplicative set $$S$$, the ring $$R \left[ { \scriptstyle { {S}^{-1}} } \right]$$ literally means localizing at that set. For $${\mathfrak{p}}\in \operatorname{Spec}R$$, the ring $$R \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right]$$ means localizing at the multiplicative set $$S \coloneqq{\mathfrak{p}}^c$$. ::: # Friday, August 20 ::: {.definition title="Sheaf"} Recall the definition of a presheaf, and the main 3 properties: 1. $$F( \emptyset) = 0$$, 2. $$\rho_{UU} = \operatorname{id}_{{\mathcal{F}}(U)}$$ 3. For all $$W \subseteq V \subseteq U$$, a cocycle condition: $\rho_{UW} = \rho_{VW} \circ \rho_{UV} .$ Write $$s_i \in {\mathcal{F}}(U_i)$$ to be a section. A presheaf is a **sheaf** if it additionally satisfies 4. When restrictions are compatible on overlaps, so ${ \left.{{s_i}} \right|_{{U_i \cap U_j}} } = { \left.{{s_j}} \right|_{{U_i \cap U_j}} } ,$ there exists a uniquely glued section $${\mathcal{F}}(\cup U_i)$$ such that $${ \left.{{s}} \right|_{{U_i}} } = s_i$$ for all $$i$$. ::: ::: {.example title="?"} Take $$C^0({-}; {\mathbb{R}})$$ the sheaf of continuous real-valued functions on a topological space. For $$f_i: U_i \to {\mathbb{R}}$$ agreeing on overlaps, there is a continuous function $$f: \cup U_i\to {\mathbb{R}}$$ restricting to $$f_i$$ on each $$U_i$$ by just defining $$f(x) = f_i(x)$$ for $$x\in U_i$$, which is well-defined by agreement of the $$f_i$$ on overlaps. ::: ::: {.example title="?"} Let $$X$$ be a topological space and $$A\in \mathsf{CRing}$$, then take the constant sheaf $$\underline{A}$$ which maps to $$A$$ iff $$U\neq \emptyset$$ and 0 otherwise. This is not a sheaf, taking $$X = {\mathbb{R}}$$ and $$A = {\mathbb{Z}}/2$$. Let $$U_1 = (0, 1)$$ and $$U_2 = (2, 3)$$ and take $$s_1 = 0$$ on $$U_1$$ and $$s_2 = 1$$ on $$U_2$$. Using that $$U_1 \cap U_2 = \emptyset$$, so they trivially agree on overlaps, but there is no constant function on $$U_1 \cup U_2$$ restricting to 1 on $$U_2$$ and 0 on $$U_1$$ ::: ::: {.definition title="Locally constant sheaves"} The **(locally) constant sheaf** $$\underline{A}$$ on any $$X\in {\mathsf{Top}}$$ is defined as $\underline{A}(U) \coloneqq\left\{{ f: U\to A {~\mathrel{\Big|}~}f \text{ is locally constant} }\right\} .$ ::: ::: {.remark} As a general principle, this is a sheaf since this property can be verified locally. ::: ::: {.example title="?"} Let $$C^0_{\mathrm{bd}}$$ be the presheaf of bounded continuous functions on $$S^1$$. This is not a sheaf, but one needs to go to infinitely many sets: take the image of $$[{1\over n}, {1\over n+1}]$$ with (say) $$f_n(x) = n$$ for each $$n$$. Then each $$f_n$$ is bounded (it's just constant), but the full collection is unbounded, so these can not glue to a bounded function. ::: ::: {.definition title="Stalks"} Let $${\mathcal{F}}\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)$$ and $$p\in X$$, then the **stalk** of $${\mathcal{F}}$$ at $$p$$ is defined as ${\mathcal{F}}_p(U) \coloneqq\lim_{U\ni p} \coloneqq\left\{{(s, U) {~\mathrel{\Big|}~}U\ni p \text{ open}, \, s\in {\mathcal{F}}(U)}\right\}/\sim ,$ where $$(s, U) \sim (t ,V)$$ iff there exists a $$W \ni p$$ with $$W \subset U \cap V$$ with $${ \left.{{s}} \right|_{{W}} } = { \left.{{t}} \right|_{{W}} }$$. An equivalence class $$[(s, U)] \in {\mathcal{F}}_p$$ is referred to as a **germ**. ::: ::: {.example title="?"} Let $$C^\omega({-}; {\mathbb{R}})$$ be the sheaf of analytic functions, i.e. those locally expressible as convergent power series. This is a sheaf because this condition can be checked locally. What is the stalk $$C_0^\omega$$ at zero? An example of a function in this germ is $$[(f(x) = {1\over 1-x}, (-1, 1))$$. A first guess is $${\mathbb{R}} { \left[ {t} \right] }$$, but the claim is that this won't work. Note that there is an injective map $$C_0^\omega \hookrightarrow{\mathbb{R}} { \left[ {t} \right] }$$ because $$f, g$$ have analytic power series expansions at zero, and if these expressions are equal then $${ \left.{{f}} \right|_{{I}} } = { \left.{{g}} \right|_{{I}} }$$ for some $$I$$ containing zero. This map won't be surjective because there are power series with a non-positive radius of convergence, for example taking $$f(t) \coloneqq\sum_{k=0}^\infty {kt}^k$$ which only converges at $$t=0$$. So the answer is that $$C_0^\omega \leq {\mathbb{R}} { \left[ {t} \right] }$$ is the subring of power series with positive radius of convergence. ::: ::: {.definition title="Local ring of the structure sheaf, $\\OO_p$"} Let $$X \in {\mathsf{Alg}}{\mathsf{Var}}$$ and $${\mathcal{O}}$$ its sheaf of regular functions. For $$p\in X$$, the stalk $${\mathcal{O}}_p$$ is the **local ring** of $$X$$ at $$p$$. ::: ::: {.example title="?"} For $$X \coloneqq{\mathbb{A}}^1_{/ {k}}$$ for $$k=\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$$, the opens are cofinite sets and $${\mathcal{O}}(U) = \left\{{f/g {~\mathrel{\Big|}~}f, g\in k[t]}\right\}$$. Consider the stalk $${\mathcal{O}}_p$$. Applying the definition, we have ${\mathcal{O}}_p \coloneqq\left\{{(f/g, U) {~\mathrel{\Big|}~}p\in U,\, g\neq 0 \text {on } U}\right\} / \sim .$ Given any $$g\in k[t]$$ with $$g(p) \neq 0$$, there is a Zariski open set $$U = V(g)^c = D_g$$, the distinguished open associated to $$g$$, where $$g\neq 0$$ on $$U$$ by definition. Thus $$p\in U$$, and so any $$f/g\in \operatorname{ff}{k[t]}$$ with $$p\neq 0$$ defines an element $$(f/g, D_g) \in {\mathcal{O}}_p$$. Concretely: ${ \left.{{f/g}} \right|_{{W}} } = { \left.{{f/g}} \right|_{{W'}} } \implies f/g = f'/g' \in \operatorname{ff}{k[t]} = k(t) ,$ and $$fg' = f'g$$ on the cofinite set $$W$$, making them equal as polynomials. We can thus write ${\mathcal{O}}_p = \left\{{f/g \in k(t) {~\mathrel{\Big|}~}g(p) \neq 0}\right\} = k[t] \left[ { \scriptstyle { {\left\langle{t-p}\right\rangle}^{-1}} } \right], \quad \left\langle{t-p}\right\rangle\in \operatorname{mSpec}k[t] ,$ recalling that $$k[t] \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right] = \left\{{f/g {~\mathrel{\Big|}~}g\not\in {\mathfrak{p}}}\right\}$$. Note that for $$X\in {\mathsf{Aff}}{\mathsf{Var}}$$, so $$X = V(f_i) = V(I)$$ for $$I$$ reduced, we have the coordinate ring $$k[X] = k[x_1, \cdots, x_{n}]/I = R$$, then $${\mathcal{O}}_p = R \left[ { \scriptstyle { {{\mathfrak{m}}_p}^{-1}} } \right]$$ where $${\mathfrak{m}}_p \coloneqq\left\{{f\in R {~\mathrel{\Big|}~}f(p) = 0}\right\}$$. ::: ::: {.warnings} This doesn't quite hold for non-algebraically closed fields. Take $$f(x) x^p-x \in {\mathbb{F}}_p[x]$$, then $$f(x) \equiv 0$$ since every element in $${\mathbb{F}}_p$$ is a root. ::: ::: {.remark} Next time: morphisms of sheaves/presheaves, and isomorphisms can be checked on stalks for sheaves. ::: # Monday, August 23 ::: {.remark} Recall that the **stalk** of a presheaf $${\mathcal{F}}$$ at $$p$$ is defined as ${\mathcal{F}}_p \coloneqq\colim_{U\ni p} {\mathcal{F}}(U) = \left\{{ (s, U) {~\mathrel{\Big|}~}s\in {\mathcal{F}}(U) }\right\}_{/ {\sim}} .$ ::: ::: {.definition title="Morphisms of presheaves"} Let $${\mathcal{F}}, {\mathcal{G}}\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)$$, then a **morphism** $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ is a collection $$\left\{{\phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)}\right\}$$ of morphisms of abelian groups for all $$U\in {\mathsf{Open}}(X)$$ such that for all $$V \subset U$$, the following diagram commutes: {=tex} \begin{tikzcd} {{\mathcal{F}}(U)} && {{\mathcal{G}}(U)} \\ \\ {{\mathcal{F}}(V)} && {{\mathcal{G}}(V)} \arrow["{\phi(U)}", from=1-1, to=1-3] \arrow["{\phi(V)}", from=3-1, to=3-3] \arrow["{\operatorname{res}(UV)}"{description}, from=1-1, to=3-1] \arrow["{\operatorname{res}'(UV)}"{description}, from=1-3, to=3-3] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1jZihVKSJdLFswLDIsIlxcbWNmKFYpIl0sWzIsMCwiXFxtY2coVSkiXSxbMiwyLCJcXG1jZyhWKSJdLFswLDIsIlxccGhpKFUpIl0sWzEsMywiXFxwaGkoVikiXSxbMCwxLCJcXHJlcyhVVikiLDFdLFsyLDMsIlxccmVzJyhVVikiLDFdXQ==) An **isomorphism** is a morphism with a two-sided inverse. ::: ::: {.remark} Note that if we regard a sheaf as a contravariant functor, a morphism is then just a natural transformation. ::: ::: {.remark} A morphism $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ defines a morphisms on stalks $$\phi_p: {\mathcal{F}}_p \to {\mathcal{G}}_p$$. ::: ::: {.example title="of a nontrivial morphism of sheaves"} Let $$X \coloneqq{\mathbb{C}}^{\times}$$ with the classical topology, making it into a real manifold, and take $$C^0({-}; {\mathbb{C}}) \in {\mathsf{Sh}}(X, {\mathsf{Ab}})$$ be the sheaf of continuous functions and let $$C^0({-}; {\mathbb{C}})^{\times}$$ the sheaf of of nowhere zero continuous continuous functions. Note that this is a sheaf of abelian groups since the operations are defined pointwise. There is then a morphism $\exp({-}): C^0({-}; {\mathbb{C}}) &\to C^0({-}; {\mathbb{C}})^{\times}\\ f &\mapsto e^f && \text{ on open sets } U\subseteq X .$ Since exponentiating and restricting are operations done pointwise, the required square commutes, yielding a morphism of sheaves. ::: ::: {.definition title="(co)kernel and image sheaves"} Let $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ be morphisms of presheaves, then define the presheaves $\ker(\phi)(U) &\coloneqq\ker(\phi(U)) \\ \operatorname{coker}^{{\mathsf{pre}}}(\phi)(U) &\coloneqq{\mathcal{G}}(U) / \phi({\mathcal{F}}(U))\\ \operatorname{im}(\phi)(U) &\coloneqq\operatorname{im}(\phi(U)) \\ .$ ::: ::: {.warnings} If $${\mathcal{F}}, {\mathcal{G}}\in {\mathsf{Sh}}(X)$$, then for a morphism $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$, the image and cokernel presheaves need not be sheaves! ::: ::: {.example title="?"} Consider $$\ker \exp$$ where $$\exp: C^0({-}; {\mathbb{C}})\to C^0({-}; {\mathbb{C}})^{\times}\in {\mathsf{Sh}}({\mathbb{C}}^{\times})$$. One can check that $$\ker \exp = 2\pi i \underline{{\mathbb{Z}}}(U)$$, and so the kernel is actually a sheaf. We also have $$\operatorname{coker}^{{\mathsf{pre}}} \exp(U) \coloneqq C^0(U; {\mathbb{C}})/ \exp(C^0(U;{\mathbb{C}})^{\times})$$. On opens, $$\operatorname{coker}^{{\mathsf{pre}}} \exp(U) = \left\{{1}\right\} \iff$$ every nonvanishing continuous function $$g$$ on $$U$$ has a continuous logarithm, i.e. $$g = e^f$$ for some $$f$$. Examples of opens with this property include any contractible (or even just simply connected) open set in $${\mathbb{C}}^{\times}$$. Consider $$U\coloneqq{\mathbb{C}}^{\times}$$ and $$z\in C^0({\mathbb{C}}^{\times}; {\mathbb{C}})^{\times}$$, which is a nonvanishing function. Then the equivalence class $$[z] \in \operatorname{coker}^{{\mathsf{pre}}} \exp({\mathbb{C}}^{\times})$$ is nontrivial, since $$z\neq e^f$$ for any $$f\in C^0({\mathbb{C}}^{\times}; {\mathbb{C}})$$, since any attempted definition of $$\log(z)$$ will have monodromy. on the other hand we can cover $${\mathbb{C}}^{\times}$$ by contractible opens $$\left\{{U_i}\right\}_{i\in I}$$ where $${ \left.{{[z]}} \right|_{{U_i}} } = 1 \in \operatorname{coker}^{{\mathsf{pre}}} \exp (U_i)$$ and similarly $${ \left.{{1}} \right|_{{\operatorname{id}}} } = 1 \in \operatorname{coker}^{{\mathsf{pre}}} \exp(U_i)$$, showing that the cokernel fails the unique gluing axiom and is not a sheaf. ::: ::: {.definition title="Sheafification"} Given any $${\mathcal{F}}\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)$$ there exists an $${\mathcal{F}}^+ \in {\mathsf{Sh}}(X)$$ and a morphism of presheaves $$\theta: {\mathcal{F}}\to {\mathcal{F}}^+$$ such that for any $${\mathcal{G}}\in {\mathsf{Sh}}(X)$$ with a morphism $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ there exists a unique $$\psi: {\mathcal{F}}^+ \to {\mathcal{G}}$$ making the following diagram commute: {=tex} \begin{tikzcd} {\mathcal{F}}&& {\mathcal{G}}\\ \\ && {{\mathcal{F}}^+} \arrow["\theta"', from=1-1, to=3-3] \arrow["\phi", from=1-1, to=1-3] \arrow["{\exists! \psi}"', from=3-3, to=1-3] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXG1jZiJdLFsyLDIsIlxcbWNmXisiXSxbMiwwLCJcXG1jZyJdLFswLDEsIlxcdGhldGEiLDJdLFswLDIsIlxccGhpIl0sWzEsMiwiXFxleGlzdHMhIFxccHNpIiwyXV0=) The sheaf $${\mathcal{F}}^+ \in {\mathsf{Sh}}(X)$$ is called the **sheafification** of $${\mathcal{F}}$$. This is an example of an adjunction of functors: $\mathop{\mathrm{Hom}}_{ \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)}({\mathcal{F}}, {\mathcal{G}}^{\mathsf{pre}}) \cong \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}({\mathcal{F}}^+, {\mathcal{G}}) ,$ where we use the forgetful functor $${\mathcal{G}}\to {\mathcal{G}}^{\mathsf{pre}}$$. This yields an adjoint pair $\adjunction{a}{b}{c}{d} .