--- sort: 030 title: Schemes flashcard: "Orals::Schemes" --- # Schemes - [x] Give several concrete examples of schemes ✅ 2022-11-25 - [ ] $V(xy)$ the union of axes, which is not irreducible. - [ ] $V(y^2-x^3)$ the cuspidal curve. - [ ] $V(x^2-y^2)$, the X-shaped space. - [ ] $V(y^2-x^3-x)$ the nodal curve - [ ] $V(y^2-ax^3-bx)$ a general elliptic curve. - [ ] The rational normal curves $\PP^1\to \PP^N$ given by the Veronese embedding $[x:y]\mapsto$ monomials of degree $d$, so $N = {n+d\choose d}$. - [ ] The twisted cubic $\im\qty{\tv{x:y} \to \tv{x^3: x^2 y : xy^2 : y^3} }$, cut out as $V(xz-y^2, yw-z^2, xw-yz)\subseteq \PP^3$ (i.e. the image of Veronese $\PP^1\to \PP^3$). Equivalently the $d=3$ rational normal curve. - [ ] The dual numbers $\spec k[\eps]/\eps^2$. - [ ] $\AA^1$ with a doubled origin (not separated) - [ ] The quadric surface $V(xy-zw) \subseteq \PP^3$ - [ ] $V(x^2+y^2-z^2)$ the double cone along the $z\dash$axis. - [ ] $\spec R$ for any ring (obviously!) - [ ] $\spec k$ is a point - [ ] $\spec R$ for $R\in \DVR$ is 2 points: a closed point and a generic point. - [ ] The Whitney umbrella $V(x^2-y^2z)$. - [x] Give several concrete examples of morphisms of schemes. ✅ 2022-11-25 - [ ] The normalization of the cuspidal curve $\eta:\AA^1\to C$ where $t\mapsto (t^2, t^3)$. - [ ] $(x, y)\mapsto (x, xy)$ on $\AA^2$ - [ ] $x\mapsto x^n$ on $\GG_m$ - [ ] $V(xy-c) \to \AA^1$ given by coordinate projection - [x] Give an example of a morphism of ringed spaces which is not induced by any ring morphism. ✅ 2022-11-25 - [ ] Take $R$ a DVR, then $\spec \ff(R) \to \spec R$ sending the unique point to the closed point. This is not induced by any morphism $R\to \ff(R)$ since it is not a morphism of *locally* ringed spaces. - [x] What is a ringed space? Locally ringed spaces? #syllabus ✅ 2022-11-28 - [ ] $(X, \OO_X)$ where $X\in \Top$ and a **structure sheaf** $\OO_X\in \Sh(X, \CRing)$. This is locally ringed if all stalks $\OO_{X, x}$ have unique maximal ideals. - [x] What is a morphism of ringed spaces? Locally ringed spaces? #syllabus ✅ 2022-11-28 - [ ] Pairs $(f, f^\sharp)$ where $f\in \Top(X, Y)$ and $f^\sharp: \OO_{Y} \to f_* \OO_X$. This is a morphism of locally ringed spaces iff $f^\sharp_x: \OO_{Y, f(x)} \to f_* \OO_{X, x}$ is a morphism of local rings, i.e. $f^\sharp_x( \mfm_{f(x)} ) \subseteq \mfm_x$. - [x] What is the definition of a scheme? #syllabus ✅ 2022-11-28 - [ ] A locally ringed space $(X, \OO_X)$ locally isomorphic to an affine scheme, i.e. $\exists \mcu \covers X$ such that $(U, \ro{\OO_X} U) \cong (\spec A, \OO_{\spec A})$ for each $U$. - [x] What is the dimension of a scheme? #syllabus ✅ 2022-11-28 - [ ] Supremum of lengths of chains of irreducible closed subsets, where the length is the number of *links* in a chain $Z_0 \subseteq Z_1 \subseteq \cdots \subseteq Z_n$. - [x] What is the Krull dimension of a ring? #syllabus ✅ 2022-11-28 - [ ] Supremum of all lengths of chains of primes ideals. - [x] What is a variety in scheme-theoretic terms? ✅ 2022-11-24 - [ ] A separated integral scheme of finite type over