*Note:
These are notes live-tex’d from a graduate course in Sheaf Cohomology taught by Valery Alexeev at the University of Georgia in Spring 2022. As such, any errors or inaccuracies are almost certainly my own.
*

dzackgarza@gmail.com

Last updated: 2022-04-22

Topic: cohomology of sheaves and derived categories. The plan:

- Sheaves (see ELC notes)
- Derived functors and coherent sheaves (see ELC notes)
- Derived categories (Gelfand-Manin, Tohoku)

References:

- Valery’s notes (see ELC)
- Gelfand-Manin,
*Methods of Homological Algebra*.

Compare (genus \(g\)) Riemann surfaces in the classical topology to (genus \(g\), projective) algebraic curves over \({\mathbb{C}}\) in the Zariski topology. Recall that \begin{align*} H^*(\Sigma_g; {\mathbb{Z}}) = \begin{cases} {\mathbb{Z}}& *=0, 2 \\ {\mathbb{Z}}^{2g} & *=1 \\ 0 & \text{else}. \end{cases} \end{align*} Note that this is a linear invariant in the sense that the constituents are free abelian groups, and we can extract a numerical invariant. For surfaces up to homeomorphism, this distinguishes them completely.

For algebraic curves, note that the topology is very different: the only closed sets are finite. In this topology, \begin{align*} H^*(X; {\mathbb{Z}})= \begin{cases} {\mathbb{Z}}& *=0 \\ 0 & \text{else}, \end{cases} \end{align*} which doesn’t see the genus at all. In fact all such curves are homeomorphic in this topology, witnessed by picking any bijection and noting that it sends closed sets to closed sets. The linear replacement: \(H^*(X; {\mathcal{O}}_X)\) for \({\mathcal{O}}_X\) the structure sheaf, which yields \begin{align*} H^*(X; {\mathcal{O}}_X) = \begin{cases} {\mathbb{C}}& *=0 \\ {\mathbb{C}}^g & *=1 \\ 0 & \text{else}. \end{cases} \end{align*} These surfaces can be parameterized by the moduli space \({ \mathcal{M}_{g} }\), which is dimension \(3g-3\) for \(g \geq 2\).

The POV in classical topology is to fix the coefficients: \({\mathbb{Z}}, {\mathbb{R}}, {\mathbb{C}}, {\mathbb{Z}}/n\), or \(R\) a general ring. A minor variation is to consider a local system \({\mathcal{L}}\), which are locally constant but may have nontrivial monodromy around loops. For example, one might have \({\mathbb{R}}\) locally, but traversing a loop induces an automorphism \(f\in \mathop{\mathrm{Aut}}({\mathbb{R}}) = {\mathbb{R}}^{\times}\). In this setting, we have a functor \(F({-}) = H({-}; R)\). For sheaf cohomology, instead fix \(X\) and take \(G({-}) = H(X; {-})\). In general, one can take sheaves of abelian groups, \({\mathcal{O}}_X{\hbox{-}}\)modules, quasicoherent sheaves, or coherent sheaves: \begin{align*} {\mathsf{Sh}}(X, {\mathsf{Ab}}{\mathsf{Grp}}) \hookrightarrow{\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}\hookrightarrow{\mathsf{QCoh}}(X) \hookrightarrow{\mathsf{Coh}}(X) .\end{align*}

We’ll be looking at three kinds of topologies:

The order topology: start with a poset and define the open sets to be the

*decreasing/lower sets*, i.e. subsets \(U_{x_0}\) that contain every element below a point \(x_0\). In other words, if \(x\in U\) and \(y\leq x\), then \(y\in U\).The Zariski topology: let \(R\) be a DVR, so \(\operatorname{Spec}R = \left\{{ \left\langle{0}\right\rangle, {\mathfrak{m}}}\right\}\). E.g. for \(R \mathrel{\vcenter{:}}={\mathbb{C}} { \left[ {t} \right] }\), \({\mathfrak{m}}= \left\langle{t}\right\rangle\), and the open sets are \(\left\{{\left\langle{0}\right\rangle}\right\}, \operatorname{Spec}R\), corresponding to the poset \({\operatorname{pt}}\to{\operatorname{pt}}\).

The classical topology, usually paracompact and Hausdorff.

One can define sheaves in all three cases, which have different properties. For posets, e.g. one can take \(C^0({-}, R)\) for \(R = {\mathbb{R}}, {\mathbb{C}}, { {\mathbb{Z}}_p }\).

Some computational tools:

- Vanishing theorems
- Riemann-Roch

Some topological notions to recall:

- \(T_0\), Kolmogorov spaces: distinct points don’t have the exact same neighborhoods, i.e. there exists a neighborhood of \(x\) not containing \(y\)
**or**a neighborhood of \(y\) not containing \(x\). - \(T_1\), Frechet spaces: points are separated, so replace “or” with “and” above.
- \(T_2\), Hausdorff spaces: points are separated by disjoint neighborhoods.
- Alexandrov spaces: arbitrary intersections of opens are open.
- Metrizability
- Paracompactness

Recall that a topology \(\tau\) on \(X\) satisfies

- \(\emptyset, X\in \tau\)
- \(A,B\in \tau \implies A \cap B \in \tau\)
- \(\displaystyle\bigcup_{j\in J} A_j \in \tau\) if \(A_j\in \tau\) for all \(j\).