$ ::: ::: {.proof title="of existence of sheafification"} We construct it directly as $${\mathcal{F}}^+ \coloneqq\left\{{s:U \to {\textstyle\coprod}_{p\in U} {\mathcal{F}}_p }\right\}$$ such that 1. $$s(p) \in {\mathcal{F}}_p$$, 2. The germs are compatible locally, so for all $$p\in U$$ there is a $$V\supseteq p$$ such that for some $$t\in {\mathcal{F}}(V)$$, $$s(p) = t_p$$ for all $$p$$ in $$V$$. ::: {.slogan} Collections of germs that are locally compatible. ::: So about any point, there should be an actual function specializing to all germs in an open set. ::: ::: {.remark} The point will be that $$\operatorname{coker}\exp$$ will be zero as a sheaf, since it'll be zero on a sufficiently small set. ::: # Wednesday, August 25 ::: {.remark} Recall the definition of sheafification: let $${\mathcal{F}}\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X; {\mathsf{Ab}}{\mathsf{Grp}})$$. Construct a sheaf $${\mathcal{F}}^+\in {\mathsf{Sh}}(X, {\mathsf{Ab}}{\mathsf{Grp}})$$ and a morphism $$\theta: {\mathcal{F}}\to {\mathcal{F}}^+$$ of presheaves satisfying the appropriate universal property: {=tex} \begin{tikzcd} {{\mathcal{F}}^+} \\ \\ {\mathcal{F}}&& {\mathcal{G}}\\ \\ {} \arrow["\psi", from=3-1, to=3-3] \arrow["\theta", from=3-1, to=1-1] \arrow["{\exists \tilde \psi}", dashed, from=1-1, to=3-3] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJcXG1jZiJdLFswLDRdLFswLDAsIlxcbWNmXisiXSxbMiwyLCJcXG1jZyJdLFswLDMsIlxccHNpIl0sWzAsMiwiXFx0aGV0YSJdLFsyLDMsIlxcZXhpc3RzIFxcdGlsZGUgXFxwc2kiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=) So any presheaf morphism to a sheaf factors through the sheafification uniquely (via $$\theta$$). Note that this is a instance of a general free/forgetful adjunction. We can construct it as ${\mathcal{F}}^+(U) \coloneqq\left\{{s:U\to {\textstyle\coprod}_{p\in U} {\mathcal{F}}_p,\quad s(p) \in {\mathcal{F}}_p, \cdots}\right\} .$ where the addition condition is that for all $$q\in U$$ there exists a $$V\nu q$$ and $$t\in {\mathcal{F}}(V)$$ such that $$t_p = s(p)$$ for all $$p\in V$$. Note that $$\theta$$ is defined by $$\theta(U)(s) = \left\{{s:p\to s_p}\right\}$$, the function assigning points to germs with respect to the section $$s$$. Idea: this is like replacing an analytic function on an interval with the function sending a point $$p$$ to its power series expansion at $$p$$. ::: ::: {.example title="?"} Recall $$\exp: C^0 \to (C^0)^{\times}$$ on $${\mathbb{C}}^{\times}$$, then $$\operatorname{coker}^{\mathsf{pre}}(\exp)(U) = \left\{{1}\right\}$$ on contractible $$U$$, using that one can choose a logarithm on such a set. However $$\operatorname{coker}^{\mathsf{pre}}(\exp)({\mathbb{C}}^{\times}) \neq \left\{{1}\right\}$$ since $$[z]\in (C^0)^{\times}({\mathbb{C}}^{\times})/\exp(C^0({\mathbb{C}}^{\times}))$$. ::: ::: {.remark} Letting $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ be a morphisms of sheaves, then we defined $$\operatorname{coker}(\phi) \coloneqq(\operatorname{coker}^{\mathsf{pre}}(\phi))^+$$ and $$\operatorname{im}(\phi) \coloneqq(\operatorname{im}^{\mathsf{pre}}(\phi))^+$$. Then $\operatorname{coker}^{\mathsf{pre}}(\exp) &\to \operatorname{coker}(\exp) \\ s\in {\mathcal{F}}(U) &\mapsto s(p) = s_p .$ The claim is that $$[z]_p = 1$$ for all $$p\in {\mathbb{C}}^{\times}$$, since we can replace $$[([z], {\mathbb{C}}^{\times})]$$ with $$([z]_U, U)$$ for $$U$$ contractible. ::: ::: {.example title="?"} A useful example to think about: $$X = \left\{{p, q}\right\}$$ with - $${\mathcal{F}}(p) = A$$ - $${\mathcal{F}}(q) = B$$ - $${\mathcal{F}}(X) = 0$$ Then local sections don't glue to a global section, so this isn't a sheaf, but it is a presheaf. The sheafification satisfies $${\mathcal{F}}^+(X) = A\times B$$. ::: ## Subsheaves ::: {.definition title="Subsheaves, injectivity, surjectivity"} $${\mathcal{F}}'$$ is a **subsheaf** of $${\mathcal{F}}$$ if - $${\mathcal{F}}'(U) \leq {\mathcal{F}}(U)$$ for all $$U$$, - $$\mathop{\mathrm{Res}}'(U, V) = { \left.{{ \mathop{\mathrm{Res}}(U, V) }} \right|_{{{\mathcal{F}}'(U)}} }$$. $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ is **injective** iff $$\ker \phi = 0$$, **surjective** if $$\operatorname{im}(\phi) = {\mathcal{G}}$$ or $$\operatorname{coker}\phi = 0$$. ::: ::: {.exercise title="?"} Check that $$\ker \phi$$ already satisfies the sheaf property. ::: ::: {.definition title="Exact sequences of sheaves"} Let $$\cdots \to {\mathcal{F}}^{i-1} \xrightarrow{\phi^{i-1}} {\mathcal{F}}^i \xrightarrow{\phi^i} {\mathcal{F}}^{i+1}\to \cdots$$ be a sequence of morphisms in $${\mathsf{Sh}}(X)$$, this is **exact** iff $$\ker \phi^i = \operatorname{im}\phi^{i-1}$$. ::: ::: {.lemma title="?"} $$\ker \phi$$ is a sheaf. ::: ::: {.proof title="?"} By definition, $$\ker(\phi)(U) \coloneqq\ker \qty{ \phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U) }$$, satisfying part (a) in the definition of presheaves. We can define restrictions $${ \left.{{\mathop{\mathrm{Res}}(U, V)}} \right|_{{\ker(\phi)(U)}} } \subseteq \ker(\phi)(V)$$. Use the commutative diagram for the morphism $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$. Now checking gluing: Let $$s_i \in \ker(\phi)(U_i)$$ such that $$\mathop{\mathrm{Res}}(s_i, U_i \cap U_j) = \mathop{\mathrm{Res}}(s_j, U_i \cap U_j)$$ for all $$i, j$$. This holds by viewing $$s_i \in {\mathcal{F}}(U_i)$$, so $$\exists ! s\in {\mathcal{F}}(\displaystyle\bigcup_i U_i)$$ such that $$\mathop{\mathrm{Res}}(s, U_i) = s_i$$. We want to show $$s\in \ker(\phi)\qty{\displaystyle\bigcup U_i}$$, so consider $t\coloneqq\phi\qty{ \displaystyle\bigcup_i U_i}(s) \in {\mathcal{G}}\qty{\displaystyle\bigcup U_i} ,$ which is zero. Now $\mathop{\mathrm{Res}}(t, U_i) = \phi(U_i)(\mathop{\mathrm{Res}}(s, U_i)) = \phi(U_i)(s_i) = 0$ by assumption, using the commutative diagram. By unique gluing for $${\mathcal{G}}$$, we have $$t=0$$, since $$0$$ is also a section restricting to $$0$$ everywhere. ::: ::: {.definition title="Quotients"} For $${\mathcal{F}}' \leq {\mathcal{F}}$$ a subsheaf, define the **quotient** $${\mathcal{F}}/{\mathcal{F}}' \coloneqq(({\mathcal{F}}/{\mathcal{F}}')^{\mathsf{pre}})^+$$ where $({\mathcal{F}}/{\mathcal{F}}')^{\mathsf{pre}}(U) \coloneqq{\mathcal{F}}(U)/ {\mathcal{F}}'(U) .$ ::: # Friday, August 27 ::: {.theorem title="Sheaf isomorphism iff isomorphism on stalks"} Let $$\phi:{\mathcal{F}}\to{\mathcal{G}}$$ be a morphism in $${\mathsf{Sh}}(X)$$, then $$\phi$$ is an isomorphism iff $$\phi_p: {\mathcal{F}}_p \to{\mathcal{G}}_p$$ is an isomorphism for all $$p\in X$$. ::: ::: {.proof title="$\\implies$"} Suppose $$\phi$$ is an isomorphism, so there exists a $$\psi: {\mathcal{G}}\to {\mathcal{F}}$$ which is a two-sided inverse for $$\phi$$. Then $$\psi_p$$ is a two-sided inverse to $$\phi_p$$, making it an isomorphism. This follows directly from the formula: $\phi_p: {\mathcal{F}}_p &\to {\mathcal{G}}_p \\ (s, U) & \mapsto (\phi(U)(s), U) .$ ::: ::: {.proof title="$\\impliedby$"} It suffices to show $$\phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)$$ is an isomorphism for all $$U$$. This is because we could define $$\psi(U):{\mathcal{G}}(U) \to {\mathcal{F}}(U)$$ and set $$\phi^{-1}(U) \coloneqq\psi(U)$$, then reversing the arrows in the diagram for a sheaf morphism again yields a commutative diagram. ::: {.claim} $$\phi(U)$$ is injective. ::: For $$s\in {\mathcal{F}}(U)$$, we want to show $$\phi(U)(s) = 0$$ implies $$s=0$$. Consider the germs $$(s, U) \in {\mathcal{F}}_p$$ for $$p\in U$$, we have $$\phi_p(s, U) = (0, U) = 0\in {\mathcal{F}}_p$$. So $$S_p = 0$$ for all $$p\in U$$. Since we have a germ, there exists $$V_p \ni p$$ open such that $${ \left.{{s}} \right|_{{V_p}} } = 0$$. Noting that $$\left\{{V_p {~\mathrel{\Big|}~}p\in U}\right\}\rightrightarrows U$$, by unique gluing we get an $$s$$ where $${ \left.{{s}} \right|_{{V_p}} } = 0$$ for all $$V_p$$, so $$s\equiv 0$$ on $$U$$. ::: {.claim} $$\phi(U)$$ is surjective. ::: Let $$t\in {\mathcal{G}}(U)$$, and consider germs $$t_p\in {\mathcal{G}}_p$$. There exists a unique $$s_p\in {\mathcal{F}}_p$$ with $$\phi_p(s_p) = t_p$$, since $$\phi_p$$ is an isomorphism of stalks by assumption. Use that $$s_p$$ is a germ to get an equivalence class $$(s_p, V)$$ where $$V \subseteq U$$. We have $$\phi(V)(s(p), V) \sim (t, U)$$, noting that $$s$$ depends on $$p$$. Having equivalent germs means there exists a $$W(p) \subseteq V$$ with $$p\in W$$ with $$\phi(W(p)) \qty{{ \left.{{s(p)}} \right|_{{W}} }} = { \left.{{t}} \right|_{{W(p)}} }$$. We want to glue these $$\left\{{ { \left.{{s(p)}} \right|_{{W(p)}} } {~\mathrel{\Big|}~}p\in U }\right\}$$ together. It suffices to show they agree on intersections. Taking $$p, q\in U$$, both $${ \left.{{s(p)}} \right|_{{W(p) \cap W(q)}} }$$ and $${ \left.{{s(q)}} \right|_{{W(p) \cap W(q)}} }$$ map to $${ \left.{{t}} \right|_{{W(p) \cap W(q)}} }$$ under $$\phi(W(p) \cap W(q) )$$. Injectivity will force these to be equal, so $$\exists ! s \in {\mathcal{F}}(U)$$ with $${ \left.{{s}} \right|_{{W(p)}} } = s(p)$$. We want to now show that $$\phi(U)(s) = t$$. Using commutativity of the square, we have $$\phi(U)(s) { \left.{{}} \right|_{{W(p)}} } = \phi(W(p)) \qty{{ \left.{{s}} \right|_{{W(p)}} } }$$. This equals $$\phi(W(p))(s(p)) = { \left.{{t}} \right|_{{W(p)}} }$$. Therefore $$\phi(U)(s)$$ and $$t$$ restrict to sections $$\left\{{w(p) {~\mathrel{\Big|}~}p\in U}\right\}$$. Using unique gluing for $${\mathcal{G}}$$ we get $$\phi(U)(s) = t$$. ::: ::: {.remark} Note: we only needed to check overlaps because of exactness of the following sequence: $0 \to{\mathcal{F}}(U) \to \prod_{i\in I} {\mathcal{F}}(U_i) \to \prod_{i ::: # Monday, August 30 ::: {.remark} Let $$R$$ be a commutative unital ring in which $$0\neq 1$$ unless $$R=0$$. The goal is to define a space $$X$$ such that $$R$$ is the ring of functions on $$X$$, imitating the correspondence between $$X$$ a manifold and $$C^0(X; {\mathbb{R}})$$. Recall that an ideal $${\mathfrak{p}}\in \operatorname{Id}(R)$$ is **prime** iff $${\mathfrak{p}}\subset A$$ is a proper subset and $$fg\in {\mathfrak{p}}\implies f\in {\mathfrak{p}}$$ or $$g\in {\mathfrak{p}}$$. ::: ::: {.definition title="Spectrum of a ring"} For $$A$$ a ring, $$\operatorname{Spec}(A)$$ is the set of prime ideals. Topologize this by setting the closed sets to be of the form $$V(I) = \left\{{ {\mathfrak{p}}\in \operatorname{Spec}(A) {~\mathrel{\Big|}~}{\mathfrak{p}}\supseteq I }\right\}$$. ::: ::: {.remark} Ideals are "contagious" under multiplication, so prime ideals have reverse contagion! It remains to prove that $$\operatorname{Spec}(A)$$ forms a topological space. ::: ::: {.example title="?"} For $$A$$ a field, $$\operatorname{Spec}(A) = \left\{{\left\langle{0}\right\rangle}\right\}$$, since any other nonzero element would be a unit and put 1 in the ideal. ::: ::: {.example title="?"} For $$k$$ an algebraically closed field, $$\operatorname{Spec}k[t] = \left\{{ \left\langle{0}\right\rangle, \left\langle{t-a}\right\rangle {~\mathrel{\Big|}~}a\in A}\right\}$$. This is a PID, so every ideal is of the form $$I = \left\langle{f}\right\rangle$$, so \[ V(\left\langle{f}\right\rangle) = \begin{cases} \operatorname{Spec}k[t] & f=0 \\ \left\langle{x-a_1, \cdots, a-a_k}\right\rangle & f = \prod_{i=1}^k (x-a_i) \end{cases} .