a field. - [x] What is an **affine scheme**? What are their morphisms? #syllabus ✅ 2022-11-24 - [ ] A locally ringed space $(X, \OO_X)$ which is isomorphic to $(R, \OO_{\spec R})$ for some $R\in \CRing$. Morphisms are morphisms of *locally* ringed spaces, and maps $\spec A\to \spec B$ biject with ring maps $B\to A$. - [x] Describe $(\spec A, \OO_{\spec A})$. #syllabus ✅ 2022-12-05 - [ ] Stalks $\OO_{\spec A, \mfp} = A_\mfp \da S\inv A, S\da A\sm \mfp$. - [ ] Sections: $\OO(U) = s: U\to \disjoint_{p\in U} A_p$ where $s(p) \in A_p$ and locally a fraction: $\forall p\in U, \exists V$ with $p\in V \subseteq U$ and $\exists a, f\in A$ such that for each $q\in V$ with $f\not\in q$, $s(q) = a/f$. - [ ] Global sections: $A$. - [x] What is a (quasi)projective scheme? #syllabus ⏫ ✅ 2022-12-05 - [ ] Projective morphisms: $f:X\to Y$ factoring through a closed immersion $X\to \PP^n\slice Y$ - [ ] Quasiprojective morphisms: factoring through an *open* immersion. - [ ] (quasi)projective iff structure morphism is. - [x] How is a non-affine **scheme** defined? ✅ 2022-11-24 - [ ] A locally ringed space in which every point $x$ admits a neighborhood which is isomorphic to an affine scheme. - [x] What is a **normal scheme**? ✅ 2022-12-04 - [ ] $X$ is normal iff irreducible and $\OO_{X, x}$ is a normal ring at each point, i.e. reduced and integrally closed in its field of fractions. - [ ] A ring $R$ is integrally closed if every $s\in S \da \ff(R)$ is integral over $R$, i.e. $\exists f\in R[x]$ with $f(s) = 0$.. - [x] What is the **normalization** of a scheme? ✅ 2022-11-25 - [ ] Cover $\spec A_i\covers X$, let $K_i$ be the integral closure of $A_i$ in its fraction field, and glue $\spec K_i$ together to get $\tilde X \to X$. - [x] Discuss the normalization. #syllabus ✅ 2022-12-08 - [ ] Exists for any reduced scheme, yields $f: \tilde X\to X$ an integral birational morphism (finite for varieties), always regular in codimension 1 so yields regular (thus smooth) curves and surfaces with only isolated singularities. Makes every finite birational morphism into $\tilde X$ an isomorphism. - [x] Give an example of a non-normal scheme. #syllabus ✅ ⏫ 2022-12-06 ✅ 2022-12-08 - [ ] The cuspidal curve $V(y^2-x^3)$, since there is a finite birational morphism $$\begin{align*}\AA^1&\to X \\ t&\mapsto (t^2, t^3)\end{align*}$$ which is not an isomorphism. Similarly, the nodal cubic $V(y^2-x(x+1))$ is not normal since its parameterization $t\mapsto (f_1(t), f_2(t))$ which is not an isomorphism (2-to-1 at the node). - [x] What is the universal property of normalization? ✅ 2022-11-25 - [ ] Every dominant $f:Z\to X$ factors through $\tilde X\to X$. - [x] What is the **reduced scheme structure** on a closed subset of a scheme? #syllabus ⏫ ✅ 2022-12-08 #unknown - [ ] For affines $Y \subseteq X = \spec A$: set $\mfa \da \Intersect_{\mfp\in Y}\mfp$ the intersection of all prime ideals in $Y$, then $\spec(A/\mfa) \cong Y$ maximally, and $A/\mfa$ is reduced since $\mfa$ is radical, this defines a reduced scheme. Alternatively take $V(\mfa)$ which has coordinate ring $A/\mfa$. - [ ] For arbitrary schemes $Y\subseteq X$, cover by affines $U_i$, intersect with $Y$ to get $Y_i$, put above reduced structure on them. These agree on triple overlaps and so one can glue the sheaves on $Y_i$ to one on $Y$. - [x] What is an **integral scheme**? #syllabus ✅ 2022-12-08 #unknown - [ ] Covered by spectra $\spec R_i$ with each $R_i$ an integral domain, so no nonzero zero divisors. - [ ] Equivalently, $X$ is reduced and irreducible. - [ ] $\OO_X(U)$ is an integral domain for all $U$. - [x] What is a consequence of a scheme being integral? ✅ 2022-12-08 - [ ] There is only one generic point, as opposed to many associated points. A section of a line bundle over a nonempty open is determined by its stalk at its generic point. - [x] What is a **regular scheme**? ⏫ ✅ 2022-12-04 - [ ] For $X$ locally Noetherian, the rings $\OO_{X, x}$ at closed points are all regular rings. - [ ] $$\dim_{\kappa(x)} \T_{X, x} = \krulldim R, R\da \OO_{X, x} \qquad\forall x\in X, k \da R/\mfm_R$$ - [x] What does it mean to be **regular in codimension 1**? ✅ 2022-12-08 - [ ] Every local ring $\OO_{X, x}$ of dimension 1 is regular. - [ ] More generally for local rings, for every prime ideal $\mfp \in \spec A$ of height 1, the localization $A_\mfp$ is regular and hence a field or DVR. - [x] What are some examples of schemes that are regular in codimension 1? ✅ 2022-11-25 - [ ] Smooth varieties over a field: the local rings of closed points are regular, hence of all points, since they are localizations of the former. - [ ] Any Noetherian normal scheme: any local ring of dimension 1 is an integrally closed domain and hence regular. - [x] What is a **reduced scheme**? #syllabus ✅ 2022-11-24 - [ ] $X$ is reduced iff $\OO_X(U)$ is a reduced ring for every $U$, or equivalently every local ring $\OO_{X, x}$ is reduced, where a ring $R$ is reduced iff no nonzero nilpotents. - [x] What is a **quasicompact morphism**? A quasicompact scheme? ⏫ ✅ 2022-12-08 - [ ] $f:Y\to X$ is quasicompact iff $\exists \mcv\covers Y$ by open affine subschemes such that each $f\inv(V_i)$ is quasicompact as a space (i.e. every open cover admits a finite subcover). - [x] What is a **locally Noetherian scheme**? #syllabus ✅ 2022-12-04 - [ ] A scheme is **locally** Noetherian iff every $x\in X$ admits an open neighborhood $U_x\ni x$ such that $U_x = \spec R_i$ with $R_i$ a Noetherian ring (ACC) - [x] What is a **Noetherian** scheme? #syllabus ✅ 2022-12-08 - [ ] A scheme is Noetherian iff locally Noetherian and quasicompact. - [x] What is a **separated scheme**? ✅ 2022-12-08 - [ ] $X$ is separated iff its structure morphism $X\to \spec \ZZ$ is a separated morphism, i.e. the diagonal $\Delta_{X/\spec \ZZ}: X\to X\fiberprod{\ZZ} X$ is a **closed immersion**. - [x] What is a **quasiseparated** scheme? ✅ 2022-12-08 - [ ] $X$ is quasiseparated iff $X\to \spec \ZZ$ is a quasiseparated morphisms, where $f:X\to Y$ is quasiseparated iff $\Delta_{X/Y}: X\to X\fiberprod{Y} X$ is a quasicompact morphism. - [x] What is a **locally factorial** scheme? (Equivalently, **factorial**) ✅ 