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Tag 0DAI

102.42. More cohomology

Exercise 102.42.1. 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 102.42.2. 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 102.42.3. 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 102.42.4. 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 102.42.5. 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 102.42.6. 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 102.39.7) is equal to $H^1(X, \mathcal{O}_X^*)$.

Exercise 102.42.7. Give an example of an affine scheme $X$ with nontrivial $\mathop{\mathrm{Pic}}\nolimits(X)$. Conclude using Exercise 102.42.6 that $H^1(X, -)$ is not the zero functor for any such $X$.

Exercise 102.42.8. 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 102.42.2.

Exercise 102.42.9. 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 102.42.10. 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.

The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 4031–4165 (see updates for more information).

\section{More cohomology}
\label{section-more-cohomology}

\begin{exercise}
\label{exercise-cohomology-coordinate-axes}
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 {\v C}ech cohomology.
\end{exercise}

\begin{exercise}
\label{exercise-compute-cohomology-punctured}
Let $A$ be a ring. Let $I = (f_1, \ldots, f_t)$
be a finitely generated ideal of $A$.
Let $U \subset \Spec(A)$ 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 $H^n(U, \widetilde{M})$.
\end{exercise}

\begin{exercise}
\label{exercise-compute-cohomology-affine-space-punctured}
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 $n$.
\end{exercise}

\begin{exercise}
\label{exercise-find-curve-genus-one-hundred}
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
$\dim_k H^1(X, \mathcal{O}_X) = 100$.
\end{exercise}

\begin{exercise}
\label{exercise-degree-2-cover}
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 injective\footnote{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.}.
\end{exercise}

\begin{exercise}
\label{exercise-pic}
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 \ref{definition-picard-group}) is
equal to $H^1(X, \mathcal{O}_X^*)$.
\end{exercise}

\begin{exercise}
\label{exercise-pic-nontrivial}
Give an example of an affine scheme $X$ with nontrivial $\Pic(X)$.
Conclude using Exercise \ref{exercise-pic} that $H^1(X, -)$ is not the
zero functor for any such $X$.
\end{exercise}

\begin{exercise}
\label{exercise-kill-cohomology-complement}
Let $A$ be a ring. Let $I = (f_1, \ldots, f_t)$ be a finitely generated ideal
of $A$. Let $U \subset \Spec(A)$ be the complement of $V(I)$.
Given a quasi-coherent $\mathcal{O}_{\Spec(A)}$-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 \ref{exercise-compute-cohomology-punctured}.
\end{exercise}

\begin{exercise}
\label{exercise-h0-complement}
Let $A$ be a ring. Let $I = (f_1, \ldots, f_t)$ be a
finitely generated ideal of $A$. Let $U \subset \Spec(A)$
be the complement of $V(I)$. Let $M$ be an $A$-module
whose $I$-torsion is zero, i.e.,
$0 = \Ker((f_1, \ldots, f_t) : M \to M^{\oplus t})$.
Show that there is a canonical isomorphism
$$
H^0(U, \widetilde{M}) = \colim \Hom_A(I^n, M).
$$
Warning: this is not trivial.
\end{exercise}

\begin{exercise}
\label{exercise-Noetherian}
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
$$
\colim \Hom_A(I^n, M) = \colim \Hom_A(I^n, M').
$$
Warning: this is not trivial. Hints: (1) try to reduce to $M$ finite,
(2) show any element of $\Ext^1_A(I^n, N)$
maps to zero in $\Ext^1_A(I^{n + m}, N)$ for some $m > 0$
if $N = M[I^\infty]$ and $M$ finite, (3) show the same thing
as in (2) for $\Hom_A(I^n, N)$, (3) consider the long exact
sequence
$$
0 \to \Hom_A(I^n, M[I^\infty]) \to \Hom_A(I^n, M) \to \Hom_A(I^n, M')
\to \Ext^1_A(I^n, M[I^\infty])
$$
for $M$ finite and compare with the sequence for $I^{n + m}$ to conclude.
\end{exercise}

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