$ Note that this is not the cofinite topology, since $$f=0$$ is a generic point. ::: # Wednesday, September 01 ::: {.example title="?"} Let $$k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$$ be algebraically closed, then $\operatorname{Spec}k[x] = \left\{{ \left\langle{x-a}\right\rangle {~\mathrel{\Big|}~}a\in k}\right\}\cup\left\langle{0}\right\rangle .$ Similarly, $\operatorname{Spec}k[x, y] = \left\{{ \left\langle{x-a, y-b}\right\rangle {~\mathrel{\Big|}~}a,b\in k}\right\} \cup\left\{{\left\langle{f}\right\rangle {~\mathrel{\Big|}~}f \text{ irreducible}}\right\} \cup\left\langle{0}\right\rangle .$ Note that both have non-closed, generic points. ::: ::: {.example title="?"} Consider $$X \coloneqq\operatorname{Spec}{ {\mathbb{Z}}_p }$$ and $$Y\coloneqq\operatorname{Spec}{\mathbb{C}} { \left[ {t} \right] }$$, then $$\operatorname{Spec}(X) = \left\{{\left\langle{p}\right\rangle, \left\langle{0}\right\rangle}\right\}$$ and $$\operatorname{Spec}(Y) = \left\{{ \left\langle{t}\right\rangle, \left\langle{0}\right\rangle }\right\}$$. Both are two point spaces, with open points $$\left\langle{0}\right\rangle$$ and closed points $$\left\langle{p}\right\rangle$$ and $$\left\langle{f}\right\rangle$$ respectively. This spaces are homeomorphic, and later we'll see that we can distinguish them as ringed spaces. ::: ::: {.proposition title="Prime spectra of rings"} Let $$A\in\mathsf{CRing}$$, then $$\operatorname{Spec}A$$ with the closed sets declared to be those of the form $$V(I) = \left\{{p \in \operatorname{Spec}(A) {~\mathrel{\Big|}~}p\supseteq I}\right\}$$. ::: ::: {.lemma title="?"} $$V(IJ) = V(I) \cup V(J)$$, so if a prime ideal $$p$$ contains $$IJ$$ then $$p\supseteq I$$ or $$p\supseteq J$$. ::: ::: {.proof title="?"} $$\impliedby$$: If $$I \subseteq P$$ or $$J \subseteq P$$, then $$IJ \subseteq I$$ and $$IJ \subseteq J$$, so $$IJ \subset p$$. $$\implies$$: Suppose $$IJ \subset p$$ but $$J \not\subset p$$, so pick $$j\in J \setminus p$$. Then for all $$i\in I$$, we have $$ij\in IJ \subseteq p$$, forcing $$i\in p$$. ::: ::: {.lemma title="?"} An arbitrary intersection satisfies $$\displaystyle\bigcap_i V(I_i) = V(\sum_i I_i)$$. ::: ::: {.proof title="?"} $$\implies$$: For $$p\in \operatorname{Spec}(A)$$, we want to show that $$p \supseteq\sum I_i$$ iff $$p \supseteq I_i$$ for all $$i$$, so $$I_i \subseteq \sum I_i \subset P$$. $$\impliedby$$: Ideals are additive groups, regardless of whether or not they're prime! ::: ::: {.proof title="of proposition"} {=tex} \envlist  - $$\emptyset$$ is closed, since $$\emptyset = V(A)$$ - $$X$$ is closed, since $$X = V(0)$$ and $$O$$ is contained in every prime ideal. - Closure under finite unions: by induction, it's enough to show that $$V(I) \cup V(J)$$ is closed. This follows from the 1st lemma above. - Closure under arbitrary unions: this follows from the 2nd lemma. ::: ::: {.proposition title="?"} $$V(I) = V(\sqrt I)$$. The proof is simple: prime ideals are radical. ::: ::: {.example title="?"} Note that $$\operatorname{Spec}{\mathbb{Z}}= \left\{{ \left\langle{p}\right\rangle, \left\langle{0}\right\rangle {~\mathrel{\Big|}~}p \text{ is prime}}\right\}$$. Note that maximal ideals are always closed points, and $$\left\langle{0}\right\rangle$$ is not a closed point. This is homeomorphic to, say $$\operatorname{Spec}\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{Q}}\mkern-1.5mu}\mkern 1.5mu[t]$$. ::: ::: {.definition title="?"} Suppose $$p \subseteq A$$ is a prime ideal, then the **localization** of $$A$$ at $$p$$, $$A \left[ { \scriptstyle { {(p^c)}^{-1}} } \right]$$ (or $$A_p$$) is defined as $A \left[ { \scriptstyle { {(p^c)}^{-1}} } \right] \coloneqq\left\{{ a/f {~\mathrel{\Big|}~}f\not\in p}\right\}_{/ {\sim }} && {a\over f}\sim {b\over g}\iff \exists \, h\in A \text{ s.t. } h(ag-bf)=0 .$ This makes the elements of $$p^c$$ invertible, and is a local ring with residue field $$\kappa = \operatorname{ff}(A/p)$$ and maximal ideal $$pA_p$$. Ideals of $$A_p$$ biject with ideals of $$A$$ contained in $$p$$. ::: ::: {.remark} Idea: $$A_p$$ should look like germs of functions at the point $$p$$. Note that localizing at the ideal $$p$$ is like deleting $${ \operatorname{cl}} _X(V(p))$$, which is also useful. We now want to construct a sheaf $${\mathcal{O}}= {\mathcal{O}}_{\operatorname{Spec}A}$$ which has stalks $$A_p$$. We'll construct something that's obviously a sheaf, at the cost of needing to work hard to prove things about it! ::: ::: {.definition title="Structure sheaf"} For $$U\in \operatorname{Spec}(A)$$ open, so $$U = V(I)^c$$, define the **structure sheaf of $$X$$** as the sheaf given ${\mathcal{O}}(U) \coloneqq\left\{{ s:U \to \displaystyle\coprod{p\in U} A_p {~\mathrel{\Big|}~}s(p) \in A_p, \text{ and } s \text{ is locally a fraction}}\right\} .$ Here *locally a fraction* means that for all $$p\in U$$ there is an open $$p\in V \subseteq U$$ and elements $$a, f\in A$$ such that 1. $$f\not\in Q$$ for any $$Q\in V$$ and 2. $$s(Q) = a/f$$ for all $$Q \in V$$. Restriction is defined for $$V \subseteq U$$ as honest function restriction on $${\mathcal{O}}(U) \to {\mathcal{O}}(V)$$. ::: ::: {.remark} Note that this is sheafifying the presheaf $$U = D_f \mapsto A_f$$. ::: ::: {.example title="?"} Let $$k \in \mathsf{Field}$$, then $$X\coloneqq\operatorname{Spec}(k) = \left\{{\left\langle{0}\right\rangle}\right\}$$ and $${\mathcal{O}}_X$$ is determined by ${\mathsf{\Gamma}\qty{X; {\mathcal{O}}_X} } = \left\{{s: \operatorname{Spec}k \to k {~\mathrel{\Big|}~}\cdots}\right\} = k ,$ since the conditions are vacuous here. ::: ::: {.example title="?"} Let $$X = \operatorname{Spec}{\mathbb{C}} { \left[ {t} \right] } = \left\{{ \left\langle{0}\right\rangle, \left\langle{1}\right\rangle}\right\}$$ and $${\mathcal{O}}_X(X) = {\mathbb{C}} { \left[ {t} \right] }$$ and $${\mathcal{O}}_X(\left\langle{0}\right\rangle) = {\mathbb{C}} { \left( {t} \right) }$$. ::: # Friday, September 03 ::: {.remark} Last time: we defined $$\operatorname{Spec}A$$ as a topological space and $${\mathcal{O}}_{\operatorname{Spec}A}$$, a sheaf of rings on $$\operatorname{Spec}A$$ which evidently satisfied the gluing condition: ${\mathcal{O}}_{\operatorname{Spec}A}(U) \coloneqq\left\{{s: U\to \displaystyle\coprod_{p\in U} A \left[ { \scriptstyle { {p}^{-1}} } \right] {~\mathrel{\Big|}~}s(p) \in A \left[ { \scriptstyle { {p}^{-1}} } \right] \forall p,\,\, s \text{ is locally a fraction}}\right\} .$ ::: ::: {.example title="?"} Set $$X\coloneqq{\mathbb{A}}^1_{/ {k}} \coloneqq\operatorname{Spec}k[t]$$ for $$k=\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$$. Take a point $$\left\langle{t}\right\rangle = \left\langle{t-0}\right\rangle \in \operatorname{Spec}k[t]$$ corresponding to 0, then $${\mathcal{O}}_{X}(X\setminus\left\{{0}\right\}) = k[t, t^{-1}]$$, i.e. functions of the form $$f/t^k$$ for any $$k$$. Generally for $$p = \left\langle{t-a_i}\right\rangle$$ we get $$s_p \in k[t] \left[ { \scriptstyle { {\left\{{t-a_i}\right\}}^{-1}} } \right]$$. Note that for $$p = \left\langle{0}\right\rangle$$, we get $$s_p \in k(t)$$. ::: ::: {.claim} $$s$$ is determined by $$s_{\left\langle{0}\right\rangle}$$, so there is an injective map ${\mathcal{O}}(k[t]\setminus\left\{{0}\right\}) &\to k(t) \\ s &\mapsto s_{\left\langle{0}\right\rangle} .$ ::: ::: {.proof title="?"} Note that $$\left\langle{0}\right\rangle$$ is in every open set, so locally near $$p$$ there exists a $$P\in V$$ and $$a,f$$ with $$f\not\in Q$$ for all $$Q$$ and $$s_Q = a/f$$ for all $$Q\in V$$. Since $$\left\langle{0}\right\rangle \in V$$, we have $$s_{\left\langle{0}\right\rangle} = a/f \in k(t)$$ and $$s_p = a/f\in A_p$$. Since $$A_p \subseteq k(t)$$, we get $$s_p = s_{\left\langle{0}\right\rangle}$$ under this inclusion. ::: ::: {.claim} $${\mathcal{O}}(\operatorname{Spec}k[t]\setminus\left\{{0}\right\}) = k[t, t^{-1}]$$. ::: ::: {.proof title="?"} We showed that the LHS is a subset of $$k(t)$$, so which subsets can be written as things that are locally fractions on the complement of zero. $$\supseteq$$: This can clearly be done in $$k[t, t^{-1}]$$ since every element is locally the fraction $$f/t^k$$. $$\subseteq$$: Suppose $$f/g$$ with $$f,g$$ coprime (this is a PID!) with a pole away from zero, so $$g\in Q$$ for some $$Q\neq \left\langle{0}\right\rangle$$. But then $$f/g$$ isn't in $$A_Q$$. ::: ::: {.remark} Note that $$X \coloneqq\operatorname{mSpec}k[t] \subseteq X' \coloneqq\operatorname{Spec}k[t]$$ as the set of closed points, and restricting $${\mathcal{O}}_{X'}$$ to $$X$$ yields the sheaf of regular functions. Having the extra generic point was useful! ::: ::: {.exercise title="?"} Show that the maximal ideals in $$\operatorname{mSpec}A$$ correspond precisely to closed points. ::: ::: {.example title="?"} Of a function that is locally but not globally a fraction. Take $$A \coloneqq k[x,y,z,w]/\left\langle{xy-zw}\right\rangle$$, which is the cone over a smooth quadric surface and $$X\coloneqq\operatorname{Spec}A$$. Then take $$U = \operatorname{Spec}(A) \setminus V(y, w) = V(y)^c \cap V(w)^c$$ and consider the section $s(p) \coloneqq \begin{cases} x/w & p\in V(w)^c \\ z/y & p\in V(y)^c. \end{cases}$ For $$p\in U$$, it makes sense to consider $$x/w$$ and $$z/y$$. Are they equal? The answer is yes because $$xy-zw = 0$$. Check that this can't be a global fraction, which is a consequence of this random open set not being the complement of localizing at a prime ideal. ::: ::: {.definition title="?"} Given $$f\in A$$, the **distinguished open** $$D(f)$$ corresponding to $$f$$ is defined as $D(f) = V(\left\langle{f}\right\rangle)^c \coloneqq\left\{{p\in \operatorname{Spec}(A) {~\mathrel{\Big|}~}f\in p}\right\}^c = \left\{{p\in \operatorname{Spec}A {~\mathrel{\Big|}~}f\not\in p}\right\} ,$ i.e. the points of $$\operatorname{Spec}(A)$$ where $$f$$ doesn't vanish. ::: ::: {.remark} The sets $$\left\{{D(f) {~\mathrel{\Big|}~}f\in A}\right\}$$ for a basis for the topology on $$\operatorname{Spec}(A)$$. This follows from writing $$V(I)^c = \displaystyle\bigcup_{f\in I} D(f)$$. ::: ::: {.theorem title="Hartshorne Prop 2.2"} Let $$A\in \mathsf{CRing}$$ be unital with $$1\neq 0$$ unless $$A=0$$ and consider $$(\operatorname{Spec}A, {\mathcal{O}})$$. Then a. For any $$p\in \operatorname{Spec}A$$, the stalk $${\mathcal{O}}_p \cong A \left[ { \scriptstyle { {p}^{-1}} } \right]$$. b. For any $$f\in A$$, $${\mathcal{O}}(D(f)) = A \left[ { \scriptstyle { {f}^{-1}} } \right]$$. c. Taking $$f=1$$, $$\Gamma(\operatorname{Spec}A, {\mathcal{O}}) = A$$. ::: ::: {.remark} Note that (b) gives the values of $${\mathcal{O}}$$ on a basis of opens, which determines the sheaf. ::: ::: {.proof title="of a"} Define a map $f_p: {\mathcal{O}}_p &\to A_p \\ (U, s) &\mapsto s(p) .$ This is well-defined since $$p\in W$$ for any $$W \subseteq U \cap V$$ for equivalent germs $$(U, s) \sim (V, t)$$. Surjectivity: given $$x=a/g \in A_p$$, we want $$(U, s)\in {\mathcal{O}}_p$$ such that $$f_p(U, s) = a/g$$, so just take $$U = D(g)$$ and $$s=a/g$$ (which makes sense!) and evidently maps to $$a/g$$. Injectivity: supposing $$f_p(U, s) = 0$$ in $$A_p$$, we want $$(U, s) = 0$$. If $$s(p) = 0$$, then there exists some $$h\in P$$ with $$h\cdot s(p) = 0$$. Since $$s(p)$$ is locally a fraction, we can find $$p\in V \subseteq U$$ with $$s=a/g$$ on $$V$$ with $$g\neq 0$$ on $$V$$, so consider $$V \cap D(h)$$. The claim is that $$s$$ is 0 here, which follows from $$h\cdot (a/g) = 0$$. ::: # Wednesday, September 08 ::: {.remark} Recall that we defined a first version of *affine schemes*, namely pairs $$(\operatorname{Spec}A, {\mathcal{O}}_A)$$ where for $$U \subseteq \operatorname{Spec}A$$ open we have $$s\in {\mathcal{O}}(U)$$ locally represented by $${ \left.