\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

103.42 More cohomology

Exercise 103.42.1. Let $k$ be a field. Let $X \subset \mathbf{P}^ n_ k$ be the “coordinate cross”. Namely, let $X$ be defined by the homogeneous equations

\[ T_ i T_ j = 0\text{ for }i > j > 0 \]

where as usual we write $\mathbf{P}^ n_ k = \text{Proj}(k[T_0, \ldots , T_ n])$. In other words, $X$ is the closed subscheme corresponding to the quotient $k[T_0, \ldots , T_ n]/(T_ iT_ j; i > j > 0)$ of the polynomial ring. Compute $H^ i(X, \mathcal{O}_ X)$ for all $i$. Hint: use Čech cohomology.

Exercise 103.42.2. Let $A$ be a ring. Let $I = (f_1, \ldots , f_ t)$ be a finitely generated ideal of $A$. Let $U \subset \mathop{\mathrm{Spec}}(A)$ be the complement of $V(I)$. For any $A$-module $M$ write down a complex of $A$-modules (in terms of $A$, $f_1, \ldots , f_ t$, $M$) whose cohomology groups give $H^ n(U, \widetilde{M})$.

Exercise 103.42.3. Let $k$ be a field. Let $U \subset \mathbf{A}^ d_ k$ be the complement of the closed point $0$ of $\mathbf{A}^ d_ k$. Compute $H^ n(U, \mathcal{O}_ U)$ for all $n$.

Exercise 103.42.4. Let $k$ be a field. Find explicitly a scheme $X$ projective over $k$ of dimension $1$ with $H^0(X, \mathcal{O}_ X) = k$ and $\dim _ k H^1(X, \mathcal{O}_ X) = 100$.

Exercise 103.42.5. Let $f : X \to Y$ be a finite locally free morphism of degree $2$. Assume that $X$ and $Y$ are integral schemes and that $2$ is invertible in the structure sheaf of $Y$, i.e., $2 \in \Gamma (Y, \mathcal{O}_ Y)$ is invertible. Show that the $\mathcal{O}_ Y$-module map

\[ f^\sharp : \mathcal{O}_ Y \longrightarrow f_*\mathcal{O}_ X \]

has a left inverse, i.e., there is an $\mathcal{O}_ Y$-module map $\tau : f_*\mathcal{O}_ X \to \mathcal{O}_ Y$ with $\tau \circ f^\sharp = \text{id}$. Conclude that $H^ n(Y, \mathcal{O}_ Y) \to H^ n(X, \mathcal{O}_ X)$ is injective1.

Exercise 103.42.6. Let $X$ be a scheme (or a locally ringed space). The rule $U \mapsto \mathcal{O}_ X(U)^*$ defines a sheaf of groups denoted $\mathcal{O}_ X^*$. Briefly explain why the Picard group of $X$ (Definition 103.39.7) is equal to $H^1(X, \mathcal{O}_ X^*)$.

Exercise 103.42.7. Give an example of an affine scheme $X$ with nontrivial $\mathop{\mathrm{Pic}}\nolimits (X)$. Conclude using Exercise 103.42.6 that $H^1(X, -)$ is not the zero functor for any such $X$.

Exercise 103.42.8. Let $A$ be a ring. Let $I = (f_1, \ldots , f_ t)$ be a finitely generated ideal of $A$. Let $U \subset \mathop{\mathrm{Spec}}(A)$ be the complement of $V(I)$. Given a quasi-coherent $\mathcal{O}_{\mathop{\mathrm{Spec}}(A)}$-module $\mathcal{F}$ and $\xi \in H^ p(U, \mathcal{F})$ with $p > 0$, show that there exists $n > 0$ such that $f_ i^ n \xi = 0$ for $i = 1, \ldots , t$. Hint: One possible way to proceed is to use the complex you found in Exercise 103.42.2.

Exercise 103.42.9. Let $A$ be a ring. Let $I = (f_1, \ldots , f_ t)$ be a finitely generated ideal of $A$. Let $U \subset \mathop{\mathrm{Spec}}(A)$ be the complement of $V(I)$. Let $M$ be an $A$-module whose $I$-torsion is zero, i.e., $0 = \mathop{\mathrm{Ker}}((f_1, \ldots , f_ t) : M \to M^{\oplus t})$. Show that there is a canonical isomorphism

\[ H^0(U, \widetilde{M}) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M). \]

Warning: this is not trivial.

Exercise 103.42.10. Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$. Let $M$ be an $A$-module. Let $M[I^\infty ]$ be the set of $I$-power torsion elements defined by

\[ M[I^\infty ] = \{ x \in M \mid \text{ there exists an }n \geq 1\text{ such that }I^ nx = 0\} \]

Set $M' = M/M[I^\infty ]$. Then the $I$-power torsion of $M'$ is zero. Show that

\[ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M'). \]

Warning: this is not trivial. Hints: (1) try to reduce to $M$ finite, (2) show any element of $\mathop{\mathrm{Ext}}\nolimits ^1_ A(I^ n, N)$ maps to zero in $\mathop{\mathrm{Ext}}\nolimits ^1_ A(I^{n + m}, N)$ for some $m > 0$ if $N = M[I^\infty ]$ and $M$ finite, (3) show the same thing as in (2) for $\mathop{\mathrm{Hom}}\nolimits _ A(I^ n, N)$, (3) consider the long exact sequence

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M[I^\infty ]) \to \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M) \to \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M') \to \mathop{\mathrm{Ext}}\nolimits ^1_ A(I^ n, M[I^\infty ]) \]

for $M$ finite and compare with the sequence for $I^{n + m}$ to conclude.

[1] There does exist a finite locally free morphism $X \to Y$ between integral schemes of degree $2$ where the map $H^1(Y, \mathcal{O}_ Y) \to H^1(X, \mathcal{O}_ X)$ is not injective.

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