Equivalently one can specify the closed sets and require closure under finite unions and arbitrary intersections.

Running examples:

- Any subset \(S \subseteq {\mathbb{R}}^n\) is Hausdorff and paracompact.
- Order topologies on posets
- Zariski topologies on varieties over \(k= \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\), e.g. \(\operatorname{mSpec}A\) for \(A\in {\mathsf{Alg}}^{\mathrm{fg}}_{/ {k}}\) or affine schemes \(\operatorname{Spec}A\).
- The discrete/initial topology \(\tau = 2^X\).
- The indiscrete topology \(\tau = \left\{{\emptyset, X}\right\}\).

Recall the separation axioms:

\(T_0\): points can be topologically distinguished. Note that the indiscrete topology s not \(T_0\) if \({\sharp}X\geq 2\).

\(T_1\): points can be separated by (not necessarily disjoint) neighborhoods. Equivalently, points are closed.

\(T_2\)/Hausdorff: points can be separated by disjoint neighborhoods.

\(T_{3.5}\)/Tychonoff:?

\(T_6\):?

Show that points are closed in \(X\) iff \(X\) is \(T_1\).

A space \(X\) is **paracompact** iff every open cover \({\mathcal{U}}\rightrightarrows X\) admits a *locally* finite refinement \({\mathcal{V}}\rightrightarrows X\), i.e. any \(x\in X\) is in only finitely many \(V_i\).

Show that any \(S \subseteq {\mathbb{R}}^n\) is paracompact, and indeed any metric space is paracompact.

Let \({\mathcal{U}}\rightrightarrows X \mathrel{\vcenter{:}}={\mathbb{R}}^d\) be an open cover and define a proposed locally open refinement in the following way:

- Write \({\mathcal{U}}\mathrel{\vcenter{:}}=\left\{{U_\alpha {~\mathrel{\Big\vert}~}\alpha\in A}\right\}\) for some index set.
- Use that \(W_n \mathrel{\vcenter{:}}={ \operatorname{cl}} _{X}({\mathbb{B}}_n(\mathbf{0}))\) is compact, and since \({\mathcal{U}}\rightrightarrows W_n\) there is a finite subcover \({\mathcal{V}}_n \mathrel{\vcenter{:}}=\left\{{U_{n, 1}, \cdots, U_{n, m}}\right\}\rightrightarrows{ \operatorname{cl}} _X({\mathbb{B}}^n(\mathbf{0}))^c\).
- Show that \({\mathcal{V}}\mathrel{\vcenter{:}}=\left\{{{\mathcal{V}}}\right\}_{n\in {\mathbb{Z}}_{\geq 0}}\) is an open refinement of \({\mathcal{U}}\).
- Why: it is a subcollection, and every \(x\in X\) is in a ball of radius \(R\approx N\mathrel{\vcenter{:}}=\ceil{{\left\lVert {x} \right\rVert}}\). So \(x\in {\mathbb{B}}_N(0)\), thus \(x\in U_{N, k}\) for some \(k\).

- Show that \({\mathcal{V}}\) is locally finite.
- Why: each \({\mathcal{V}}_n\) misses the \({\mathbb{B}}_{k<n}(0)\), so each \(x\not\in \displaystyle\bigcup_{k\geq N} {\mathcal{V}}_n\) if \(N\) is defined as above. So \(x\) is in only finitely many \({\mathcal{V}}_n\).

Paracompact spaces admit a POU – for \({\mathcal{U}}\rightrightarrows X\), a collection \(A\) of function \(f_\alpha: X\to {\mathbb{R}}\) where for all \(\alpha \in \mathop{\mathrm{supp}}f_\alpha = { \operatorname{cl}} \qty{\left\{{f\neq 0}\right\}}\), for all \(x\in X\), there exists a \(V\ni x\) such that for only finitely many \(\alpha, { \left.{{f_\alpha}} \right|_{{V}} } \not\equiv 0\), and \(\sum_{\alpha\in A} f_{ \alpha}(x) = 1\).

Recall the order topology: for \((P, \leq )\) a poset, so

- \(x\leq y, y\leq x\implies x=y\),
- \(x\leq y\leq z\implies x\leq z\)
- \(x\leq x\)

Define

- Open sets to be increasing sets, so \(x\in U, x\leq y \implies y\in U\),
- Closed sets to be decreasing sets, so \(x\in U, x\geq y \implies y\in U\)

Note that this is a free choice!

Show that the order topology is closed under arbitrary unions *and* intersections of opens.

Show that the order topology is not \(T_1\) by showing \({ \operatorname{cl}} _P\qty{\left\{{x}\right\}} = Z_{\leq}(x) \mathrel{\vcenter{:}}=\left\{{y\in P{~\mathrel{\Big\vert}~}y\leq x}\right\}\).

For \(k\) an infinite field, \({\mathbb{A}}^1_{/ {k}}\) is the cofinite topology and thus not Hausdorff.

Recall the definition of a **poset**.

Given a polytope, one can take its face poset \(\mathrm{FP} (P) = \left\{{F \leq P}\right\}\) where \(F_1 \leq F_2\) iff \(F_1 \subseteq F_2\) for the faces \(F_i\). More generally, one can take a complex of polytopes, i.e. a collection of polytopes that only intersect at faces. An example of a complex is the fan of a toric variety.

Similarly, one can take **cones** \(\sum c_i \mathbf{v}_i \subseteq {\mathbb{R}}^d\) for some positive coefficients.