{{s}} \right|_{{V}} } = a/f$$ for $$V \subseteq U$$ where $$a, f\in A$$ and $$V(f) \cap V = \emptyset$$, so $$f$$ doesn't vanish on $$V$$. Note that the $$D(f)$$ form a topological basis for $$\operatorname{Spec}A$$, and the gluing condition is difficult, i.e. $${\mathcal{O}}(U)$$ may be hard to compute. We proved that $$OO_{{\mathfrak{p}}} = A \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right]$$ last time, and today we're showing - $${\mathcal{O}}(D(f)) = A \left[ { \scriptstyle { {f}^{-1}} } \right]$$, - $${\mathsf{\Gamma}\qty{(} }\operatorname{Spec}A, {\mathcal{O}}) \cong A$$. ::: ::: {.proof title="of b and c"} $$b\implies c$$: Take $$f=1\in A$$, then $${\mathcal{O}}(\operatorname{Spec}A) = {\mathcal{O}}(D(1)) = A$$ using (b), so the only hard part is showing (b). To prove (b), by definition of $${\mathcal{O}}$$ there is a ring morphism $\psi: A \left[ { \scriptstyle { {f}^{-1}} } \right] &\to {\mathcal{O}}(D(f)) \\ {a\over f^n} &\mapsto {a\over f^n} .$ Note that this is just a careful statement, since the morphisms on stalks $$\psi_{{\mathfrak{p}}}: A \left[ { \scriptstyle { {f}^{-1}} } \right] \to A \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right]$$ by not be injective in general. ::: {.claim} $$\psi$$ is bijective. ::: ::: {.proof title="of injectivity"} Suppose $$\psi(s) = 0$$, we then want to show $$s=0$$. Write $$s = a/f^n$$, then for all $${\mathfrak{p}}\in D(f)$$ we know $$a/f^n = 0 \in A \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right]$$. So for each $${\mathfrak{p}}$$ there is some $$h_{{\mathfrak{p}}} \not\in{\mathfrak{p}}$$ where $h_{{\mathfrak{p}}}(a\cdot 1 - f^n\cdot 0) = 0 && \text{in } A$ in $$A$$. Consider the ideal $${\mathfrak{a}}\coloneqq\operatorname{Ann}(a) \coloneqq\left\{{b\in A {~\mathrel{\Big|}~}ab=0 \in A}\right\} \ni h_{{\mathfrak{p}}}$$. So take the closed subset $$V({\mathfrak{a}})$$, which does not contain $${\mathfrak{p}}$$ since $${\mathfrak{a}}\not\subseteq{\mathfrak{p}}$$. Now iterating over all $${\mathfrak{p}}\in D(f)$$, we get $$V({\mathfrak{a}}) \cap D(f) = \emptyset$$. So $$V({\mathfrak{a}}) \subseteq V(f) = D(f)^c$$, thus $$f\in \sqrt{{\mathfrak{a}}}$$ and $$f^m a = 0$$ for some $$m$$. Then $$f^m(a\cdot 1 - f^n\cdot 0) = 0$$ in $$A$$, so $$a/f^n = 0$$ in $$A \left[ { \scriptstyle { {f}^{-1}} } \right]$$. ::: ::: {.proof title="of surjectivity"} **Step 1**: Expressing $$s\in {\mathcal{O}}(D(f))$$ nicely locally. By definition of $${\mathcal{O}}_{D(f)}$$, there exist $$V_i$$ with $${ \left.{{s}} \right|_{{V_i}} } = a_i/g_i$$ for $$a_i, g_i\in A$$. We'd like $$g_i = h_i^{m_i}$$ for some $$m_i$$, so $$g$$ is a power of $$h_i$$, but this may not be true a priori. Fix $$V_i = D(h_i)$$, then $$a_i / g_i\in {\mathcal{O}}(D(h_i))$$ implies that $$g_i\not\in {\mathfrak{p}}$$ for any $${\mathfrak{p}}\in D(h_i)$$. This implies that $$D(h_i) \subseteq D(g_i)$$, and taking complements yields $$V(h_i) \supseteq V(g_i)$$, and $$h_i \in \sqrt{\left\langle{g_i}\right\rangle}$$ and $$h_i^{n} = g_i$$. Writing $$g_i = h_i^n/c$$ we have $$a_i/g_i = ca_i / h_i^n$$. Note that $$D(h_i) = D(h_i^n)$$. Now replace $$a_i$$ with $$ca_i$$ and $$g_i$$ with $$h_i$$ to get ${ \left.{{s}} \right|_{{D(h_i)}} } = a_i / h_i .$ **Step 2**: Quasicompactness of $$D(f)$$. Note that $$\left\{{D(h_i)}\right\}_{i\in I} \rightrightarrows D(f)$$, so take a finite subcover $$\left\{{D(h_i)}\right\}_{i\leq m}$$. Proof of quasicompactness: since $$D(f) \supseteq\displaystyle\bigcup_{i\in I} D(h_i)$$, we get $V(f) \subseteq \displaystyle\bigcap_{i\in I} V(h_i) = V\qty{ \sum h_i} .$ So $$f^u \in \sum h_i$$, and up to reordering we can conclude $$f^u = \sum_{i\leq m} b_i h_i$$ for some $$b_i \in A$$. Then $$D(f) \subseteq \displaystyle\bigcup_{i\leq m} D(h_i)$$. ::: {.remark} Since we can write $$\operatorname{Spec}A = D(1)$$, it is quasicompact. ::: **Step 3**: Showing surjectivity. Next time. ::: ::: # Friday, September 10 ## Sections and Stalks of $${\mathcal{O}}_{\operatorname{Spec}A}$$ as Localizations {#sections-and-stalks-of-mathcalo_operatornamespeca-as-localizations} ::: {.remark} Last time: any affine scheme is quasicompact, so for each ring $$A$$ and an open cover $${\mathcal{U}}\rightrightarrows D(f)$$ then there is a finite subcover $$\left\{{D(h_i)}\right\}\rightrightarrows D(f)$$. We were looking at proposition: for the ringed space $$(\operatorname{Spec}A, {\mathcal{O}}_A)$$, - $${\mathcal{O}}_{\mathfrak{p}}\cong A \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right]$$ for all $${\mathfrak{p}}\in \operatorname{Spec}A$$, - $${\mathcal{O}}(d(f))\cong A \left[ { \scriptstyle { {f}^{-1}} } \right]$$ for all $$f\in A$$, - $${\mathsf{\Gamma}\qty{\operatorname{Spec}A; {\mathcal{O}}_A} } \cong A$$. Note that $${\mathcal{O}}_A$$ is uniquely characterized by these properties. ::: ::: {.remark} We can write $$D(1) = \operatorname{Spec}A$$ and write $$\left\{{ D(h_i) }\right\} \rightrightarrows\operatorname{Spec}A$$ to obtain $$1^n = \sum b_i h_i$$. This is analogous to a partition of unity, where $$b_i h_i$$ vanishes on $$D(h_i)^c = V(h_i)$$ ::: ::: {.proof title="of 2.2b"} Let $$\psi:A \left[ { \scriptstyle { {f}^{-1}} } \right] \hookrightarrow{\mathcal{O}}(D(f))$$ where we just take restrictions of functions. We know this is injective, and we want to show surjectivity. **Step 1**: Let $$s\in {\mathcal{O}}(D(f))$$. For each open $$D(h_i)$$, write $${ \left.{{s}} \right|_{{D(h_i)}} } = a_i /h_i$$ for some $$a_i \in A$$. **Step 2**: By quasicompactness, write $$f^n = \sum_{1 \leq i\leq m} b_i h_i$$. **Step 3**: Glue the $$a_i/h_i$$ into an element $$a/f$$ of $$A \left[ { \scriptstyle { {f}^{-1}} } \right]$$. *Part 1*: For any $$1\leq i\neq j\leq m$$, $$D(h_i h_j) = D(h_i) \cap D(h_j)$$. Note that $$a_i/h_i = a_j/h_j$$ in $${\mathcal{O}}(D(h_i h_j))$$, and these are elements of $$A \left[ { \scriptstyle { {h_i h_j}^{-1}} } \right]$$ since $$a_i /h_i = a_ih_j/h_i h_j$$. Using injectivity of $$\psi$$ for $$h_i h_j$$, we get an equality $$a_i/h_i = a_j/h_j$$ in $$A_{h_i h_j}$$. Then for $$\ell$$ large enough, $$(h_i h_j)^\ell( a_i h_j - a_j h_i) = 0 \in A$$. Regrouping yields $$h_j^{n+1}(h_i^n a_i) - h_i^{n+1}(h_j a_j) = 0$$, so ${a_i h_i^n \over h_i ^{n+1}} = {a_j h_j^r \over h_j^{n+1}} \coloneqq{\tilde a_i \over \tilde h_i} = {\tilde a_j \over \tilde h_j} ,$ noting that $$D(\tilde h_i) = D(\tilde h_i)$$ since $$\tilde h_i$$ is a power of $$h_i$$. Now use POU gluing to write $$f^n = \sum_i \tilde b_i \tilde h_i$$ and $$a \coloneqq\sum \tilde a_i \tilde h_i\in A$$ be a global function on $$D(f)$$. Then (claim) $$a_j/f^n = \tilde a_j/\tilde h_j$$ on $$D(\tilde h_j)$$. We can rewrite $\tilde h_j a = \sum_i \tilde b_i \tilde a_i \tilde h_j = \sum_i \tilde b_i \tilde a_j \tilde h_i .$ But then $$a/f^n = { \left.{{s}} \right|_{{D(\tilde h_i)}} }$$, so $$s= a/f^n$$. ::: ::: {.example title="?"} Consider $${\mathbb{P}}^1_{/ {k}}$$ as a scheme -- we know the space, and the claim is that we can glue sheaves of affines to obtain a structure sheaf for it. Cover $${\mathbb{P}}^1$$ by $$U_0 = {\mathbb{P}}^1\setminus\left\{{\infty}\right\} \cong {\mathbb{A}}^1$$ and $$U_1 = {\mathbb{P}}^1\setminus\left\{{0}\right\} \cong {\mathbb{A}}^1$$. The gluing data is the following: {=tex} \begin{tikzcd} && {{\mathbb{P}}^1_{/ {k}} } \\ \\ {{\mathbb{A}}^1} & {U_0} && {U_1\cong {\mathbb{A}}^1} & {{\mathbb{A}}^1} \\ \\ & {{\mathbb{A}}^1\setminus\left\{{0}\right\}} & {U_0 \cap U_1 \cong D(t) \subseteq {\mathbb{A}}^1} & {{\mathbb{A}}^1\setminus\left\{{0}\right\}} \arrow["{\phi_0}", hook, from=3-2, to=1-3] \arrow["{\phi_1}"', hook', from=3-4, to=1-3] \arrow[hook', from=5-3, to=3-2] \arrow[hook, from=5-3, to=3-4] \arrow[hook, two heads, from=5-3, to=5-4] \arrow[hook, two heads, from=5-3, to=5-2] \arrow[hook, from=5-2, to=3-1] \arrow[hook', two heads, from=3-1, to=3-2] \arrow[hook, from=5-4, to=3-5] \arrow[hook', two heads, from=3-5, to=3-4] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=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) Here the transition maps are $\phi_1 \circ \phi_0^{-1}: \phi_0( U_0 \cap U_1) &\to \phi_1(U_0 \cap U_1) \\ t &\mapsto t^{-1} .$ What is the map on sheaves? We need a map $${ \left.{{{\mathcal{O}}}} \right|_{{U_0\setminus\left\{{0}\right\}}} } { { \, \xrightarrow{\sim}\, }}{ \left.{{{\mathcal{O}}}} \right|_{{U_1 \setminus\left\{{\infty}\right\}}} }$$. ::: ::: {.definition title="Ringed Space"} A **ringed space** $$(X, {\mathcal{O}}_X) \in {\mathsf{Top}}\times {\mathsf{Sh}}(X, \mathsf{Ring})$$ as a topological space with a sheaf of rings. A morphism is a pair $$(f, f^\#) \in C^0(X, Y) \times \in \mathop{\mathrm{Mor}}_{\mathsf{Sh}}({\mathcal{O}}_Y, f_* {\mathcal{O}}_X)$$. ::: ::: {.example title="?"} The scheme $$(\operatorname{Spec}A, {\mathcal{O}}_{\operatorname{Spec}A})$$ is a ringed space. ::: ::: {.example title="?"} Consider $${\mathbb{R}}$$ in the Euclidean topology, then $$({\mathbb{R}}, C^0({-}, {\mathbb{R}}))$$ with the sheaf of continuous functions is a ringed space. Then consider the morphism given by projection onto the first coordinate of $${\mathbb{R}}^2$$: $(\pi, \pi^\#): ({\mathbb{R}}^2, C^0({-}, {\mathbb{R}})) &\to ({\mathbb{R}}, C^{\infty }({-}, {\mathbb{R}})) \\ (x, y) &\mapsto x .$ For $$\pi^\#$$, we can consider $$\pi_* C^0({-}, {\mathbb{R}})(U) \coloneqq C^0(\pi^{-1}(U)) = C^0(U\times {\mathbb{R}})$$, so $\pi^\#: C^\infty(U, {\mathbb{R}}) &\to C^0(U\times {\mathbb{R}}) \\ f &\mapsto f\circ \pi ,$ which is a well-defined map of rings since smooth functions are continuous. ::: ::: {.warnings} Not every scheme is built out of affine opens! ::: # Monday, September 13 ## Affine Schemes ::: {.definition title="Restricted sheaves"} Let $$(X, {\mathcal{O}}_X) \in \mathsf{RingSp}$$ and $$U \subseteq X$$ be open, then for $$V \subseteq U$$ open, define the restricted sheaf $${\mathcal{O}}_{X}{ \left.{{}} \right|_{{V}} }(V) \coloneqq{\mathcal{O}}_X(V)$$. ::: ::: {.warnings} ${\mathsf{Sh}}_{/ {X}} \ni{ \left.{{{\mathcal{O}}_X}} \right|_{{U}} }\neq {\mathcal{O}}_X(U) \in \mathsf{Ring} .$ ::: ::: {.remark} Recall the definition of a ringed space $$(X, {\mathcal{O}}_X)$$. The quintessential example: $$X$$ a smooth manifold and $${\mathcal{O}}_X \coloneqq C^{\infty}({-}; {\mathbb{R}})$$ the sheaf of smooth functions. For defining morphisms, consider a map $$f:X\to Y$$, then an alternative way of defining $$f$$ to be smooth is that there is a pullback $f^*: C^0(V, {\mathbb{R}}) &\to C^0(U, {\mathbb{R}}) \\ g &\mapsto g \circ f$ for $$U \subseteq X, V \subseteq Y$$, and that $$f^*$$ in fact restricts to $$f^*: C^\infty(V; {\mathbb{R}}) \to C^\infty(U; {\mathbb{R}})$$, i.e. preserving smooth functions. ::: ::: {.definition title="Morphisms of ringed spaces"} A morphism of ringed spaces is a pair $(M, {\mathcal{O}}_M) \xrightarrow{(\varphi, \varphi^\#)} (N, {\mathcal{O}}_N) .$ where $$\varphi \in C^0(M, N)$$ and $$\varphi^\# \in \mathop{\mathrm{Mor}}_{{\mathsf{Sh}}_{/ {N}} }({\mathcal{O}}_N, \varphi_* {\mathcal{O}}_M)$$. This is an **isomorphism** of ringed spaces if 1. $$\varphi$$ is a homeomorphism, and 2. $$\varphi^\#$$ is an isomorphism of sheaves. ::: ::: {.remark} In the running example, $\varphi^\#(U): {\mathcal{O}}_N(U) \to \varphi_* {\mathcal{O}}_M(M) = {\mathcal{O}}_M(\varphi^{-1}(U)) .$ This implies that maps of ringed spaced here induce smooth maps, and so there is an embedding $${\mathsf{sm}}{\mathsf{Mfd}}_{/ {{\mathbb{R}}}} \hookrightarrow\mathsf{RingSp}$$. ::: ::: {.remark} We'll try to set up schemes the same way one sets up smooth manifolds. A topological manifold is a space locally homeomorphic to $${\mathbb{R}}^n$$, and a smooth manifold is one in which it's locally isomorphic as a ringed space to $$({\mathbb{R}}^n, C^\infty({-}; {\mathbb{R}}))$$ with its sheaf of smooth functions. ::: ::: {.definition title="Smooth manifolds, alternative definition"} A **smooth manifold** is a ringed space $$(M, {\mathcal{O}}_M)$$ that is locally isomorphic to $$({\mathbb{R}}^d, C^\infty({-}; {\mathbb{R}}))$$, i.e. there is an open cover $${\mathcal{U}}\rightrightarrows M$$ such that $(U_i, { \left.{{{\mathcal{O}}_M}} \right|_{{U_i}} }) \cong ({\mathbb{R}}^n, C^{\infty}({-}; {\mathbb{R}})) .$ ::: ::: {.example title="?"