2022-12-08 - [ ] All local rings are UFDs. Implies that every Weil divisor is Cartier. - [x] What is a **formal scheme**? ✅ 2022-11-25 - [ ] A locally ringed space locally isomorphic to $(\Spf A, \OO_{\Spf A})$ where $\Spf A$ is the formal spectrum of a ring -- roughly the completion of a Noetherian scheme $X_i$ along a closed subscheme $Y_i$, where one takes $Y$ as the space and $\varprojlim_n \OO_X/\mci_Y^n$. - [x] What is a **smooth scheme** of relative dimension $n$? ✅ 2022-12-08 - [ ] $f:X\to Y$ flat, $\dim(X')= \dim Y' + n$ whenever $X', Y_i$ are irreducible with $f(X')\subseteq Y'$, and $\dim_{\kappa(x)} (\Omega_{X/Y}\tensor \kappa(x)) = n$ for all closed and generic points. - [ ] For $X$ integral, the last condition can be replaced with $\Omega_{X/Y}$ is locally free of rank $n$. - [x] What are some examples of smooth schemes? ✅ 2022-12-08 - [ ] $\AA^n/S, \PP^n/S$ are smooth of relative dimension $n$. - [ ] Over a field $k=\kbar$, smooth iff regular of dimension $n$. - [ ] For an irreducible separated scheme over a field $k =\kbar$, smooth iff nonsingular as a variety. - [x] What is an example of a non-smooth scheme? ✅ 2022-12-08 - [ ] $V(x^2)\subseteq \AA^1\slice k$, - [ ] $\spec E$ over $\spec k$ where $E$ is a nonseparable extension e.g. $\FF_p(t^{1\over p})$ over $\FF_p(t)$, - [ ] the cuspidal curve $V(y^2-x^3)$. - [x] What is a **Gorenstein** scheme? ✅ 2022-12-04 - [ ] Locally Noetherian whose local rings are Gorenstein ($R$ such that $R$ has finite injective dimension in $\rmod$). - [x] What are some consequences of being Gorenstein? ✅ 2022-12-04 - [ ] The canonical line bundle $K_X$ exists. - [x] What is a **local complete intersection**? ✅ 2022-12-08 - [ ] For a closed subscheme $Y$ of $\PP^n/k$: the ideal $I$ of $Y$ in $k[x_0,\cdots, x_n]$ can be generated by $r\da\codim_{\PP^n}(Y)$ elements. Equivalent to $Y = \Intersect_{i=1}^r H_i$ for some hypersurfaces $H_i$ and thus $\mci_Y = \sum_{i=1}^r \mci_{H_i}$. - [ ] For an arbitrary closed subscheme $Y\leq X$ of a smooth variety: $Y$ is lci in $X$ iff $\mci_Y$ can be locally generated by $r\da \codim_{X}(Y)$ elements at every point $y\in Y$. - [x] What is a **Cohen-Macaulay** scheme? ✅ 2022-11-25 - [ ] All local rings are Cohen-Macaulay: $A$ where $\mathrm{depth} A = \dim A$, where the depth is the maximal length of a regular sequence whose generates are in $\mfm_A$, and $\ts{x_1,\cdots, x_n}$ is a regular sequence for $M\in \mods{A}$ is $x_1$ is not a zero divisor in $M$ and $x_{i+1}$ is not a zero divisor in $M/I_iM$ where $I_i \da \gens{x_1,\cdots, x_i}$. - [x] What is a **scheme-theoretic intersection**? ✅ 2022-11-24 - [ ] For relative schemes $X_i\to Y$, the intersection of $X_1$ and $X_2$ is the fiber product $X_1 { \underset{\scriptscriptstyle {Y} }{\times} } X_2$. For affines $X_1 = R/J_1$ and $X_2 = R/J_2$, this is realized by $$\operatorname{Spec}(R/J_1 \otimes_R R/J_2) \cong \operatorname{Spec}R/(J_1 + J_2).