Conversely, given a poset \(I\), one can associate a simplicial complex \({ {\left\lvert {I} \right\rvert} }\), the geometric realization. Any chain \(i_{n_1} < \cdots i_{n_k}\) is sent to a face and glued.

Consider a polytope \(P\), taking the face poset \(\mathrm{FP} (P)\), and its geometric realization \({ {\left\lvert { \mathrm{FP} (P)} \right\rvert} }\). A square has

- \({\sharp}P_2 = 1\)
- \({\sharp}P_1 = 4\)
- \({\sharp}P_0 = 1\)

Note that one can take the geometric realization of a category by using the nerve to first produce a poset.

With the right choices, there exists a continuous map \({ {\left\lvert {I} \right\rvert} } \to I\) where \(I\) is given the order topology. Pulling back sheaves on the latter yields constructible sheaves on convex objects, which are locally constant on the interior components.

A first version of the Zariski topology: let \(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu \in \mathsf{Field}\) and let \(R\in{\mathsf{Alg}}^{\mathrm{fg}}_{/ {k}}\) be of the form \(R = k[x_1, \cdots, x_{n}]/\left\langle{f_a}\right\rangle\). We can consider \(X\mathrel{\vcenter{:}}=\operatorname{mSpec}R \subseteq {\mathbb{A}}^n_{/ {k}}\) as the points \(\mathbf{x}\in k{ {}^{ \scriptscriptstyle\times^{n} } }\) such that \(f_a(\mathbf{x}) = 0\). Recall Noether’s theorem – the \(f_a\) can be replaced with a finite collection. The closed subsets are of the form \(V(g_b)\). Note that this is \(T_1\) since points are closed: given \(\mathbf{p} = {\left[ {p_0,\cdots, p_n} \right]}\), take \(f(\mathbf{p}) = \prod_{i\leq n} (x-p_i)\) so that \(V(f) = \left\{{\mathbf{p}}\right\}\). These points biject with maximal ideals in \(R\).

An improved version of the Zariski topology: \(X = \operatorname{Spec}R\), including prime ideals. The points are as before, and additionally for every irreducible subvariety \(Z \subseteq X\), there is a generic point \(\eta_Z\). This adds new points which can’t be described in coordinates.

Note that this generalizes to arbitrary (associative, commutative) rings. For rings that aren’t finitely generated, one loses the coordinate interpretation. These generally won’t embed into \({\mathbb{A}}^n_{/ {k}}\) for any \(n\), but can be embedded into (say) \({\mathbb{A}}^1_{/ {R}}\). Use that a closed embedding \(X\hookrightarrow Y\) corresponds precisely to a surjection of associated rings \(R_Y \twoheadrightarrow R_X\).

Let \(U \subseteq \Omega \subseteq {\mathbb{C}}\) and consider \(C^0(\Omega, {\mathbb{C}}) \mathrel{\vcenter{:}}=\mathop{\mathrm{Hom}}_{\mathsf{Top}}(\Omega, {\mathbb{C}})\) – this forms a sheaf of abelian groups, \({\mathbb{C}}{\hbox{-}}\)algebras, rings, sets, etc.

We’ll refer to this as \({\mathcal{O}}^\text{cts}_X\).

Some properties:

For every \(\iota: V \subseteq U \implies\) there is a restriction ma \begin{align*} {\mathcal{F}}(\iota): {\mathcal{F}}(U) &\to {\mathcal{F}}(V) \\ f &\mapsto { \left.{{f}} \right|_{{V}} } .\end{align*}

\({\mathcal{F}}({ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }) = { \mathscr{1}_{\scriptscriptstyle \uparrow} }\), so e.g. for rings \({ \mathscr{1}_{\scriptscriptstyle \uparrow} }= \left\{{0}\right\}\) is the zero ring.

(Sheaf 1, uniqueness): if \({\mathcal{U}}\rightrightarrows U\) and \(s_1, s_2 \in {\mathcal{F}}({\mathcal{U}})\), then \({ \left.{{s_1}} \right|_{{U_i}} } = { \left.{{s_2}} \right|_{{U_i}} } \implies s_1 = s_2\).

(Sheaf 2, existence): if \(s_i\in {\mathcal{F}}(U_{ij})\) and \({ \left.{{s_1}} \right|_{{U_{ij}}} } = { \left.{{s_2}} \right|_{{U_{ij}}} }\), then there is a global section \(s\in {\mathcal{F}}(U_1 \cup U_2)\).

Other examples of sheaves:

- \({\mathcal{O}}^\text{cts}_X\). One can check the sheaf properties directly.
- \({\mathcal{O}}^\text{hol}_X = {\mathcal{O}}^{\mathrm{an}}_X\) the holomorphic (complex differentiable) and thus analytic (locally equal to a convergent power series) functions on \(X\).
- Given a fixed continuous map \(f: Y\to X\), setting \({\mathcal{F}}(U) = \left\{{s: U\to Y}\right\}\) the set of continuous sections of \(f\).

Some examples of sheaves:

- For \(X \subseteq {\mathbb{C}}\) open, consider \({\operatorname{pr}}_1: X\times {\mathbb{C}}\to X\) and consider the space of continuous sections \({\mathcal{O}}_X^\text{cts}(U) \mathrel{\vcenter{:}}=\mathop{\mathrm{Hom}}_{\mathsf{Top}}(U, U\times {\mathbb{C}})\).