} An example of a morphism of ringed spaces that is not an isomorphism: take $$({\mathbb{R}}, C^0) \to ({\mathbb{R}}, C^\infty)$$ given by $$(\operatorname{id}, \operatorname{id}^\#)$$ where $$\operatorname{id}^\#: C^\infty \to \operatorname{id}_* C^0$$ is given by $$\operatorname{id}^\#(U): C^\infty(U) \to C^0(U)$$ is the inclusion of continuous functions into smooth functions. ::: ::: {.remark} We'll define schemes similarly: build from simpler pieces, namely an open cover with isomorphisms to affine schemes. A major difference is that there may not exist a *unique* isomorphism type among all of the local charts, i.e. the affine scheme can vary across the cover. ::: ::: {.remark} Recall that for $$A$$ a ring we defined $$(\operatorname{Spec}A, {\mathcal{O}}_{\operatorname{Spec}A})$$, where $$\operatorname{Spec}A \coloneqq\left\{{\text{Prime ideals } {\mathfrak{p}}{~\trianglelefteq~}A}\right\}$$, equipped with the Zariski topology generated by closed sets $$V(I) \coloneqq\left\{{{\mathfrak{p}}{~\trianglelefteq~}A {~\mathrel{\Big|}~}I\supseteq{\mathfrak{p}}}\right\}$$. We then defined ${\mathcal{O}}_{\operatorname{Spec}A}(U) \coloneqq\left\{{s: U\to \displaystyle\coprod_{{\mathfrak{p}}\in U} A \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right] {~\mathrel{\Big|}~} s({\mathfrak{p}}) \in A \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right], \, s\text{ locally a fraction} }\right\} .$ We saw that 1. We can identify stalks: $${\mathcal{O}}_{\operatorname{Spec}A, {\mathfrak{p}}} = A \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right]$$ 2. We can identify sections on distinguished opens: ${\mathcal{O}}_{\operatorname{Spec}A}(D_f) = A \left[ { \scriptstyle { {f}^{-1}} } \right] = \left\{{a/f^k {~\mathrel{\Big|}~}a\in A, k\in {\mathbb{Z}}_{\geq 0}}\right\} ,$ where $$D_f \coloneqq V(f)^c = \left\{{{\mathfrak{p}}\in \operatorname{Spec}A {~\mathrel{\Big|}~}f\not\in {\mathfrak{p}}}\right\}$$. As a corollary, we get $${\mathcal{O}}_{\operatorname{Spec}A}(\operatorname{Spec}A) = A$$, noting $$\operatorname{Spec}A = d_1$$ and $$A \left[ { \scriptstyle { {1}^{-1}} } \right] = A$$. Thus we can recover the ring $$A$$ from the ringed space $$(X, {\mathcal{O}}_X) \coloneqq(\operatorname{Spec}A, {\mathcal{O}}_{\operatorname{Spec}A})$$ by taking global sections, i.e. $${\mathsf{\Gamma}\qty{\operatorname{Spec}A; {\mathcal{O}}_{\operatorname{Spec}A}} } = A$$. ::: ## Affine Varieties ::: {.remark} Let $$k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$$ and set $${\mathbb{A}}^n_{/ {k}} = k^n$$ whose regular functions are given by $$k[x_1, \cdots, x_{n}]$$, regarded as maps to $$k$$. ::: ::: {.definition title="Affine variety"} An affine variety is any set of the form $X \coloneqq V(f_1,\cdots, f_n) = \left\{{p\in {\mathbb{A}}^n_{/ {k}} {~\mathrel{\Big|}~}f_1(p) = \cdots = f_m(p) = 0}\right\}$ for $$f_i \in k[x_1, \cdots, x_{n}]$$, ::: ::: {.remark} Writing $$I = \left\langle{f_1,\cdots, f_m}\right\rangle$$, we have $$X = V(\sqrt I)$$. Letting $$I(S) = \left\{{f\in k[x_1, \cdots, x_{n}]{~\mathrel{\Big|}~}{ \left.{{f}} \right|_{{S}} } = 0}\right\}$$, then by the Nullstellensatz, $$IV(I) = \sqrt{I}$$. This gives a bijection between affine varieties in $${\mathbb{A}}^n_{/ {k}}$$ and radical ideals $$I {~\trianglelefteq~}k[x_1, \cdots, x_{n}]$$. ::: ::: {.definition title="Coordinate rings of affine varieties"} The **coordinate ring** of an affine variety $$X$$ is $$k[X] \coloneqq k[x_1, \cdots, x_{n}]/I(X)$$, regarded as polynomial functions on $$X$$. ::: ::: {.remark} We quotient here because if the difference of functions is in $$I(X)$$, these functions are equal when restricted to $$X$$. For example, $$y=x$$ in $$k[x, y]/ \left\langle{x-y}\right\rangle$$, which are different functions where for $$X\coloneqq\Delta$$, we have $${ \left.{{x}} \right|_{{\Delta}} } = { \left.{{y}} \right|_{{\Delta}} }$$. ::: ::: {.remark} As an application of the Nullstellensatz, there is a correspondence $\adjunction{I({-})}{V({-})}{\left\{{\text{Points } p\in X}\right\}}{\operatorname{mSpec}k[X]}$ ::: ::: {.remark} Why is an affine variety $$X$$ an example of an affine scheme $$\operatorname{Spec}k[X]$$? These don't have the same underlying topological space: $\tau(X) &\coloneqq\left\{{V(I) \coloneqq\left\{{p\in X {~\mathrel{\Big|}~}f_i(p) = 0 \,\, \forall f_i \in I}\right\} {~\mathrel{\Big|}~}I{~\trianglelefteq~}k[X]}\right\} \\ \tau(\operatorname{mSpec}k[X]) &\coloneqq\left\{{ V(I) \coloneqq\left\{{ {\mathfrak{m}}\in \operatorname{mSpec}k[X] {~\mathrel{\Big|}~}{\mathfrak{m}}\supseteq I}\right\} {~\mathrel{\Big|}~}I{~\trianglelefteq~}k[X] }\right\} .$ However, they are closely related: ${ \left.{{\tau(\operatorname{mSpec}k[X])}} \right|_{{\operatorname{Spec}k[X]}} } = \tau(X_{\mathrm{zar}}) ,$ i.e. the space $$\operatorname{Spec}k[X]$$ with the restricted topology from $$\operatorname{mSpec}k[X]$$ is homeomorphic to $$X$$ with the Zariski topology. I.e. restricting to *closed points* recovers regular functions on $$X$$. ::: ::: {.warnings} Defining things that are locally isomorphic to schemes would work for objects but not morphisms! ::: # Wednesday, September 15 ::: {.remark} Last time: for $${\mathsf{Aff}}{\mathsf{Var}}$$ we considered $$X = V(I) \subseteq {\mathbb{A}}^n_{/ {k}}$$, and for $${\mathsf{Aff}}{\mathsf{Sch}}$$ we considered $$\operatorname{Spec}k[X]$$ where $$k[X] \coloneqq k[x_1, \cdots, x_{n}]/ I(X)$$. Both had the Zariski topology, and $$X = \operatorname{mSpec}k[X] \subseteq \operatorname{Spec}k[X]$$. We had structure sheaves $${\mathcal{O}}_{\operatorname{Spec}k[X]}$$, and for $$X$$, we have $$U' \coloneqq U \cap\operatorname{mSpec}k[X]$$. On $$\operatorname{mSpec}k[X]$$, we have the notion of a regular function, and $${\mathcal{O}}_X(U') = {\mathcal{O}}_{\operatorname{Spec}k[X]}(U')$$. The previous setup only worked for rings finitely generated over a field, whereas for schemes, we can take things like $$\operatorname{Spec}{\mathbb{Z}}$$, so they're much more versatile (e.g. for number theory). ::: ::: {.example title="?"} Compare $${\mathbb{A}}^2_{/ {k}} \in {\mathsf{Aff}}{\mathsf{Var}}$$ to $$\operatorname{Spec}k[x, y]$$. Note that $$\left\langle{0}\right\rangle \in \operatorname{Spec}k[x, y]$$, and taking its closure yields ${ \operatorname{cl}} (\left\langle{0}\right\rangle) &= \displaystyle\bigcap_{V(I)\supseteq\left\langle{0}\right\rangle } V(I) \\ &= \displaystyle\bigcap_{V(I)\ni 0 } V(I) \\ &= V(0) \\ &= \operatorname{Spec}k[x, y] ,$ so $$0$$ is a dense point! {=tex} \begin{tikzpicture} \fontsize{20pt}{1em} \node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Schemes/sections/figures}{2021-09-15_18-17.pdf_tex} }; \end{tikzpicture}  But there are prime ideals of height $$>1$$. For example, any irreducible subvariety of $${\mathbb{A}}^2$$ yields a generic point. {=tex} \todo[inline]{Krull's dimension theorem?}  ::: ::: {.exercise title="?"} Try to draw $$\operatorname{Spec}{\mathbb{Z}}$$ and $$\operatorname{Spec}{\mathbb{Z}}[t]$$. ::: ::: {.remark} We'll now try a naive definition of schemes, which we'll find won't quite work. ::: ::: {.definition title="A wrong definition of a scheme!"} A **scheme** is a ringed space $$(X, {\mathcal{O}}_X)$$ which is locally an affine scheme, i.e. there exists an open cover $${\mathcal{U}}\rightrightarrows X$$ such that there is a collection of rings $$A_i$$ with $(U_i, { \left.{{{\mathcal{O}}_{X}}} \right|_{{U_i}} } ) { { \, \xrightarrow{\sim}\, }}(\operatorname{Spec}A_i, {\mathcal{O}}_{\operatorname{Spec}A_i}) .$ ::: ::: {.remark} This produces the right objects, but not the correct morphisms. This is a subtle issue! With this definition, a morphism of schemes would be a morphism of ringed spaces $$(f, f^\#)$$ with $$f\in {\mathsf{Top}}(X, Y)$$ and $$f^\# \in {\mathsf{Sh}}_{/ {Y}} ({\mathcal{O}}_Y, f_* {\mathcal{O}}_X)$$, where $$f^\#$$ is supposed to capture "pullback of functions". The issue: $$f^\#$$ may not notice that $$p \xrightarrow{f} f(p)$$! In particular, it may not preserve the fact that $$f(p) = 0$$. {=tex} \begin{tikzpicture} \fontsize{42pt}{1em} \node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Schemes/sections/figures}{2021-09-15_11-49.pdf_tex} }; \end{tikzpicture}  Hartshorne exercises for how this issue can actually arise. ::: ::: {.remark} Let $$(f, f^\#)$$ be a map of ringed spaces, then there is an induced map $f_p^\#: {\mathcal{O}}_{Y, f(p)} &\to {\mathcal{O}}_{X, p} \\ (U, s) &\mapsto (f^{-1}(U), f^\#(U)(s)) .$ ::: ::: {.definition title="Locally ringed space"} A **locally ringed space** $$(X, {\mathcal{O}}_X)$$ is a ringed space for which the stalks $${\mathcal{O}}_{X, p} \in \mathsf{Loc}\mathsf{Ring}$$ are local rings, i.e. there exists a unique maximal ideal. ::: ::: {.example title="of a locally ringed space"} For $$(X, {\mathcal{O}}_X) \coloneqq(\operatorname{Spec}A, {\mathcal{O}}_{\operatorname{Spec}A})$$, we saw that $${\mathcal{O}}_{X, p} = A \left[ { \scriptstyle { {p}^{-1}} } \right]$$, which is local. ::: ::: {.definition title="Morphisms of locally ringed spaces"} A **morphism of locally ringed spaces** is a morphism of ringed spaces $(f, F^\#): (X, {\mathcal{O}}_X) \to (Y, {\mathcal{O}}_Y)$ such that $$f^\#_p: {\mathcal{O}}_{Y, f(p)} \to {\mathcal{O}}_{X, p}$$ is a homomorphism of local rings, i.e. $$f^\#({\mathfrak{m}}_{f(p)}) \subseteq {\mathfrak{m}}_p$$. > Here we should also require that $$f^\# \neq 0$$. ::: ::: {.remark} Morally: this extra condition enforces that pulling back functions vanishing at $$f(p)$$ yields functions which vanish at $$p$$. ::: ::: {.remark} Alternatively one could require that $$(f^\#)^{-1}({\mathfrak{m}}_p) = {\mathfrak{m}}_{f(p)}$$, and (claim) this is equivalent to the above definition. Use that $$(f^\#)^{-1}({\mathfrak{m}}_p)$$ is a prime ideal containing $${\mathfrak{m}}_p$$. ::: ::: {.example title="of a locally ringed space"} Take $$(X, {\mathcal{O}}_X) \coloneqq({\mathbb{R}}, C^0({-}; {\mathbb{R}}))$$. Why this is in $$\mathsf{Loc}\mathsf{RingSp}$$: write a stalk as $C_p^0 = \left\{{(f, I) {~\mathrel{\Big|}~}I\ni p \text{ an interval}, f\in {\mathsf{Top}}(I, {\mathbb{R}})}\right\}_{/ {\sim}} .$ Why is this local? Take $${\mathfrak{m}}_p \coloneqq\left\{{(f, I) {~\mathrel{\Big|}~}f(p) = 0}\right\}$$, which is maximal since $$C_p^0/{\mathfrak{m}}\cong {\mathbb{R}}$$ is a field. Then $${\mathfrak{m}}_p^c = \left\{{(f, I) {~\mathrel{\Big|}~}f(p) \neq 0}\right\}$$, and any continuous function that isn't zero at $$p$$ is nonzero in some neighborhood $$J \supseteq I$$, so $${ \left.{{f}} \right|_{{J}} }\neq 0$$ anywhere. Then $$(f, I) \sim ({ \left.{{f}} \right|_{{J}} }, J)$$, which is invertible in the ring, so any element in the complement is a unit. ::: ::: {.example title="?"} Consider $({\mathbb{R}}, C^0({-}; {\mathbb{R}})) \xrightarrow{(f, f^\#)} ({\mathbb{R}}, C^\infty({-}; {\mathbb{R}})) .$ Take $$f = \operatorname{id}$$ and the inclusion $f^\# : C^\infty({-}; {\mathbb{R}})\hookrightarrow\operatorname{id}_* C^0({-}; {\mathbb{R}}) = C^0({-}; {\mathbb{R}}) .$ Then $f_p^\#: C_p^\infty({-}; {\mathbb{R}}) \to C_p^0({-}; {\mathbb{R}}) .$ satisfies $$f_p^\#({\mathfrak{m}}^\infty_{\operatorname{id}(p)}) = {\mathfrak{m}}^0_p$$. We also have $$(f_p^\#)^{-1}({\mathfrak{m}}_p^0) = {\mathfrak{m}}_p^\infty$$, since the germs are just equal. ::: ::: {.definition title="Scheme"} A **scheme** $$(X, {\mathcal{O}}_X)$$ is a locally ringed space which is locally isomorphic to $$(\operatorname{Spec}A, {\mathcal{O}}_{\operatorname{Spec}A})$$ in $$\mathsf{Loc}\mathsf{RingSp}$$. A **morphism of schemes** is a morphism in $$\mathsf{Loc}\mathsf{RingSp}$$. ::: ::: {.remark} Note that we can restrict to opens, since this doesn't change the stalks. ::: ::: {.remark} As a proof of concept that this is a good notion, we'll see that there's a fully faithful contravariant functor $$\operatorname{Spec}: \mathsf{CRing}\to {\mathsf{Sch}}$$, so $\operatorname{Spec}(\mathop{\mathrm{Mor}}_\mathsf{Ring}(B, A)) = \mathop{\mathrm{Mor}}_{\mathsf{Sch}}( \operatorname{Spec}A, \operatorname{Spec}B) .