$$If $X_i \hookrightarrow Y$ are closed subschemes, then the intersection is again a closed subscheme of $Y$.. - [x] What is an **open subscheme**? ✅ 2022-12-08 - [ ] A scheme $U$ with $\abs{U} \subseteq \abs{X}$ is an open subspace where $\OO_{U} \cong \ro{\OO_X}U$. - [x] What is the **residue field** at a point $x\in X$? ✅ 2022-11-24 - [ ] $\kappa(x) \da A_\mfp / \mfp A_\mfp$ where $U = \spec A \ni x$ and $\mfp\in \spec A$ corresponds to $x$. - [x] What is the scheme-theoretic definition of an **affine variety**? ✅ 2022-11-25 - [ ] Integral noetherian separated schemes of finite type over a field - [x] What is the **scheme associated to a variety**? ✅ 2022-12-04 - [ ] Use the functor $t: \Var\slice k\to \Sch\slice k$, then $V$ is homeomorphic to the closed points in $\abs{t(V)}$ and $\OO_V \cong \ro{ \OO_{t(V)}}{t(V)^\cl}$, i.e. the regular functions on $V$ are restrictions to closed points of the structure sheaf of $t(V)$. - [ ] Generally $t(V)$ is the set of nonempty irreducible closed subsets of $V$, define the images of such sets to be closed sets to form a topology. - [x] What is the scheme-theoretic definition of a (smooth) **curve**? ✅ 2022-12-08 - [ ] An integral separated scheme of dimension 1 of finite type over a field $k$. - [ ] Smooth iff all local rings are regular. - [x] What is the **Proj construction**? #syllabus ✅ 2022-12-05 - [ ] For $S$ a graded ring,write $S_+ \da \bigoplus_{d\geq 1} S_d$ and let $\Proj S$ be the set of homogeneous prime ideals of $S$ which do not completely contains $S_+$. Define $V(\mfp) \da\ts{P\in \Proj S \st P \contains \mfp}$. - [ ] For the structure sheaf, for each $p\in \Proj S$ define $S_p$ to be the elements of degree zero in the localization $S[T\inv]$ where $T$ is the system of all homogeneous elements of $S$ which are not in $p$, and define $\OO(U)$ to be functions $s: U\to \Disjoint_{p\in U} S_p$ with $s(p) \in S_p$ and $s$ is locally a fraction: $s(q) = a/f$ with $a,f\in S$ homogeneous of the same degree and $f\not\in q$. - [x] What is $\AA^n\slice R$ as a scheme? ✅ 2022-11-25 - [ ] $\spec R[x_1,\cdots, x_n]$ or $\AA^n\slice \ZZ \fiberprod{\ZZ} \spec R$. - [x] How are morphisms $X\to \AA^n\slice S$ characterized? ✅ 2022-11-25 - [ ] A choice of $n$ sections $h_1,\cdots,h_n \in H^0(X; \OO_X)$. - [x] How is $\PP^n/S$ defined for $S$ a scheme? ✅ 2022-11-25 - [ ] Define $\PP^n\slice \ZZ \da \Proj \ZZ[x_0,\cdots, x_n]$ and $\PP^n\slice S \da \PP^n\slice \ZZ \fiberprod{\ZZ} S$. - [x] What is the **reduced scheme** associated to a scheme? ✅ 2022-11-25 - [ ] For $(X, \OO_X)\in \Sch$, sheafify $U\mapsto \OO_X(U)^\red$ where $R^\red \da R/\nilrad{R}$ is the quotient by the ideal of nilpotent elements to get $\OO_{X}^\red$. - [x] What is an **irreducible** scheme? #syllabus ✅ 2022-12-06 - [ ] Irreducible as a topological space, i.e. does not admit a cover by two proper nonempty closed subsets. - [x] What is the **reduced induced scheme structure** on $Z\subseteq X$ a closed subset? ✅ 2022-12-08 - [ ] For any $T\subseteq X$ closed, construct a unique closed $Z\subseteq X$ such that $\abs{Z} = \abs{T}$ and $Z$ s reduced. For affine $X\subseteq \spec A$, take $\mci \da\Intersect_{\mfm\in X}\mfm$ and the subscheme it defines; for