Analytic functions \({\mathcal{O}}_X^{\mathrm{an}}\)

\({\mathcal{O}}_X^\text{cts}\) where \({\mathbb{C}}\) is given the discrete topology instead of the Euclidean topology. The opens in \(U\times {\mathbb{C}}\) are of the form \(U\times V\) for \(V \subseteq {\mathbb{C}}\) any set at all:

- Constant sheaves \(\underline{{\mathbb{C}}}(U)\) defined as the locally constant continuous \({\mathbb{C}}{\hbox{-}}\)valued functions on \(U\).

Recall the sheaf properties:

- \(U\to F(U)\) and \(\iota_{U, V} \mapsto \mathop{\mathrm{Res}}{F(V),F(U)}\).
- \({ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }\mapsto F({ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }) = { \mathscr{1}_{\scriptscriptstyle \uparrow} }\).
- Sheaf conditions:
- Unique gluing: \({\mathcal{U}}\rightrightarrows X\) with \(\mathop{\mathrm{Res}}_{X, U_i} s = \mathop{\mathrm{Res}}_{X, U_i} t \implies s=t\in F(X)\)
- Existence of gluing: \(\left\{{s_i\in F(U_i)}\right\}\) with \(\mathop{\mathrm{Res}}_{U_i, U_{ij}} s_i = \mathop{\mathrm{Res}}_{U_j, U_{ij}} s_j\) implies \(\exists ! s\in F(X)\) with \(\mathop{\mathrm{Res}}_{X, U_i} s = s_i\) for all \(i\).

Recall that a **basis** of a topology is a collection \(B_i\) where every \(U \in \tau_X\) can be written as \(\displaystyle\bigcup_{i\in I} B_i\) for some index set \(I = I(X)\). Some examples:

- For \(X\in {\mathsf{Alg}}{\mathsf{Var}}_{/ {k}}\), the distinguished opens \(D(f) = \left\{{f\neq 0}\right\}\) and \(Z(f) = \left\{{f=0}\right\}\).
- For \(X =\operatorname{Spec}R \in {\mathsf{Aff}}{\mathsf{Sch}}_{/ {k}}\), take \(D(f) = \left\{{{\mathfrak{p}}\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}f\neq 0 \in R/{\mathfrak{p}}}\right\} = \left\{{{\mathfrak{p}}\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}f\not\in {\mathfrak{p}}}\right\}\)
- Note that \({\mathcal{O}}_{\operatorname{Spec}R}(D(f)) = R \left[ { \scriptstyle { {f}^{-1}} } \right]\).

Formulate the sheaf condition with a basis instead of arbitrary opens.

Hint: keep all of the same conditions, but since intersections may not be basic opens, write \(B_\alpha \cap B_\beta = \cup_k B_k\).

Some upcoming standard notions:

- Stalks \(F_x\)
- Sheafification \(F \mapsto F^+\)

A less standard topic:

- The espace etale or “flat space” of \(F\).

Recall that \begin{align*} F_x = \colim_{U\ni x} F(U) = \left\{{(U, s\in F(U))}\right\} / \sim && (U, s) \sim (V, t) \iff \exists W \supseteq U, V,\, \mathop{\mathrm{Res}}_{U, W}s = \mathop{\mathrm{Res}}_{V, W} t .\end{align*}

Example: \({\mathcal{O}}_{X, p}^{\mathrm{an}}= \left\{{f(z) \mathrel{\vcenter{:}}=\sum c_k (z-p)^k {~\mathrel{\Big\vert}~}f\text{ has a positive radius of convergence} }\right\}\). Note that \({\mathcal{O}}_{X, p}^\text{cts}\) doesn’t have such a nice description, since continuous functions can be distinct while agreeing on a small neighborhood. Similarly, \(\underline{{\mathbb{C}}}_p = {\mathbb{C}}\), since locally constant is actually constant on a small enough neighborhood.

Recall that morphisms of (pre)sheaves are natural transformations of functors. There is a forgetful functor \(\mathop{\mathrm{Forget}}: {\mathsf{Sh}}(X) \to \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)\), which has a left adjoint \(({-})^+: \underset{ \mathsf{pre} } {\mathsf{Sh} }(X) \to {\mathsf{Sh}}(X)\). There is a description of \(F^+(U)\) as collections of local compatible sections of \(F\) modulo equivalence – compatibility here means that if \({\mathcal{U}}\rightrightarrows X\), then writing \(U_{ij} = \cup V_k\) we have \(\mathop{\mathrm{Res}}_{X, V_k}s_i = \mathop{\mathrm{Res}}_{X, V_k}s_j\) for all \(i, j\).

Last time: definitions of presheaves and sheaves. There is an adjunction \begin{align*} \adjunction{({-})^+}{\mathop{\mathrm{Forget}}}{ \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)}{{\mathsf{Sh}}(X)} .\end{align*}

Recall that constant sheaves for \(A\in \mathsf{D}\) are defined as \(\underline{A}({-}) \mathrel{\vcenter{:}}=\mathop{\mathrm{Hom}}_{\mathsf{Top}}({-}, A)\) where \(A\) is equipped with the discrete topology.

What is \(\Gamma(\underline{A}, X)\) for \(X\mathrel{\vcenter{:}}=\left\{{1/n}\right\}_{n\in {\mathbb{Z}}_{\geq 0}} \subseteq {\mathbb{R}}\)? So \(A(U) \neq A^{{\sharp}\pi_0 U}\) in general, since there may not be a notion of connected components for an arbitrary topological space.