$ ::: # Appendix ::: {.remark} A bunch of stuff I always forget! ::: ::: {.definition title="Classical AG"} {=tex} \envlist  - A **section** is just an element $$s\in {\mathcal{F}}(U)$$. - A **stalk** of a (pre)sheaf $${\mathcal{F}}$$ at a point $$p$$ is defined as ${\mathcal{F}}_p \coloneqq\colim_{p\ni U_i} ({\mathcal{F}}(U_i), \operatorname{res}_{ij}) .$ - A **germ** $$\tilde f_p$$ at a point $$p$$ is an element in a stalk $${\mathcal{F}}_p$$. It can concretely be described as $\tilde f_p = [(U\ni p, s\in {\mathcal{F}}(U))]/\sim && (U, s)\sim (V, t) \iff \exists W \subseteq U \cap V,\, { \left.{{s}} \right|_{{W}} } = { \left.{{t}} \right|_{{W}} } .$ ::: ::: {.definition title="Colimit of a diagram"} Given a diagram $$J$$ in a category $$\mathsf{C}$$, regard it as a functor $$F: \mathsf{J}\to \mathsf{C}$$ where $$\mathsf{J}$$ is the diagram category of $$J$$. Then the **colimit** of $$J$$ is defined as the initial object in the category of co-cones over $$F$$. - A **co-cone** of $$F$$ is an $$N\in {\operatorname{Ob}}(\mathsf{C})$$ and a family of morphisms $$\left\{{ \psi_X: F(X)\to N{~\mathrel{\Big|}~}X\in {\operatorname{Ob}}(\mathsf{J})}\right\}$$. - The **category of co-cones** over $$F$$ is the comma category $$F \downarrow \Delta$$, where $$\Delta: \mathsf{C} \to {\mathsf{Fun}}(\mathsf{J}, \mathsf{C})$$ is the diagonal functor sending $$N\in {\operatorname{Ob}}(\mathsf{C})$$ to the constant functor to $$N$$: $\Delta(N):\mathsf{J} &\to \mathsf{C} \\ X &\mapsto N .$ - The **comma category** generalizes slice categories: given categories and functors $\mathsf{A} \mapsto{S} \mathsf{C} \mapsfrom{T} \mathsf{B} ,$ the comma category $$S\downarrow T$$ is given by triples $$(A, B, h: S(A)\to T(B))$$ making the obvious diagrams commute: {=tex} \begin{tikzcd} {A_0} && {A_1} &&& {S(A_0)} && {S(A_1)} \\ && {} \\ {B_0} && {B_1} &&& {T(B_0)} && {T(B_1)} \arrow["{S(f)}", from=1-6, to=1-8] \arrow["{h_1}", from=1-8, to=3-8] \arrow["{h_1}"', from=1-6, to=3-6] \arrow["{T(g)}"', from=3-6, to=3-8] \arrow["f", from=1-1, to=1-3] \arrow["g", from=3-1, to=3-3] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbNSwwLCJTKEFfMCkiXSxbNSwyLCJUKEJfMCkiXSxbNywwLCJTKEFfMSkiXSxbNywyLCJUKEJfMSkiXSxbMCwwLCJBXzAiXSxbMiwwLCJBXzEiXSxbMCwyLCJCXzAiXSxbMiwyLCJCXzEiXSxbMiwxXSxbMCwyLCJTKGYpIl0sWzIsMywiaF8xIl0sWzAsMSwiaF8xIiwyXSxbMSwzLCJUKGcpIiwyXSxbNCw1LCJmIl0sWzYsNywiZyJdXQ==) Taking $$\mathsf{C} = A$$, $$S = \operatorname{id}_{\mathsf{A}}$$, and $$\mathsf{B} \coloneqq{\operatorname{pt}}$$ to be a 1-object category with only the identity morphism forces $$X\coloneqq T({\operatorname{pt}}) \in {\operatorname{Ob}}(\mathsf{A})$$ to be a single object and $$(\mathsf{A} \downarrow X)$$ is the usual slice category over $$X$$. ::: # Problem Sets ## Problem Set 1 ::: {.remark} All problems are sourced from Hartshorne. ::: ## Chapter 2, Section 1 ::: {.remark} List of useful facts used: - Morphisms of sheaves commute with restrictions: if $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ then for any $$s\in {\mathcal{F}}(U)$$ and $$V \subseteq U$$, $$\mathop{\mathrm{Res}}(U, V)(\phi(s)) = \phi(\mathop{\mathrm{Res}}(U, V)(s))$$. - $$\phi$$ is an isomorphism iff $$\phi_p$$ are all isomorphisms. - Elements of stalks $${\mathcal{F}}_p:$$ equivalence classes $$[U, s\in {\mathcal{F}}(U)]$$. - The induced map on stalks: $$\phi_p([U, s]) \coloneqq[U, \phi(U)(s)]$$. - A surjection of sheaves need not induce a surjection on sections. - The colimit diagram: {=tex} \begin{tikzcd} & \bullet \\ \vdots && \vdots \\ {U_1} && {F(U_1)} \\ &&&& {\forall O} && {\colim_{i} F(U_i)} \\ {U_2} && {F(U_2)} \\ \vdots && \vdots \\ & \bullet \arrow["f", from=5-1, to=3-1] \arrow["{F(f)}"', from=3-3, to=5-3] \arrow["{\psi_2}"', from=5-3, to=4-7] \arrow["{\psi_1}", from=3-3, to=4-7] \arrow["{\psi'_1}"', from=3-3, to=4-5] \arrow["{\psi'_2}", from=5-3, to=4-5] \arrow["{\exists !}", dashed, from=4-5, to=4-7] \arrow[dotted, no head, from=1-2, to=7-2] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=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) - Colimits are initial co-cones, where $$I$$ is initial if $$I\to X$$ for any $$X$$. AKA direct limits. - Filtered colimits commute with finite limits. - In particular, monomorphisms are pullbacks, so finite limits, and stalks are filtered colimits. So injections of sheaves induce injections on stalks. ::: ::: {.remark} Recommended problems: - 1.1 - 1.2 - 1.3 - 1.4 - 1.5 ::: ::: {.problem title="1.1"} Let $$A$$ be an abelian group, and define the *constant presheaf* associated to $$A$$ on the topological space $$X$$ to be the presheaf $$U \mapsto A$$ for all $$U \neq \emptyset$$, with restriction maps the identity. Show that the constant sheaf $${\mathcal{A}}$$ defined in Hartshorne is the sheafification of this presheaf. ::: ::: {.solution} Let $$X\in {\mathsf{Top}}$$ be a space. Recapping the definitions, define the constant presheaf as $\underline{A}^{\mathsf{pre}}(U) \coloneqq \begin{cases} A & U\neq \emptyset \\ 0 & \text{else}. \end{cases} \quad \operatorname{res}^1(U, V) \coloneqq \begin{cases} \operatorname{id}_A & U\neq \emptyset \\ 0 & \text{else}. \end{cases} .$ Then define the constant *sheaf* as $\underline{A}(U) \coloneqq\mathop{\mathrm{Hom}}_{{\mathsf{Top}}}(U, A)\quad \operatorname{res}^2(U, V)(f) \coloneqq{ \left.{{f}} \right|_{{V}} } .$ We're then tasked with finding a morphism of sheaves $\Psi: (\underline{A}^{\mathsf{pre}})^+ \xrightarrow{\sim} \mathop{\mathrm{Hom}}_{{\mathsf{Top}}}({-}, A) ,$ which we'll also want to have an inverse morphism and this define an isomorphism in $${\mathsf{Sh}}(X)$$. We'll use the implicitly stated fact in Hartshorne that $$\mathop{\mathrm{Hom}}_{{\mathsf{Top}}}(U, A) = A^{\oplus n}$$ where $$n \coloneqq\# \pi_0(X)$$ is the number of connected components of $$U$$. Suppose first that $$n=1$$, so $$X$$ is connected, and define the following morphism of groups: \Psi_U: (\underline{A}^{\mathsf{pre}})(U) = A &\longrightarrow\mathop{\mathrm{Hom}}_{\mathsf{Top}}(U, A)\\ a_0 &\mapsto \left\{ { \begin{aligned} \varphi_{a_0}: U \to A \\ x \mapsto a_0, \end{aligned} } \right. which maps a group element $$a_0$$ to the constant function on $$U$$ sending every point to $$a_0 \in A$$. The claim is that the following diagram commutes in the category $$\underset{ \mathsf{pre} } {\mathsf{Sh} }(X)$$ (in both directions) for all $$U$$ and $$V$$: {=tex} \begin{tikzcd} && {f(U)} && f \\ && {a_0} && { \begin{aligned} \varphi_{a_0}: U &\to A \\ x &\mapsto a_0 \end{aligned} } \\ U && {(\underline{A}^{\mathsf{pre}})(U) = A} && {\mathop{\mathrm{Hom}}_{\mathsf{Top}}(U, A)} \\ \\ V && {(\underline{A}^{\mathsf{pre}})(V) = A} && {\mathop{\mathrm{Hom}}_{\mathsf{Top}}(V, A)} \\ && {a_0} && { \begin{aligned} \varphi_{a_0}: V &\to A \\ x &\mapsto a_0 \end{aligned} } \\ && {f(V)} && f \arrow[hook, from=5-1, to=3-1] \arrow["{\Psi_U}", from=3-3, to=3-5] \arrow["{\Psi_V}", from=5-3, to=5-5] \arrow["{\operatorname{res}^1(U, V)}"', from=3-3, to=5-3] \arrow["{\operatorname{res}^2(U, V)}", from=3-5, to=5-5] \arrow[maps to, from=2-3, to=2-5] \arrow[maps to, from=1-5, to=1-3] \arrow[maps to, from=7-5, to=7-3] \arrow[maps to, from=6-3, to=6-5] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=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) Here we've specified simultaneously what $$\Psi$$ and $$\Psi^{-1}$$ prescribe on opens $$U, V$$, and abuse notation slightly by writing $$\mathop{\mathrm{Hom}}_{{\mathsf{Top}}}({-}, A)$$ for the sheaf it represents and its underlying presheaf. - That this commutes follows readily, since running the diagram counterclockwise yields $$\operatorname{res}^1(U, V) = \operatorname{id}_A$$, so the composition $(A \xrightarrow{\operatorname{res}^1(U, V)} A \xrightarrow{\Psi_V} \mathop{\mathrm{Hom}}(V, A)) = (A \xrightarrow{\Psi_V} \mathop{\mathrm{Hom}}(V, A))$ sends an element $$a_0\in A$$ to the constant function $$\varphi_{a_0, V}: V\to A$$. Running the diagram clockwise yields $(A \xrightarrow{\Psi_U} \mathop{\mathrm{Hom}}(U, A) \xrightarrow{\operatorname{res}^2(U, V)} \mathop{\mathrm{Hom}}(V, A)) ,$ which sends $$a_0$$ to the constant function $$\varphi_{a_0, U}: U\to A$$ sending everything to $$a_0$$, which then gets sent to $${ \left.{{\varphi_{a_0, U}}} \right|_{{V}} }: V\to A$$ sending everything to $$a$$. Since $${ \left.{{\varphi_{a_0}}} \right|_{{V}} }(x) = \varphi_{a_0, V}(x) = a$$ for every $$x\in U$$, these functions are equal. - That the reverse maps $$\Psi_U^{-1}$$ are well-defined follows from the fact that $$U$$ is connected: the continuous image of a connected set is connected. Since $$A$$ is given the discrete topology, any subset with 2 or more elements in disconnected, so each function $$f\in \mathop{\mathrm{Hom}}(U, A)$$ is necessarily a constant function and $$f(U) = \left\{{a}\right\}$$ is a singleton. - $$\Psi_U, \Psi_U^{-1}$$ clearly compose to the identity in either order, so $$\Psi_U$$ defines an isomorphism of abelian groups. As a consequence, we get a well-defined morphism of presheaves $$\underline{A}^{\mathsf{pre}}({-}) \to { \left.{{ \mathop{\mathrm{Hom}}({-}, A)}} \right|_{{ \underset{ \mathsf{pre} } {\mathsf{Sh} }}} }$$, and by the sheafification adjunction we can lift this to a morphism of sheaves: $\adjunction{{\mathcal{F}}\mapsto {\mathcal{F}}^+ }{{\mathcal{G}}\mapsto { \left.{{{\mathcal{F}}}} \right|_{{ \underset{ \mathsf{pre} } {\mathsf{Sh} }}} } }{ \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)}{{\mathsf{Sh}}(X)} ,$ which reads $\mathop{\mathrm{Hom}}_{ \underset{ \mathsf{pre} } {\mathsf{Sh} }}({\mathcal{F}}, { \left.{{{\mathcal{G}}}} \right|_{{ \underset{ \mathsf{pre} } {\mathsf{Sh} }}} }) &\xrightarrow{\sim} \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}}({\mathcal{F}}^+, {\mathcal{G}}) \\ \Psi &\mapsto \tilde \Psi ,$ and since $$\Psi$$ was an isomorphism, so is $$\tilde \Psi$$. > It remains to handle the $$n\geq 2$$, case when (say) $$U = U_1 {\textstyle\coprod}U_2$$ has more than 1 connected component. Actually, is it even true that adjunctions preserve isomorphisms...? Todo: help?? ------------------------------------------------------------------------ Alternatively, consider the map $$\Psi$$ defined on presheaves -- by the universal property, we get some sheaf morphism $$\tilde\Psi$$, which we can show is an isomorphism by showing its induced map on stalks is an isomorphism. This amounts to showing the following map is a group isomorphism: $\Psi_p: (\underline{A}^{\mathsf{pre}}({-}))_p \xrightarrow{\sim} \mathop{\mathrm{Hom}}_{\mathsf{Top}}({-}, A)_p .$ First we identify the LHS: $(\underline{A}^{\mathsf{pre}}({-}))_p \coloneqq\colim_{U\ni p} \underline{A}^{\mathsf{pre}}(U) = \colim_{U\ni p} A = A .$ (todo: show $$A$$ satisfies the universal property for a colimit) Identifying the RHS, we have equivalence classes $$[U\ni p, s: U\to A]$$ - Injectivity: that $$\Psi_p$$ is injective follows from the fact that $$\ker \psi_p \coloneqq\left\{{a\in A {~\mathrel{\Big|}~}\Psi_p(a) = e}\right\}$$, where $$e$$ is the identity in the right-hand side stalk, which is represented by the class $$[U, f_e:U\to A]$$ where $$f_e(x) \coloneqq e_A$$, the identity of $$A$$, for every $$x\in U$$. - Surjectivity: that $$\Psi_p$$ is surjective follows from the fact that every fixed $$f: U\to A$$ for $$A$$ discrete is constant on connected components. Use that $$p$$ is contained in a connected component $$U_1 \ni p$$, then $$[U, f] \sim [U_1, { \left.{{f}} \right|_{{U_1}} }] \coloneqq[U_1, g]$$ to get that $$g$$ is now a constant function of $$U_1$$. So $$g(x) = a$$ for some $$a\in A$$, so $$g = \Psi_p(a)$$ is in the image. ------------------------------------------------------------------------ Alternatively: - Show that $$\underline{A}$$ satisfies the universal property of $$(\underline{A}^{\mathsf{pre}})^+$$: we need to produce a morphism $$\theta: (\underline{A}^{\mathsf{pre}}) \to \underline{A}$$ such that for any $${\mathcal{G}}\in {\mathsf{Sh}}(X)$$ and morphism of presheaves $$\varphi: \underline{A}^{\mathsf{pre}}\to { \left.