arbitrary schemes do this locally and glue. - [x] What is the **scheme-theoretic image**? #syllabus ✅ 2022-11-25 - [ ] For $f:X\to Y$, the smallest closed subscheme of $Y$ through which $f$ factors. - [ ] Can take $\mci \da \ker\qty{\OO_Y\to f_* \OO_X}$ if this is qcoh, or if $X$ is reduced take $\bar{f(X)}$. - [ ] E.g. if $f: A\to B$ inducing $\spec B\to \spec A$, take $\spec (A/I)\leq \spec A$ as a closed subscheme. - [x] What is an $n{\hbox{-}}$fold point over $k$? - [ ] A scheme $\operatorname{Spec}R\to \operatorname{Spec}k$ with one point such that $R$ is a $k{\hbox{-}}$algebra of dimension $n$. - [x] What is the **Riemann-Roch theorem**? ✅ 2022-11-24 - [ ] For $C$ a curve, $$\chi(C; \mcl) = \deg(\mcl) + (1-g)$$ - [x] How is the **adjunction formula** derived? ✅ 2022-12-06 - [ ] Start with $f: C\embeds S$ a curve in a surface over $\spec k$ to get ![](attachments/Pasted%20image%2020221206180456.png) $$\begin{align*}\T_C \injects f^* \T_S \surjects N_{C/S} \quad &\underset{\text{dualize}}\leadsto\quad N\dual_{C/S} \da I_C/I_C^2 \injects f^* \Omega^1_{S/k} \surjects \Omega^{1}_{C/k} \\ &\implies \det f^* \Omega^1_{S/k} \cong \det \Omega^1_{C/k} \tensor \det N\dual_{C/S} \\&\implies f^* \omega_S \cong \omega_C \tensor \det N\dual_{C/S} \\ &\implies\omega_C = f^*\omega_S \tensor \det N_{C/S} \end{align*}$$ Now use that $I_C = \OO_Y(C)\inv$ and $I_C/I_C^2 = \OO_Y(C)\inv \tensor \OO_C$ to get $$N_{C/S} = (I_C/I_C^2)\dual = (\OO_S(C)\inv\tensor \OO_C)\dual = \OO_S(C) \tensor \OO_C$$ and so $$\omega_C = f^* \omega_S \tensor \OO_S(C) \tensor \OO_C \implies \omega_C = \qty{\omega_S \tensor \OO_S(C) }\mid_C \implies K_C = (K_S + C)\mid_S$$ - [x] What is the **adjunction formula**? ⏫ ✅ 2022-12-06 #syllabus - [ ] $$\omega_C = \qty{\omega_S \tensor \OO_S(C) }\mid_C \implies K_C = (K_S + C)\mid_S$$ $$f^* \omega_Y = \omega_C \tensor \OO_Y(C) \tensor\OO_C\implies K_C = (K_Y + C)\mid_C$$ - [x] For a smooth curves $C$ on a surface $S$, use the adjunction formula to express the genus in terms of intersection numbers. ✅ 2022-12-06 - [ ] $$2g-2 = C.(C+K_C)$$ - [ ] Use $\omega_C \cong \omega_S \tensor \OO_S(C) \tensor \OO_C$, take degrees, and use that $\deg(\omega_C) = 2g-2$ and $\deg_C(\OO_S(D)\tensor \OO_C) = \size(C\intersect D)$ to get $$\deg(\omega_S \tensor \OO_S(C)\tensor \OO_C) = \deg(\OO_S(C+K_C)\tensor \OO_C) = \size(C \intersect (C+K_C)).$$ - [x] For $C\injects S$ a curve in a surface, what is $\omega_C$? ✅ 2022-12-08 - [ ] Idea: differential forms with (no?) poles along $C$, and base change to $\OO_C$: $$\omega_S = (\omega_S \tensor \OO_S(C))\tensor \OO_C.$$ - [x] What is the degree formula for a curve of degree $d$ in $\PP^2$? ✅ 2022-12-08 - [ ] By adjunction, $$2g-2 = C.(C+K_C) = d(d-3)\implies g ={1\over 2}(d-1)(d-2).$$ - [x] What is **Serre vanishing**? #syllabus ✅ 2022-12-08 - [ ] For $X$ a Noetherian scheme, $X$ is affine $\iff H^{i>0}(X; \mcf) = 0$ for all $\mcf\in \QCoh(X) \iff H^1(X; \mci) = 0$ for all coherent sheaves of ideals on $X$. - [x] What is a **dominant morphism**? ✅ 2022-12-08 - [ ] For schemes: dense image.