Is it true that for any \(X\in {\mathsf{Top}}\) there is a unique decomposition \(X = {\textstyle\coprod}_{i\in I} U_i\) into connected components?

Hint: form a poset of such decompositions ordered by refinement and apply Zorn’s lemma.

Consider the following poset with a prescribed topology, and applying some functor \(F\):

For this to be a sheaf, this forces

- \(F(\emptyset) = { \mathscr{1}_{\scriptscriptstyle \uparrow} }\)
- \(F_{12} \cong F_1 \oplus F_2\) by the universal property of \(\oplus\) if this is to be a sheaf.
- \(F_3\) can be anything mapping to \(F_{12}\).

What are the stalks?

- \(F_x = F(X)\) for \(x=3\), since \(X\) is the smallest open set containing \(3\).
- \(F_{x_i} = F_i\) for \(x_i = 1, 2\).

Consider now a poset in the order topology:

Now \(F\) is a sheaf iff \(F_{124}\cong F_1 { \underset{\scriptscriptstyle {F_4} }{\times} }F_2\) is the fiber product.

A map \(\pi: Y\to X \in {\mathsf{Top}}\) is a **sheaf space** if it is a local homeomorphism, so every \(y\in Y\) admits a neighborhood \(U_y\ni y\) where \({ \left.{{\pi}} \right|_{{U_y}} }: U_y\to \pi(U_y)\) is a homeomorphism onto its image.

Some examples:

- \(X\times A\to X\) for \(A\) discrete.

One possibility: “jumping.” Take \(Y \mathrel{\vcenter{:}}= X\displaystyle\coprod_{X\setminus\left\{{0}\right\}} X\) for \(X\subseteq {\mathbb{R}}\), which is a version of the line with two zeros. Then \(Y\to X\) is a sheaf space, since it is a local homeomorphism.

The other possibility is “skipping”:

These two definitions of sheaf coincide: for new to old, given \(Y \xrightarrow{\pi} X\) apply \(\mathrm{ContSec}_\pi \subseteq \mathop{\mathrm{Hom}}_{\mathsf{Top}}(X, Y)\). In the other direction, define \(Y\mathrel{\vcenter{:}}=\displaystyle\coprod_{x\in X} F_x\) and prove it is a local homeomorphism.

Next time: direct/inverse image, shriek functors, sheaves of modules.

Recall the definitions of presheaves and sheaves, and sheafification as an adjoint to \(\mathop{\mathrm{Forget}}: {\mathsf{Sh}}(X)\to \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)\). For \(F\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)\) we concretely construct its sheafification \(F^+\) using the sheaf space \(\pi: Y\mathrel{\vcenter{:}}=\displaystyle\coprod_{x\in X} F_x \to X\).

What are the sections of \(\pi\)? For a basic open \(U \subseteq X \ni x\), the fiber is \(\pi^{-1}(x) = F_x \mathrel{\vcenter{:}}=\colim_{V\ni x} F(V)\), which receives a map \(\mathop{\mathrm{Res}}_{U, x}: F(U) \to F_x\). Writing \(s\in F(U)\), define \(s_x \mathrel{\vcenter{:}}=\mathop{\mathrm{Res}}_{U, x}(s)\), and set \(W_{s, U} \mathrel{\vcenter{:}}=\left\{{s_x {~\mathrel{\Big\vert}~}x\in U}\right\}\) to be \(\pi^{-1}(U)\). Then define \(F^+\) to be the continuous sections of \(Y \xrightarrow{\pi} X\). What does such a section look like? For \(t:U\to \pi^{-1}(U)\) and \(x\in U\), the vertical fiber is \(F_x\). For a basic open \(V\ni X\) in the base, there is a basic open \(W_{s, V}\) in \(Y\) for \(s\in F(V)\):

There are maps \(s_{ij}: U_{ij}\to \pi^{-1}(U_{ij})\), but note that \(\mathop{\mathrm{Res}}(U_i, U_{ij}) s_i\) does not necessarily equal \(\mathop{\mathrm{Res}}(U_j,U_{ij}) s_j\) in \(F(U_{ij})\) – instead, there is an open cover \(U_{ij} = \displaystyle\bigcup V_{\alpha}\) with \(\mathop{\mathrm{Res}}(U_i, V_\alpha) s_i = \mathop{\mathrm{Res}}(U_j, V_\alpha) s_j\) for each \(\alpha\).

Todo

For \(f\in {\mathsf{Top}}(X, Y)\) we have the following constructions:

- The direct image \(f_*: {\mathsf{Sh}}(X) \to {\mathsf{Sh}}(Y)\), which is easy with the sheaf definition, and
- The inverse image \(f^{-1}: {\mathsf{Sh}}(Y)\to {\mathsf{Sh}}(X)\) which is easier with the sheaf space definition.

Recall the definition of a morphism of sheaves as a natural transformation.

For sheaves of abelian groups and \(\phi: F\to G\) a morphism of sheaves, there are notions of \(\ker \phi, \operatorname{coker}\phi, \operatorname{im}\phi\), and extension of a sheaf by zero.

To show these exist as presheaves, one only has to show existence of the following blue morphisms of abelian groups:

Write \((\operatorname{coker}\phi)^-\) and \((\operatorname{im}\phi)^-\) for these presheaves.