{{{\mathcal{G}}}} \right|_{{{\mathsf{pre}}}} }$$ we can produce a unique morphism $$\tilde \varphi$$ of sheaves making the following diagram commute: {=tex} \begin{tikzcd} {\underline{A}^{{\mathsf{pre}}}} && {{ \left.{{{\mathcal{G}}}} \right|_{{{\mathsf{pre}}}} }} && {\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)} \\ \\ {\underline{A}} && {\mathcal{G}}&& {\in{\mathsf{Sh}}(X)} \arrow["\varphi", from=1-1, to=1-3] \arrow["\theta"', from=1-1, to=3-1] \arrow["{{ \left.{{{-}}} \right|_{{{\mathsf{pre}}}} }}"', from=3-3, to=1-3] \arrow["{\exists! \tilde \varphi}"', dashed, from=3-1, to=3-3] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwyLCJcXHVse0F9Il0sWzAsMCwiXFx1bHtBfV57XFxwcmV9Il0sWzIsMCwiXFxyb3tcXG1jZ317XFxwcmV9Il0sWzIsMiwiXFxtY2ciXSxbNCwwLCJcXGluIFxcUHJlc2goWCkiXSxbNCwyLCJcXGluXFxTaChYKSJdLFsxLDIsIlxcdGhldGEiXSxbMSwwLCJcXHRoZXRhIiwyXSxbMywyLCJcXHJve1xcd2FpdH17XFxwcmV9IiwyXSxbMCwzLCJcXGV4aXN0cyEgXFx0aWxkZSBcXHRoZXRhIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d) - To define $$\tilde \varphi$$, it suffices to define morphisms of the form $\tilde\varphi(U): \underline{A}(U) &\to{\mathcal{G}}(U) \\ f & \mapsto \tilde\varphi(U)(f)$ - Take a map $$f\in \underline{A}(U) \coloneqq\mathop{\mathrm{Hom}}_{\mathsf{Top}}(U, A)$$. Write $$U \coloneqq{\textstyle\coprod}U_i$$ as a union of connected components. Use that $$f$$ is constant on connected components since $$A$$ is discrete, so $$f(U_i) = a_i$$ for some elements $$a_i \in A \in {\mathsf{Ab}}{\mathsf{Grp}}$$. - Plug the $$U_i$$ into $$\underline{A}^{\mathsf{pre}}$$ to get morphisms $\varphi(U_i): \underline{A}^{\mathsf{pre}}(U_i)= A \to { \left.{{{\mathcal{G}}}} \right|_{{{\mathsf{pre}}}} }(U_i) && \in {\mathsf{Ab}}{\mathsf{Grp}}$ - Write $$b_i \coloneqq\varphi(U_i)(a_i) \in { \left.{{{\mathcal{G}}}} \right|_{{{\mathsf{pre}}}} }(U_i) = {\mathcal{G}}(U_i)$$. {=tex} \begin{tikzcd} {a_i} &&& {b_i \coloneqq\varphi(U_i)(a_i)} \\ & {\underline{A}^{{\mathsf{pre}}}(U_i) = A} && {{ \left.{{{\mathcal{G}}}} \right|_{{{\mathsf{pre}}}} }(U_i)} && {\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)} \\ \\ & {\mathop{\mathrm{Hom}}(U_i, A)} && {{\mathcal{G}}(U_i)} && {\in{\mathsf{Sh}}(X)} \\ & f && {b_i} && {} \arrow["{\varphi(U_i)}", from=2-2, to=2-4] \arrow["\theta"', from=2-2, to=4-2] \arrow["{{ \left.{{{-}}} \right|_{{{\mathsf{pre}}}} }}"', from=4-4, to=2-4] \arrow["{\exists! \tilde \varphi}"', dashed, from=4-2, to=4-4] \arrow[curve={height=-24pt}, dashed, maps to, from=5-2, to=1-1] \arrow[dashed, maps to, from=1-1, to=1-4] \arrow[maps to, from=5-2, to=5-4] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=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) - Since $${\mathcal{G}}$$ is in fact a sheaf, by unique gluing there exists a unique element $$b \in {\mathcal{G}}(U)$$ such that $${ \left.{{b}} \right|_{{U_i}} } = b_i$$. So define $$\tilde\varphi(U)(f) \coloneqq b$$. - Now define the map $$\theta: \underline{A}^{\mathsf{pre}}(U_i) \to \mathop{\mathrm{Hom}}(U_i, A)$$ sending $$a_i$$ to the constant function $$f_{i}(x)\coloneqq a_i$$. Since $$\underline{A}$$ is a sheaf, there is a well defined $$F\in \mathop{\mathrm{Hom}}(U, A)$$ such that $${ \left.{{F}} \right|_{{U_i}} } = f_i$$. So for $$a\in \underline{A}^{\mathsf{pre}}(U)$$ set $$\theta(a) = F \in \underline{A}(U)$$. - This makes the relevant diagram commute: if $$a\in A = \underline{A}^{\mathsf{pre}}(U)$$, then $$b\coloneqq\phi(U)(a) \in {\mathcal{G}}(U)$$. On the other hand, $$\theta(a)$$ is the constant function $$f_a: x\mapsto a$$ (on every connected component of $$U$$), and setting $$F \coloneqq\tilde\phi(f_a)\in {\mathcal{G}}(U)$$, we have $$F \coloneqq b$$. ::: ::: {.problem title="1.2"} (a) For any morphism of sheaves $$\varphi: {\mathcal{F}}\rightarrow {\mathcal{G}}$$, show that for each point $$p$$ that $$\ker (\varphi)_{p}=$$ $$\operatorname{ker}\left(\varphi_{p}\right)$$ and $$\operatorname{im}(\varphi)_{p} = \operatorname{im}\left(\varphi_{p}\right)$$. (b) Show that $$\varphi$$ is injective (resp. surjective) if and only if the induced map on the stalks $$\varphi_{p}$$ is injective (resp. surjective) for all $$p$$. (c) Show that a sequence of sheaves and morphisms $\cdots {\mathcal{F}}^{i-1} \xrightarrow{\varphi^{i-1}} {\mathcal{F}}^i \xrightarrow{\varphi^{i}} {\mathcal{F}}^{i+1} \to \cdots$ is exact if and only if for each $$P \in X$$ the corresponding sequence of stalks is exact as a sequence of abelian groups. ::: ::: {.proof title="of 1, kernels"} {=tex} \envlist  - Write $$K\in {\mathsf{Sh}}(X)$$ for the kernel sheaf $$U \mapsto \ker \qty{ {\mathcal{F}}(U) \xrightarrow{\phi(U)} {\mathcal{G}}(U) }$$, - We then want to show $$K_p = \ker\qty{{\mathcal{F}}_p \xrightarrow{\phi_p} {\mathcal{G}}_p}$$, an equality of sets in $${\mathsf{Ab}}{\mathsf{Grp}}$$. So we just do it! - Addendum: this works because both are subsets of the same abelian group, $${\mathcal{F}}_p$$. - We can write $\phi_p: {\mathcal{F}}_p &\to {\mathcal{G}}_p \\ [U, s] &\mapsto [U, \phi(U)(s)] ,$ and note that the zero element in a stalk is the equivalence class $$[U, 0]$$ where $$0\in {\mathsf{Ab}}{\mathsf{Grp}}$$ is the zero object. Thus $\ker \phi_p &\coloneqq\left\{{ x\in {\mathcal{F}}_p {~\mathrel{\Big|}~}\phi_p(x) = 0 \in {\mathcal{G}}_p }\right\} \\ & = \left\{{ [U, s] \in {\mathcal{F}}_p {~\mathrel{\Big|}~}[U, \phi(U)(s)] = [U, 0] }\right\} \\ & = \left\{{ [U, s] \in {\mathcal{F}}_p {~\mathrel{\Big|}~}\phi(U)(s) = 0 }\right\} \\ & = \left\{{ [U, s] \in {\mathcal{F}}_p {~\mathrel{\Big|}~}s \in \ker \phi(U) }\right\} \\ &= \left\{{ [U, s] {~\mathrel{\Big|}~}s\in \ker{\qty{{\mathcal{F}}(U) \xrightarrow{\phi(U)} {\mathcal{G}}(U) } } }\right\} \\ &\coloneqq\left\{{[U, s] {~\mathrel{\Big|}~}s\in K(U)}\right\} \\ &\coloneqq K_p .$ ::: ::: {.proof title="of 1, images"} {=tex} \envlist  - Write $${\mathcal{I}}$$ for the sheaf $$\operatorname{im}\phi$$ which sends $$U\mapsto \operatorname{im}\qty{{\mathcal{F}}(U) \xrightarrow{\phi(U)} {\mathcal{G}}(U)}$$. - We want to show $${\mathcal{I}}_p = \operatorname{im}\qty{{\mathcal{F}}_p \xrightarrow{\phi_p} {\mathcal{G}}_p}$$, where both are subsets of $${\mathcal{G}}_p$$. - So we show set equality: $\operatorname{im}(\phi_p) &= \left\{{ y\in {\mathcal{G}}_p {~\mathrel{\Big|}~}\exists x\in {\mathcal{F}}_p,\, \phi_p(x) = y }\right\} \\ &= \left\{{ [U, t] \in {\mathcal{G}}_p {~\mathrel{\Big|}~}\exists [U, s] \in {\mathcal{F}}_p,\, \phi_p([U, s]) = [U, t] }\right\} \\ &= \left\{{ [U, t] \in {\mathcal{G}}_p {~\mathrel{\Big|}~}\exists s\in {\mathcal{F}}(U),\, \phi(U)(s) = t }\right\} \\ &= \left\{{ [U, t] \in {\mathcal{G}}_p {~\mathrel{\Big|}~}t\in \operatorname{im}\qty{{\mathcal{F}}(U) \xrightarrow{\phi(U)} {\mathcal{G}}(U) }}\right\} \\ &\coloneqq\left\{{ [U, t] \in {\mathcal{G}}_p {~\mathrel{\Big|}~}t \in {\mathcal{I}}(U) }\right\} \\ &\coloneqq{\mathcal{I}}_p .$ ::: ::: {.proof title="of 2, injectivity"} $$\implies$$: - Use that injectivity of a morphism $$\phi$$ of sheaves is *defined* to hold exactly when $$\ker \phi = 0$$ is the constant zero sheaf. - Now use (1): $0 = \ker(\phi) \implies 0 = \ker(\phi)_p = \ker(\phi_p) && \forall p .$ - If $$\ker \phi = 0$$, so $$\phi$$ is injective, then $$\ker \phi_p = 0$$ for all $$p$$, so $$\ker \phi_p$$ is injective. $$\impliedby$$: - Conversely, suppose $$\ker \phi_p = 0$$ for all $$p$$; we want to show $$\ker \phi(U) = 0$$ for all $$U$$. - So fix $$U\ni p$$, we want to show $s\in K(U) \coloneqq\ker\qty{{\mathcal{F}}(U) \xrightarrow{\phi(U)} {\mathcal{G}}(U)} \implies s = 0 \in {\mathcal{F}}(U) .$ - We have $$\phi(U)(s) = 0$$, so $\phi_p([U, s]) \coloneqq[U, \phi(U)(s)] = [U, 0] \in {\mathcal{G}}_p \implies [U, s] \in \ker (\phi_p) .$ - By injectivity of $$\phi_p$$, we have $$[U, s] = 0 \in {\mathcal{F}}_p$$. - So there is some open $$U_p$$ with $$U \supseteq U_p \ni p$$ and $$\mathop{\mathrm{Res}}(U, U_p)(s) = 0$$ in $${\mathcal{F}}(U_p)$$. - Then $$\left\{{U_p }\right\}_{p\in U} \rightrightarrows U$$, and since $${\mathcal{F}}$$ is a sheaf, by existence of gluing these glue to an $$F \in {\mathcal{F}}(U)$$ with $$\mathop{\mathrm{Res}}(U, U_p)(F) = 0$$ for each $$p$$. By uniqueness of gluing, $$0 = F = s$$. ::: ::: {.proof title="of 2, surjectivity"} $$\implies$$: - Suppose $$\phi$$ is surjective, then by definition $$\operatorname{im}\phi = {\mathcal{G}}$$ is an equality of sheaves. - So $$(\operatorname{im}\phi)(U) = {\mathcal{G}}(U)$$ for all $$U$$. - Let $$[U, t]\in {\mathcal{G}}_p$$, so $$t\in {\mathcal{G}}(U)$$. - Then $$t\in (\operatorname{im}\phi)(U)$$, so there exists an $$s\in {\mathcal{F}}(U)$$ such that $$\phi(U)(s) = t$$. - Then $$[U, s] \mapsto [U, \phi(U)(s)] = [U, t]$$, under $$\phi_p$$, making $$\phi_p$$ surjective. $$\impliedby$$: - Suppose $$\phi_p$$ is surjective for all $$p$$, fix $$U$$, and let $$t \in {\mathcal{G}}(U)$$. We want to produce an $$s\in {\mathcal{F}}(U)$$ such that $$\phi(U)(s) = t$$. - For $$p\in U$$, the image of $$t$$ in the stalk of $${\mathcal{G}}$$ is of the form $$[U_p, t_p] \in {\mathcal{G}}_p$$ where $$t_p \in {\mathcal{G}}(U_p)$$. - Since $$\phi_p: {\mathcal{F}}_p \twoheadrightarrow{\mathcal{G}}_p$$, we can find some pair $$[U_p, s_p]$$ mapping to $$[U_p, t_p]$$ under $$\phi_p$$, so $$\phi(U_p)(s_p) = t_p$$. - Note that $$\mathop{\mathrm{Res}}(U, U_p)(t) = t_p$$. - Note: may need to pull back to some $$\tilde U_p$$, then take a common refinement in both germs? - Now $$\left\{{U_p}\right\}_{p\in U}\rightrightarrows U$$, so using existence of gluing for $${\mathcal{F}}$$ we have some $$s\in {\mathcal{F}}(U)$$ with $$\mathop{\mathrm{Res}}(U, U_p)(s) = s_p$$ for all $$p$$. - Claim: $$\phi(U)(s) = t$$. $\mathop{\mathrm{Res}}(U, U_p)( \phi(s) ) &= \phi(\mathop{\mathrm{Res}}(U, U_p)(s)) \\ &= \phi(s_p) \\ &= t_p \\ &= \mathop{\mathrm{Res}}(U, U_p)(t) && \forall p\in U ,$ so $$\phi(s) = t$$ by uniqueness of gluing of $${\mathcal{G}}$$. ::: ::: {.proof title="of 3, exactness"} $$\implies$$: Assuming exactness of sheaves, $\ker({\mathcal{F}}^{i+1}) = \operatorname{im}({\mathcal{F}}^{i}) \iff \ker({\mathcal{F}}^{i+1})_p = \operatorname{im}({\mathcal{F}}^{i})_p && \forall p .$ $$\impliedby$$: Assuming exactness on stalks, write $\ker({\mathcal{F}}^{i+1})_p &= \ker({\mathcal{F}}^{i+1}_p) && \text{by 1 } \\ &= \operatorname{im}({\mathcal{F}}^{i}_p) && \text{exactness, by assumption} \\ &= \operatorname{im}({\mathcal{F}}^{i})_p && \text{by 1} .$ ::: ::: {.problem title="1.3"} (a) Let $$\varphi: {\mathcal{F}}\to{\mathcal{G}}$$ be a morphism of sheaves on $$X$$. Show that $$\varphi$$ is surjective if and only if the following condition holds: For every open set $$U \subseteq X$$, and for every $$s\in {\mathcal{G}}(U)$$, there is a cover $$\left\{{U_i}\right\}$$ of $$U$$ and elements $$t_i \in {\mathcal{F}}(U_i)$$ such that $$\varphi(t_i) = { \left.{{s}} \right|_{{U_i}} }$$ for all $$i$$. (b) Give an example of a surjective morphism of sheaves $$\varphi: {\mathcal{F}}\rightarrow {\mathcal{G}}$$, and an open set $$U$$ such that $$\varphi(U): {\mathcal{F}}(U) \rightarrow {\mathcal{G}}(U)$$ is not surjective. ::: ::: {.proof title="of 1"} $$\implies$$: - If $$\phi: {\mathcal{F}}\twoheadrightarrow{\mathcal{G}}$$, then $$\phi_p: {\mathcal{F}}_p \twoheadrightarrow{\mathcal{G}}_p$$ for all $$p$$, since $$\operatorname{im}(\phi_p) = (\operatorname{im}\phi)_p = {\mathcal{G}}_p$$, using problem 1.2. - Fix $$U \subseteq X$$ and $$s\in {\mathcal{G}}(U)$$, we want - To produce a cover $$\left\{{U_i}\right\}\rightrightarrows U$$, - To find $$t_i\in {\mathcal{F}}(U_i)$$, and - To show that $$\phi(t_i) = \mathop{\mathrm{Res}}(U, U_i)(s)$$ for all $$i$$. - Fix $$p$$, and take the image of $$s$$ in the stalk of $${\mathcal{G}}$$ to get $$[U_p, s_p] \in {\mathcal{G}}_p$$ with $$s_p \in {\mathcal{G}}(U_p)$$ and $$\mathop{\mathrm{Res}}(U, U_p)(s) = s_p$$. Note that $$\left\{{U_p}\right\}_{p\in U}\rightrightarrows U$$. - By surjectivity on stalks, these pull back to $$[U_p, t_p]\in {\mathcal{F}}_p$$ with $$t_p \in {\mathcal{F}}(U_p)$$ and $$\phi_p([U_p, t_p]) \coloneqq[U_p, \phi(U_p)(t_p)] = [U_p, s_p]$$. - Then $$s_p \in \operatorname{im}({\mathcal{F}}(U_p) \xrightarrow{\phi(U_p)} {\mathcal{G}}(U_p ))$$ and $$\phi(t_p) = s_p = \mathop{\mathrm{Res}}(U, U_p)(s)$$. $$\impliedby$$: - If $$\left\{{U_i}\right\}\rightrightarrows U$$ with $$\phi(t_i) = \mathop{\mathrm{Res}}(U, U_i)(s)$$ for all $$i$$, then the $$t_i$$ glue to a unique section $$t\in {\mathcal{F}}(U)$$ since $${\mathcal{F}}$$ is a sheaf. - Moreover $$\mathop{\mathrm{Res}}(U, U_i)( \phi(t) ) = \phi(\mathop{\mathrm{Res}}(U, U_i)(t)) = \phi(t_i) = \mathop{\mathrm{Res}}(U, U_i)(s)$$ for all $$i$$, and by unique gluing for $${\mathcal{G}}$$ we have $$\phi(t) = s$$. - So $$\phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)$$ is surjective for all $$U$$, making $$\operatorname{im}(\phi(U)) = {\mathcal{G}}(U)$$ - So $$\operatorname{im}\phi = {\mathcal{G}}$$ as sheaves since they make the same assignment to every open set $$U$$, making $$\phi: {\mathcal{F}}\to{\mathcal{G}}$$ surjective by definition. ::: ::: {.proof title="of 2"} {=tex} \envlist  - Take $$X \coloneqq\left\{{a,b,c}\right\}$$ a 3-point space with the topology $$\tau_X \coloneqq\left\{{\emptyset, \left\{{a}\right\}, \left\{{b}\right\}, \left\{{a,b}\right\}, X}\right\}$$. - Take $${\mathcal{F}}\coloneqq\underline{A}$$ for some nontrivial $$A\in {\mathsf{Ab}}{\mathsf{Grp}}$$. We have the stalks - $${\mathcal{F}}_a = A$$ - $${\mathcal{F}}_b = A$$ - $${\mathcal{F}}_c = A$$ - Take $${\mathcal{G}}\coloneqq\underline{A}(a) \times \underline{A}(b)$$, the skyscraper sheaves at $$a$$ and $$b$$ respectively, where $\underline{A}(x)(U) \coloneqq \begin{cases} A & x\in U \\ 0 & \text{ else} . \end{cases}$ Note that the stalks are given by $$\underline{A}(x)_x = A$$ and $$\underline{A}(x)_y = 0$$ for $$y\neq x$$, so - $${\mathcal{G}}_a = A\times 0$$ - $${\mathcal{G}}_b = 0 \times A$$ - $${\mathcal{G}}_c = 0 \times 0$$. - Now define $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ by specifying $$\phi(U):{\mathcal{F}}(U) \to {\mathcal{G}}(U)$$ for all $$U$$ in the following way: {=tex} \begin{tikzcd} & {{\mathsf{Open}}(X)} &&&& {\underline{A}} &&&& {\underline{A}(a) \times \underline{A}(b)} \\ & {X = \left\{{a,b,c}\right\}} &&&& A &&&& {A^{\times 2}} \\ & {\left\{{a,b}\right\}} &&&& A &&&& {A^{\times 2}} \\ {\left\{{a}\right\}} && {\left\{{b}\right\}} && A && A && {A\times 0} && {0\times A} \\ & \emptyset &&&& 0 &&&& 0 \arrow[from=5-2, to=4-1] \arrow[from=5-2, to=4-3] \arrow[from=4-1, to=3-2] \arrow[from=4-3, to=3-2] \arrow[from=3-2, to=2-2] \arrow[from=2-6, to=3-6] \arrow[from=3-6, to=4-5] \arrow[from=3-6, to=4-7] \arrow[from=4-7, to=5-6] \arrow[from=4-5, to=5-6] \arrow[from=2-10, to=3-10] \arrow[from=3-10, to=4-9] \arrow[from=3-10, to=4-11] \arrow[from=4-11, to=5-10] \arrow[from=4-9, to=5-10] \arrow[color={rgb,255:red,92;green,214;blue,214}, curve={height=-18pt}, dashed, from=2-6, to=2-10] \arrow[color={rgb,255:red,92;green,214;blue,214}, curve={height=-18pt}, dashed, from=3-6, to=3-10] \arrow[color={rgb,255:red,92;green,214;blue,214}, curve={height=-18pt}, dashed, from=4-5, to=4-9] \arrow[color={rgb,255:red,92;green,214;blue,214}, curve={height=-18pt}, dashed, from=4-7, to=4-11] \arrow[color={rgb,255:red,92;green,214;blue,214}, curve={height=-18pt}, dashed, from=5-6, to=5-10] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=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) - Note that the induced maps on stalks are surjective, since $$\phi_p: A \to A, 0$$ is either the identity or the zero map. But e.g. for $$\left\{{a, b}\right\}$$ we have $$A\mapsto A^{\times 2}$$, which can not be surjective. > Question: what is this map? Apparently its image is the diagonal...? ::: ::: {.problem title="1.4"} (a) Let $$\varphi: {\mathcal{F}}\to {\mathcal{G}}$$ be a morphism of presheaves such that $$\varphi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)$$ is injective for each $$U$$. Show that the induced map $$\varphi^+: {\mathcal{F}}^+ \to {\mathcal{G}}^+$$of associated sheaves is injective. (b) Use part (a) to show that if $$\varphi: {\mathcal{F}}\to{\mathcal{G}}$$ is a morphism of sheaves, then $$\operatorname{im}\varphi$$ can be naturally identified with a subsheaf of $${\mathcal{G}}$$, as mentioned in the text. ::: ::: {.proof title="of a"} {=tex} \envlist  - $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ is injective iff $$\phi_p:{\mathcal{F}}_p \to {\mathcal{G}}_p$$ is injective for all $$p$$. - Sheafification induces a map $$\phi^+: {\mathcal{F}}^+_p \to {\mathcal{G}}^+_p$$ - The sheafification has the same stalks, so $${\mathcal{F}}^+_p = {\mathcal{F}}_p$$ and $${\mathcal{G}}^+_p = {\mathcal{G}}_p$$. - So in fact $$\phi^+_p = \phi_p$$. Since $$\phi^+_p$$ is thus injective on all stalks, $$\phi^+$$ is injective on sheaves. ::: ::: {.proof title="of b"} {=tex} \envlist  - Noting that on opens $$(\operatorname{im}\phi)(U) \subseteq {\mathcal{G}}(U)$$ is an inclusion of abelian groups, so define a morphism of sheaves by $$\iota(U): (\operatorname{im}\phi)(U) \to {\mathcal{G}}(U)$$ using this inclusion. - By definition, it suffices to show $$\ker \iota = 0$$ as a sheaf. - By 1.2.2, it suffices to show $$(\ker \iota)_p = 0$$ on all stalks. - By 1.2.1, $$(\ker \iota)_p = \ker(\iota_p)$$, so it suffices to show $$\iota_p$$ is injective for all $$p$$. - Now use that $\ker(\iota_p) = \colim_{U\ni p} (\ker \phi)(\iota(U)) = \colim_{U\ni p} 0 = 0 ,$ since all of the $$\iota(U)$$ are injective, so 0 satisfies the universal property for this colimit. So we're done. ::: ::: {.problem title="1.5"} Show that a morphism of sheaves is an isomorphism if and only if it is both injective and surjective. ::: ::: {.proof title="?"} {=tex} \envlist  Problem: surjections of sheaves don't induce surjections ons sections! - $$\phi:{\mathcal{F}}\to{\mathcal{G}}$$ being injective means that $$(\ker \phi) = 0$$ as sheaves, and surjective means $$(\operatorname{im}\phi) = {\mathcal{G}}$$. - Thus $$\phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)$$ is injective, since $$(\ker \phi)(U) = 0(U) = 0$$, and surjective since $$\operatorname{im}(\phi(U)) = (\operatorname{im}\phi)(U) = {\mathcal{G}}(U)$$. This $$\phi(U)$$ is an isomorphism in abelian groups, and has an left and right inverse $$\phi^{-1}(U): {\mathcal{G}}(U) \to {\mathcal{F}}(U)$$. - So we have a diagram: {=tex} \begin{tikzcd} {{\mathcal{F}}(U)} && {{\mathcal{G}}(U)} && {{\mathcal{F}}(U)} \\ \\ {{\mathcal{F}}(V)} && {{\mathcal{G}}(V)} && {{\mathcal{F}}(V)} \arrow["{\phi(U)}", from=1-1, to=1-3] \arrow["{\phi(V)}", from=3-1, to=3-3] \arrow["{\mathop{\mathrm{Res}}_{{\mathcal{F}}}(U, V)}"', from=1-1, to=3-1] \arrow["{\mathop{\mathrm{Res}}_{{\mathcal{G}}}(U, V)}"{description}, from=1-3, to=3-3] \arrow["{\phi(U)^{-1}}", from=1-3, to=1-5] \arrow["{\phi(V)^{-1}}"', from=3-3, to=3-5] \arrow["{\mathop{\mathrm{Res}}_{{\mathcal{F}}}(U, V)}", from=1-5, to=3-5] \arrow["{\operatorname{id}_{{\mathcal{F}}(U)}}"{description}, curve={height=-30pt}, from=1-1, to=1-5] \arrow["{\operatorname{id}_{{\mathcal{F}}(V)}}"{description}, curve={height=30pt}, from=3-1, to=3-5] \end{tikzcd}  > [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJcXG1jZihVKSJdLFswLDIsIlxcbWNmKFYpIl0sWzIsMCwiXFxtY2coVSkiXSxbMiwyLCJcXG1jZyhWKSJdLFs0LDAsIlxcbWNmKFUpIl0sWzQsMiwiXFxtY2YoVikiXSxbMCwyLCJcXHBoaShVKSJdLFsxLDMsIlxccGhpKFYpIl0sWzAsMSwiXFxSZXNfe1xcbWNmfShVLCBWKSIsMl0sWzIsMywiXFxSZXNfe1xcbWNnfShVLCBWKSIsMV0sWzIsNCwiXFxwaGkoVSlcXGludiJdLFszLDUsIlxccGhpKFYpXFxpbnYiLDJdLFs0LDUsIlxcUmVzX3tcXG1jZn0oVSwgVikiXSxbMCw0LCJcXGlkX3tcXG1jZihVKX0iLDEseyJjdXJ2ZSI6LTV9XSxbMSw1LCJcXGlkX3tcXG1jZihWKX0iLDEseyJjdXJ2ZSI6NX1dXQ==) - Both squares form a morphism of sheaves, so the right square assembles to $$\phi^{-1}: {\mathcal{G}}\to{\mathcal{F}}$$ - Moreover $$(\phi^{-1}\circ \phi)({\mathcal{F}})(U) = \operatorname{id}_{{\mathcal{F}}(U)}$$ and similarly in the other order, so $$\phi^{-1}\circ \phi= \operatorname{id}_{{\mathcal{F}}}$$ Similarly $$(\phi\circ \phi^{-1})({\mathcal{G}})(U) = \operatorname{id}_{{\mathcal{G}}(U)}$$ and $$(\phi^{-1}\circ \phi) = \operatorname{id}_{{\mathcal{G}}}$$. - Then by definition an isomorphism of sheaves is a morphism with a two-sided inverse, so we're done. ::: ## Problem Set 2 ## II.1 ::: {.exercise title="II.1.8"} For any open $$U \subseteq X$$ show that the functor ${\mathsf{\Gamma}\qty{U, {-}} }: {\mathsf{Sh}}(X) \to {\mathsf{Ab}}{\mathsf{Grp}}$ is left-exact, but need not be exact. ::: ::: {.exercise title="II.1.14"} Let $${\mathcal{F}}\in {\mathsf{Sh}}(X)$$ and $$s\in {\mathcal{F}}(U)$$ be a section, and define $\mathop{\mathrm{supp}}s &\coloneqq\left\{{p\in U {~\mathrel{\Big|}~}s_p \neq 0}\right\} \subseteq U \\ \mathop{\mathrm{supp}}{\mathcal{F}}&\coloneqq\left\{{p\in X{~\mathrel{\Big|}~}{\mathcal{F}}_p\neq 0}\right\} \subseteq U ,$ where $$s_p$$ denotes the germ of $$s$$ in the stalk $${\mathcal{F}}_p$$. Show that $$\mathop{\mathrm{supp}}s$$ is closed in $$U$$ but $$\mathop{\mathrm{supp}}{\mathcal{F}}$$ need not be. ::: ::: {.exercise title="II.1.17"} Let $$X\in {\mathsf{Top}}, A\in {\mathsf{Ab}}{\mathsf{Grp}}, p\in X$$ and define the skyscraper sheaf as $\iota_p(A)(U) \coloneqq \begin{cases} A & p\in U \\ 0 & \text{else}. \end{cases} .$ Show that the stalk $$\iota_p(A)_q = A$$ when $$q\in { \operatorname{cl}} _X(\left\{{p}\right\})$$ and 0 otherwise, and that there is an equality of sheaves $$\iota_p(A) = \iota_*(\underline{A})$$ where $$\iota: { \operatorname{cl}} _X(\left\{{p}\right\}) \hookrightarrow X$$ is the inclusion. ::: ## II.2 ::: {.exercise title="II.2.1"} Let $$A\in \mathsf{Ring}$$ and $$X\coloneqq\operatorname{Spec}(A)$$, and for $$f\in A$$ let $$D(f) \coloneqq V(\left\langle{f}\right\rangle)^c$$. Show that there is an isomorphism of ringed spaces $(D(f), { \left.{{{\mathcal{O}}_X}} \right|_{{D(f)}} }) \xrightarrow{\sim} \operatorname{Spec}(A_f) .$ ::: ::: {.exercise title="II.2.3"} Note that $$(X, {\mathcal{O}}_X)\in {\mathsf{Sch}}$$ is **reduced** iff $${\mathcal{O}}_X(U)$$ has no nilpotents, and for $$A\in \mathsf{Ring}$$ define $$A^{{ \text{red} }}\coloneqq A/\sqrt{0}$$ to be $$A$$ modulo its ideal of nilpotents. a. Show that $$X$$ is reduced iff for every $$p\in X$$, the local ring $${\mathcal{O}}_{X, p}$$ has no nilpotents. b. Let $${\mathcal{O}}_X^{{ \text{red} }}$$ be the sheafification of $$U \mapsto {\mathcal{O}}_X(U)^{ \text{red} }$$. Show that $$X_{ \text{red} }\coloneqq(X, {\mathcal{O}}_X^{ \text{red} })$$ is a scheme, and there is a morphism of schemes $$X_{ \text{red} }\xrightarrow{{ \text{red} }} X$$ which induces a homeomorphism $${\left\lvert {X_{ \text{red} }} \right\rvert}\to {\left\lvert {X} \right\rvert}$$ on underlying topological spaces. c. Let $$X \xrightarrow{f} Y\in {\mathsf{Sch}}$$ with $$X$$ reduced. Show that there is a unique morphism $$X \xrightarrow{g} Y_{ \text{red} }$$ such that $$f$$ is the composition $(X \xrightarrow{f} Y) = (X \xrightarrow{g} Y_{ \text{red} }\xrightarrow{{ \text{red} }} Y ) .$ ::: ::: {.exercise title="II.2.5"} Describe $$\operatorname{Spec}{\mathbb{Z}}$$ and show it is terminal in $${\mathsf{Sch}}$$, i.e. each $$X\in {\mathsf{Sch}}$$ admits a unique morphism $$X\to \operatorname{Spec}{\mathbb{Z}}$$. ::: ::: {.exercise title="II.2.7"} Let $$X\in {\mathsf{Sch}}$$ and for $$x\in X$$ let $${\mathcal{O}}_x$$ be the local ring at $$x$$ and $${\mathfrak{m}}_x$$ its maximal ideal. Let $$\kappa(x) \coloneqq{\mathcal{O}}_x/{\mathfrak{m}}_x$$ be the residue field at $$x$$. Then for $$k$$ any field, show that giving a morphism $$\operatorname{Spec}(k) \to X \in {\mathsf{Sch}}$$ is equivalent to giving a point $$x\in X$$ and an inclusion $$\kappa(x) \hookrightarrow k$$. $x^2 - y^q = 1 && x^p - y^2 = 1 .$ $x^2 - y^q = 1 && x^p - y^2 = 1 .$ :::