\(\ker \phi\) is a sheaf.

Axiom 1: use that \(F\) is a sheaf and \(\ker \phi_U \subseteq F(U)\) can be viewed as an inclusion. Axiom 2: write \(s_i\in \ker \qty{F(U_i) \xrightarrow{\phi_{U_i}} F(U_j) }\), then there exists a unique \(s\in F(U)\). Then check that \(s\in \ker\qty{F(U) \to G(U)}\) by noting that if \(s\mapsto t\) then \({ \left.{{t}} \right|_{{U_i}} } = 0\) for all \(i\), making \(t\equiv 0\) by the sheaf property of \(G\).

Define \begin{align*} \operatorname{coker}\phi &\mathrel{\vcenter{:}}=( (\operatorname{coker}\phi)^-)^+ \\ \operatorname{im}\phi &\mathrel{\vcenter{:}}=( (\operatorname{im}\phi)^-)^+ .\end{align*}

Take \(X = {\mathbb{C}}\) and consider \(\exp: \mathop{\mathrm{Hol}}(X) \to G\) the sheaf of nowhere zero holomorphic functions. Then on \(U_i \in {\mathbb{C}}\setminus\left\{{0}\right\}\), take \(z\in G\). Then \(z = \exp(f_i)\) in each \(U_i\) with \(f_i \in \mathop{\mathrm{Hol}}(X)\), so \(f_i = \log(z)\) locally and \(z = \exp(\log z)\), but there is no global \(f\in \mathop{\mathrm{Hol}}(X)\) with \(\exp(f) = z\). So \(z\in \ker \phi_i(\mathop{\mathrm{Hol}}(U_i) \to G(U_i))\) but \(z\not \in \ker \exp\). For the same reason, \(z = 0\) in \(\operatorname{coker}\phi_i\) since it’s locally in the image. but \(z\neq 0 \in \operatorname{coker}\exp\) since it’s not globally in the image.

Recall last time: presheaf vs sheaf properties, images, kernel, cokernel. We can state the uniqueness sheaf axiom as the following: if \(s\in F(U)\) with \({ \left.{{s}} \right|_{{U_i}} } = 0\) for \({\mathcal{U}}\rightrightarrows U\), then \(s = 0\) in \(F(U)\).

- \({\mathcal{F}}\mathrel{\vcenter{:}}=(\operatorname{im}\phi)^-\) satisfies uniqueness.
- \({\mathcal{G}}\mathrel{\vcenter{:}}=(\operatorname{coker}\phi)^-\) satisfies existence.
- \({\mathcal{F}}\) fails existence \(\iff {\mathcal{G}}\) fails uniqueness
- \({\mathcal{F}}\) fails uniqueness iff \({\mathcal{G}}\) fails existence.

The presheaf image and cokernel can sometimes fail to be a sheaf: use \(\mathop{\mathrm{Hol}}(X) \xrightarrow{\exp} \mathop{\mathrm{Hol}}(X)^{\times}\). The kernel presheaf \((\ker \phi)^-\) is already a sheaf.

Show the following:

- A sheaf \({\mathcal{F}}\) is the zero sheaf iff \({\mathcal{F}}_p =0\) for all \(p\).
- \(\ker(\phi)_p = \ker(\phi_p)\), which is \(\ker({\mathcal{F}}_p \xrightarrow{\phi_p} {\mathcal{G}}_p)\) the kernel of the induced map.
- \(\operatorname{coker}(\phi)_p = \operatorname{coker}(\phi_p) \mathrel{\vcenter{:}}=\operatorname{coker}({\mathcal{F}}_p \xrightarrow{\phi_p} {\mathcal{G}}_p)\).
- \(\phi: {\mathcal{F}}\to {\mathcal{G}}\) is injective iff \(\phi_p: {\mathcal{F}}_p \to {\mathcal{G}}_p\) is injective for all \(p\).
- \(\phi: {\mathcal{F}}\to {\mathcal{G}}\) is surjective iff \(\phi_p: {\mathcal{F}}_p \to {\mathcal{G}}_p\) is surjective for all \(p\).

Hints:

- Suppose \(s\neq 0\) in \(F(U)\), does there exist a \(p\) with \(s_p = 0\)?
- Use that \(s_p \in (\ker \phi)_p\) can be regarded as \(s\in \ker(F(V) \to G(V))\) mod equivalence.

If there exists an injective morphism \(\phi:{\mathcal{F}}\to {\mathcal{G}}\), we regard \({\mathcal{F}}\leq {\mathcal{G}}\) as a **subsheaf** and define the **quotient sheaf** \({\mathcal{F}}/{\mathcal{G}}\mathrel{\vcenter{:}}=\operatorname{coker}({\mathcal{F}}\xrightarrow{\phi} {\mathcal{G}})\).

Show by example that \(({\mathcal{F}}/{\mathcal{G}})^-\) need not be a sheaf.

Note that for \(\phi: {\mathcal{F}}\to {\mathcal{G}}\), the image \(\operatorname{im}\phi\) is a secondary notion in additive categories, and can instead be defined as either

- \(\operatorname{coker}(\ker \phi \to {\mathcal{F}})\)
- \(\ker({\mathcal{G}}\to \operatorname{coker}\phi)\)

These need not coincide in general.

Defining the direct image: easier using the sheaf axioms. For \(f\in {\mathsf{Top}}(X, Y)\), define \(f_*: {\mathsf{Sh}}(X) \to {\mathsf{Sh}}(Y)\) by \begin{align*} f_* {\mathcal{F}}(U) \mathrel{\vcenter{:}}={\mathcal{F}}(f^{-1}(U)), \qquad \in {\mathsf{Sh}}(Y) \text{ for } {\mathcal{F}}\in {\mathsf{Sh}}(X) .\end{align*}

For the preimage: easier to use the espace étalé. As a special case, consider \(\iota: S \hookrightarrow Y\) where \(S\) is a subspace of \(Y\) (with the subspace topology). Then for \({\mathcal{F}}\in {\mathsf{Sh}}(Y)\), we can now define sections not only on open subsets \(U\) but arbitrary subsets \(S\) as \begin{align*} {\mathcal{F}}(S) \mathrel{\vcenter{:}}=(\iota^{-1}{\mathcal{F}})(S) .\end{align*}

Last time:

- Morphisms of sheaves \(\phi\),
- \(\ker \phi\) (already a sheaf),
- \((\operatorname{im}\phi)^-, (\operatorname{coker}\phi)^-\) (need to sheafify),
- All defined to commute with taking stalks: \((\ker \phi)_p = \ker(\phi_p)\), etc

- \((\operatorname{im}\phi)^-\) may fail the existence axioms for sheaves, using \(\exp: {\mathcal{O}}^{\mathrm{an}}\to ({\mathcal{O}}^{\mathrm{an}})^{\times}\) for \(X\) a complex analytic space,
- \((\operatorname{coker}\phi)^-\) may fail the uniqueness axioms for sheaves,
- \((\operatorname{im}\phi)^-\) satisfies existence \(\iff (\operatorname{coker}\phi)^-\) satisfies uniqueness,
- For \({\mathcal{F}}\hookrightarrow{\mathcal{G}}\) injective, the presheaf quotient \(({\mathcal{G}}/{\mathcal{F}})^-\) may fail to be a sheaf.

For \(X\in {\mathsf{Alg}}{\mathsf{Var}}_{/ {k}}\) for \(k= { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }\), let \({\mathcal{O}}_X\) be its regular algebraic functions. Take \(X = {\mathbb{P}}^1\) and \(U \mathrel{\vcenter{:}}={\mathbb{A}}^1\setminus\left\{{{\operatorname{pt}}}\right\} \subseteq {\mathbb{A}}^1 \subseteq {\mathbb{P}}^1\setminus\left\{{a_1,\cdots, a_k}\right\}\). Then \({\mathcal{O}}_X(U) = k[x] \left[ { \scriptstyle { {f}^{-1}} } \right]\) for \(f(x) \mathrel{\vcenter{:}}=\prod (x-a_k)\), \({\mathcal{O}}_X(X) = k\), \(K_X(U) = k(x)\), and \(K_X^{\times}(U) = k(x)\setminus\left\{{0}\right\}\) if \(U \neq \emptyset\). Define **Cartier divisors** as global sections of the sheaf \(\mathop{\mathrm{Cart}}\operatorname{Div}\mathrel{\vcenter{:}}= K_X^{\times}/{\mathcal{O}}_X^{\times}\). Recall that Weil divisors are finite sums of codimension 1 subvarieties, and these notions coincide for nonsingular varieties.

For \(p\in {\mathbb{A}}^1 \subseteq {\mathbb{P}}^1\), we have \begin{align*} (K_X^{\times}/{\mathcal{O}}_X^{\times})_p = {K_{X, p} \over {\mathcal{O}}_{X, p}} = {k(x) \over \left\{{f/g {~\mathrel{\Big\vert}~}f(p)\neq 0 , g(p) \neq 0}\right\}} \cong {\mathbb{Z}} ,\end{align*} using that any element in the quotient can be written as \(h(x) = (x-p)^n g(x)\) for some \(g\in {\mathcal{O}}_{X, p}^{\times}\). Here \(\mathop{\mathrm{Cart}}\operatorname{Div}(X) = \sum n_p P\) are all finite sums with \(n_p\in {\mathbb{Z}}\). The claim is that sheaf existence fails for this quotient – there are local sections that do not glue. Here

- \(K^{\times}({\mathbb{P}}^1) = k(x)^{\times}\)
- \({\mathcal{O}}^{\times}({\mathbb{P}}^1) = k^{\times}\)
- \(K^{\times}({\mathbb{P}}^1)/{\mathcal{O}}^{\times}({\mathbb{P}}^1) = {k(x)^{\times}\over k^{\times}}\)

For any \(s\) in the quotient, we can associated \((s)_0 - (s)_\infty = \sum n_p P\), but not every Cartier divisor is of this form – these are the *principal* divisors. This form a group \({\operatorname{Pic}}(X) = \mathop{\mathrm{Cart}}\operatorname{Div}(X) / \mathop{\mathrm{Prin}}\mathop{\mathrm{Cart}}\operatorname{Div}(X)\), which may not be trivial. This proof generalizes to locally Noetherian schemes, not necessarily reducible, with no embedded components.

Note that \({\operatorname{Pic}}(X)\) is also the group of invertible sheaves on \(X\), and for irreducible algebraic varieties these coincide. Use the SES \(0\to {\mathcal{O}}^{\times}\to K^{\times}\to K^{\times}/{\mathcal{O}}^{\times}\to 0\) to obtain \begin{align*} 1 \to H^0({\mathcal{O}}^{\times}) \to H^0(K^{\times}) \to \mathop{\mathrm{Prin}}\mathop{\mathrm{Cart}}\operatorname{Div}(X) \to H^1({\mathcal{O}}^{\times})\cong \text{invertible sheaves}/\sim \to 0, ,\end{align*} where \(H^1(K^{\times})\) vanishes since it’s a constant sheaf on an irreducible scheme in the Zariski topology.

\((\operatorname{im}\phi)^-\) satisfies existence \(\iff (\operatorname{coker}\phi)^-\) satisfies uniqueness.

\(\implies\): Let \(s\in \operatorname{coker}(F(U) \to G(U))\) and write \(U = \cup U_i\). We want to show that \(s_{U_i}\) implies \(s\in \operatorname{coker}(F(U_i) \to G(U_i))\) for all \(i\). Note that \(s = 0\) in \(\operatorname{coker}(F(U) \to G(U))\) iff \(s\in \operatorname{im}(F(U) \to G(U))\)

Direct image sheaf: for \({\mathcal{F}}\in {\mathsf{Sh}}(X), {\mathcal{G}}\in {\mathsf{Sh}}(Y), f\in {\mathsf{Top}}(X, Y)\), the \(f_* \in [{\mathsf{Sh}}(X), {\mathsf{Sh}}(Y)]\) is defined by \(f_* {\mathcal{F}}(U) \mathrel{\vcenter{:}}={\mathcal{F}}(f^{-1}U)\). The inverse image functor \(f^{-1}\in [{\mathsf{Sh}}(Y), {\mathsf{Sh}}(X)]\) is slightly more complicated. An easy case: if \(\iota: S \hookrightarrow Y\) is a subspace, then it is just restriction: \((\iota^{-1}G)(S) \mathrel{\vcenter{:}}= G(S)\).

Idea for sheaf space: there are strictly horizontal neighborhoods as the homeomorphic preimages of small opens in the base. So for \(\text{Ét}_{{\mathcal{G}}} \xrightarrow{\pi} Y\) the sheaf space of \({\mathcal{G}}\), define the inverse image as \begin{align*} \text{Ét}_{\iota^{-1}{\mathcal{G}}} \mathrel{\vcenter{:}}=\pi^{-1}(S) \subseteq \text{Ét}_{{\mathcal{G}}} ,\end{align*} and define a basis of sections in the following way: for \(s\in {\mathcal{G}}(U)\), set \(t(U) \mathrel{\vcenter{:}}= s(U) \cap\pi^{-1}(S) \in \text{Ét}_{{\mathcal{G}}}\) to be sections of \(\text{Ét}_{\iota^{-1}{\mathcal{G}}}\). Declare these to be a basis of opens, i.e. take the subspace topology for \(\pi^{-1}(S) \subseteq \text{Ét}_{{\mathcal{G}}}\) in the sheaf topology on the total space. More generally, for \(f\in {\mathsf{Top}}(X, Y)\), set \begin{align*} \text{Ét}_{f^{-1}{\mathcal{G}}} \mathrel{\vcenter{:}}=\text{Ét}_{\mathcal{G}}{ \underset{\scriptscriptstyle {Y} }{\times} } X .\end{align*}

The fibers are identical:

The topology on \(\text{Ét}_{f^{-1}{\mathcal{G}}}\) is the coarsest topology for which \(\pi^*\) and \(f^*\) are continuous. This is generated by \(\qty{ f^{-1}(s)(f^{-1}U)} \cap(\pi^*)^{-1}(W)\) for \(W \subseteq X\) open. Define \(f^{-1}(s) \in f^{-1}{\mathcal{G}}(f^{-1}(U)) \mathrel{\vcenter{:}}=(f^{-1}U){ \underset{\scriptscriptstyle {U} }{\times} } s(U)\). This makes the pullback continuous both vertically and horizontally.

\begin{align*} (f^{-1}{\mathcal{G}})_y = {\mathcal{G}}_{f(y)} .\end{align*}

\begin{align*} f^{-1}{\mathcal{G}}\mathrel{\vcenter{:}}=\qty{V\mapsto \colim_{U, V \subseteq f^{-1}(U)} {\mathcal{G}}(U)}^+ .\end{align*}

How to prove this coincides with the previous definition:

- Show the stalks are isomorphic,
- Show that there is a map of presheaves \((f^{-1}{\mathcal{G}}) \to f^{-1}{\mathcal{G}}\),
- Show that the map induces an isomorphism on stalks, and lift using the universal property of sheafification.

Try to prove this by commuting limits.

Recall that \(K^{\times}/{\mathcal{O}}^{\times}\cong \bigoplus _{x\in X} (\iota_*)_* \iota_*^{-1}\underline{{\mathbb{Z}}}\) which had stalks \({\mathbb{Z}}\) but was not constant – check that the local sections differ.

For \(S\hookrightarrow Y\), does every section of \({\mathcal{G}}\) over \(S\) extend to \(Y\)?

Extending by zero: for \(i: U \hookrightarrow X\) an open subspace and \({\mathcal{F}}\in {\mathsf{Sh}}(U)\), define \(i_!{\mathcal{F}}\in {\mathsf{Sh}}(X)\). If the target category has a zero object, define this in the sheaf space by extending the zero section: