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Hartshorne-Lichtenbaum vanishing

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@@ -1940,10 +1940,13 @@ \section{Cohomological dimension}
\begin{lemma}
\label{lemma-cd-dimension}
Let $I \subset A$ be a finitely generated ideal of a ring $A$.
Then $\text{cd}(A, I) \leq \dim(A)$.
If $M$ is a finite $A$-module, then
$H^i_{V(I)}(M) = 0$ for $i > \dim(\text{Supp}(M))$.
In particular, we have $\text{cd}(A, I) \leq \dim(A)$.
\end{lemma}
\begin{proof}
We first prove the second statement.
Recall that $\dim(A)$ denotes the Krull dimension. By
Lemma \ref{lemma-cd-local} we may assume $A$ is local.
If $V(I) = \emptyset$, then the result is true.
@@ -1958,8 +1961,21 @@ \section{Cohomological dimension}
\ref{cohomology-proposition-cohomological-dimension-spectral}
implies $H^i(U, \mathcal{F}) = 0$ for $i \geq \dim(A)$
which implies what we want by Lemma \ref{lemma-cd}.
In the Noetherian case we can use Grothendieck's Cohomology, Proposition
In the Noetherian case the reader may use
Grothendieck's Cohomology, Proposition
\ref{cohomology-proposition-vanishing-Noetherian}.
\medskip\noindent
We will deduce the first statement from the second.
Let $\mathfrak a$ be the annihilator of the finite $A$-module $M$.
Set $B = A/\mathfrak a$. Recall that $\Spec(B) = \text{Supp}(M)$, see
Algebra, Lemma \ref{algebra-lemma-support-closed}.
Set $J = IB$. Then $M$ is a $B$-module
and $H^i_{V(I)}(M) = H^i_{V(J)}(M)$, see
Dualizing Complexes, Lemma
\ref{dualizing-lemma-local-cohomology-and-restriction}.
Since $\text{cd}(B, J) \leq \dim(B) = \dim(\text{Supp}(M))$
by the first part we conclude.
\end{proof}
\begin{lemma}
@@ -2039,6 +2055,233 @@ \section{Cohomological dimension}
\section{Hartshorne-Lichtenbaum vanishing}
\label{section-Hartshorne-Lichtenbaum-vanishing}
\noindent
This and much else besides can be found in \cite{CD}.
\begin{lemma}
\label{lemma-cd-top-vanishing}
Let $A$ be a Noetherian ring of dimension $d$. Let $I \subset I' \subset A$
be ideals. If $I'$ is contained in the Jacobson radical
of $A$ and $\text{cd}(A, I') < d$, then $\text{cd}(A, I) < d$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-cd-dimension} we know $\text{cd}(A, I) \leq d$.
We will use Lemma \ref{lemma-isomorphism} to show
$$
H^d_{V(I')}(A) \to H^d_{V(I)}(A)
$$
is surjective which will finish the proof. Pick
$\mathfrak p \in V(I) \setminus V(I')$. By our assumption
on $I'$ we see that $\mathfrak p$ is not a maximal ideal of $A$.
Hence $\dim(A_\mathfrak p) < d$. Then
$H^d_{\mathfrak pA_\mathfrak p}(A_\mathfrak p) = 0$
by Lemma \ref{lemma-cd-dimension}.
\end{proof}
\begin{lemma}
\label{lemma-cd-top-vanishing-some-module}
Let $A$ be a Noetherian ring of dimension $d$. Let $I \subset A$
be an ideal. If $H^d_{V(I)}(M) = 0$ for some finite $A$-module
whose support contains all the irreducible components of
dimension $d$, then $\text{cd}(A, I) < d$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-cd-dimension} we know $\text{cd}(A, I) \leq d$.
Thus for any finite $A$-module $N$ we have $H^i_{V(I)}(N) = 0$
for $i > d$. Let us say property $\mathcal{P}$ holds for the
finite $A$-module $N$ if $H^d_{V(I)}(N) = 0$.
One of our assumptions is that $\mathcal{P}(M)$ holds.
Observe that $\mathcal{P}(N_1 \oplus N_2)
\Leftrightarrow (\mathcal{P}(N_1) \wedge \mathcal{P}(N_2))$.
Observe that if $N \to N'$ is surjective, then
$\mathcal{P}(N) \Rightarrow \mathcal{P}(N')$ as we
have the vanishing of $H^{d + 1}_{V(I)}$ (see above).
Let $\mathfrak p_1, \ldots, \mathfrak p_n$ be the
minimal primes of $A$ with $\dim(A/\mathfrak p_i) = d$.
Observe that $\mathcal{P}(N)$ holds if the support
of $N$ is disjoint from $\{\mathfrak p_1, \ldots, \mathfrak p_n\}$
for dimension reasons, see Lemma \ref{lemma-cd-dimension}.
For each $i$ set $M_i = M/\mathfrak p_i M$.
This is a finite $A$-module annihilated by $\mathfrak p_i$
whose support is equal to
$V(\mathfrak p_i)$ (here we use the assumption on the support of $M$).
Finally, if $J \subset A$ is an ideal, then we have $\mathcal{P}(JM_i)$
as $JM_i$ is a quotient of a direct sum of copies of $M$.
Thus it follows from Cohomology of Schemes, Lemma
\ref{coherent-lemma-property-higher-rank-cohomological}
that $\mathcal{P}$ holds for every finite $A$-module.
\end{proof}
\begin{lemma}
\label{lemma-top-coh-divisible}
Let $A$ be a Noetherian local ring of dimension $d$. Let $f \in A$
be an element which is not contained in any minimal prime of
dimension $d$. Then $f : H^d_{V(I)}(M) \to H^d_{V(I)}(M)$
is surjective for any finite $A$-module $M$ and any ideal $I \subset A$.
\end{lemma}
\begin{proof}
The support of $M/fM$ has dimension $< d$ by our assumption on $f$.
Thus $H^d_{V(I)}(M/fM) = 0$ by Lemma \ref{lemma-cd-dimension}.
Thus $H^d_{V(I)}(fM) \to H^d_{V(I)}(M)$ is surjective.
Since by Lemma \ref{lemma-cd-dimension} we know $\text{cd}(A, I) \leq d$
we also see that the surjection $M \to fM$, $x \mapsto fx$
induces a surjection $H^d_{V(I)}(M) \to H^d_{V(I)}(fM)$.
\end{proof}
\begin{lemma}
\label{lemma-cd-bound-dualizing}
Let $A$ be a Noetherian local ring with
normalized dualizing complex $\omega_A^\bullet$.
Let $I \subset A$ be an ideal.
If $H^0_{V(I)}(\omega_A^\bullet) = 0$, then $\text{cd}(A, I) < \dim(A)$.
\end{lemma}
\begin{proof}
Set $d = \dim(A)$. Let $\mathfrak p_1, \ldots, \mathfrak p_n \subset A$
be the minimal primes of dimension $d$.
Recall that the finite $A$-module
$H^{-i}(\omega_A^\bullet)$ is nonzero only for
$i \in \{0, \ldots, d\}$ and that the support
of $H^{-i}(\omega_A^\bullet)$ has dimension $\leq i$, see
Lemma \ref{lemma-sitting-in-degrees}.
Set $\omega_A = H^{-d}(\omega_A^\bullet)$.
By prime avoidence (Algebra, Lemma \ref{algebra-lemma-silly})
we can find $f \in A$, $f \not \in \mathfrak p_i$
which annihilates $H^{-i}(\omega_A^\bullet)$ for $i < d$.
Consider the distinguished triangle
$$
\omega_A[d] \to \omega_A^\bullet \to
\tau_{\geq -d + 1}\omega_A^\bullet \to \omega_A[d + 1]
$$
See Derived Categories, Remark
\ref{derived-remark-truncation-distinguished-triangle}.
By Derived Categories, Lemma \ref{derived-lemma-trick-vanishing-composition}
we see that $f^d$ induces the zero endomorphism of
$\tau_{\geq -d + 1}\omega_A^\bullet$.
Using the axioms of a triangulated category, we find a map
$$
\omega_A^\bullet \to \omega_A[d]
$$
whose composition with $\omega_A[d] \to \omega_A^\bullet$ is
multiplication by $f^d$ on $\omega_A[d]$.
Thus we conclude that $f^d$ annihilates $H^d_{V(I)}(\omega_A)$.
By Lemma \ref{lemma-top-coh-divisible} we conlude $H^d_{V(I)}(\omega_A) = 0$.
Then we conclude by Lemma \ref{lemma-cd-top-vanishing-some-module}
and the fact that $(\omega_A)_{\mathfrak p_i}$ is nonzero
(see for example
Dualizing Complexes, Lemma
\ref{dualizing-lemma-nonvanishing-generically-local}).
\end{proof}
\begin{lemma}
\label{lemma-inverse-system-symbolic-powers}
Let $(A, \mathfrak m)$ be a complete Noetherian local domain. Let
$\mathfrak p \subset A$ be a prime ideal of dimension $1$.
For every $n \geq 1$ there is an $m \geq n$ such that
$\mathfrak p^{(m)} \subset \mathfrak p^n$.
\end{lemma}
\begin{proof}
Recall that the symbolic power $\mathfrak p^{(m)}$ is defined as the
kernel of $A \to A_\mathfrak p/\mathfrak p^mA_\mathfrak p$.
Since localization is exact we conclude that in the short exact sequence
$$
0 \to \mathfrak a_n \to A/\mathfrak p^n \to A/\mathfrak p^{(n)} \to 0
$$
the support of $\mathfrak a_n$ is contained in $\{\mathfrak m\}$.
In particular, the inverse system $(\mathfrak a_n)$ is Mittag-Leffler
as each $\mathfrak a_n$ is an Artinian $A$-module.
We conclude that the lemma is equivalent to the requirement
that $\lim \mathfrak a_n = 0$. Let $f \in \lim \mathfrak a_n$.
Then $f$ is an element of $A = \lim A/\mathfrak p^n$
(here we use that $A$ is complete)
which maps to zero in the completion $A_\mathfrak p^\wedge$
of $A_\mathfrak p$. Since $A_\mathfrak p \to A_\mathfrak p^\wedge$
is faithfully flat, we see that $f$ maps to zero in $A_\mathfrak p$.
Since $A$ is a domain we see that $f$ is zero as desired.
\end{proof}
\begin{proposition}
\label{proposition-Hartshorne-Lichtenbaum-vanishing}
\begin{reference}
\cite[Theorem 3.1]{CD}
\end{reference}
Let $A$ be a Noetherian local ring with completion $A^\wedge$.
Let $I \subset A$ be an ideal such that
$$
\dim V(IA^\wedge + \mathfrak p) \geq 1
$$
for every minimal prime $\mathfrak p \subset A^\wedge$ of dimension $\dim(A)$.
Then $\text{cd}(A, I) < \dim(A)$.
\end{proposition}
\begin{proof}
Since $A \to A^\wedge$ is faithfully flat we have
$H^d_{V(I)}(A) \otimes_A A^\wedge = H^d_{V(IA^\wedge)}(A^\wedge)$
by Dualizing Complexes, Lemma \ref{dualizing-lemma-torsion-change-rings}.
Thus we may assume $A$ is complete.
\medskip\noindent
Assume $A$ is complete. Let $\mathfrak p_1, \ldots, \mathfrak p_n \subset A$
be the minimal primes of dimension $d$. Consider the complete local ring
$A_i = A/\mathfrak p_i$. We have $H^d_{V(I)}(A_i) = H^d_{V(IA_i)}(A_i)$
by Dualizing Complexes, Lemma
\ref{dualizing-lemma-local-cohomology-and-restriction}.
By Lemma \ref{lemma-cd-top-vanishing-some-module}
it suffices to prove the lemma for $(A_i, IA_i)$.
Thus we may assume $A$ is a complete local domain.
\medskip\noindent
Assume $A$ is a complete local domain. We can choose a prime ideal
$\mathfrak p \supset I$ with $\dim(A/\mathfrak p) = 1$.
By Lemma \ref{lemma-cd-top-vanishing}
it suffices to prove the lemma for $\mathfrak p$.
\medskip\noindent
By Lemma \ref{lemma-cd-bound-dualizing} it suffices to show that
$H^0_{V(\mathfrak p)}(\omega_A^\bullet) = 0$.
Recall that
$$
H^0_{V(\mathfrak p)}(\omega_A^\bullet) =
\colim \text{Ext}^0_A(A/\mathfrak p^n, \omega_A^\bullet)
$$
By Lemma \ref{lemma-inverse-system-symbolic-powers}
we see that the colimit is the same as
$$
\colim \text{Ext}^0_A(A/\mathfrak p^{(n)}, \omega_A^\bullet)
$$
Since $\text{depth}(A/\mathfrak p^{(n)}) = 1$ we see that
these ext groups are zero by Lemma \ref{lemma-sitting-in-degrees}
as desired.
\end{proof}
\begin{lemma}
\label{lemma-affine-complement}
Let $(A, \mathfrak m)$ be a Noetherian local ring.
Let $I \subset A$ be an ideal. Assume $A$ is excellent,
normal, and $\dim V(I) \geq 1$. Then $\text{cd}(A, I) < \dim(A)$.
In particular, if $\dim(A) = 2$, then $\Spec(A) \setminus V(I)$ is affine.
\end{lemma}
\begin{proof}
By More on Algebra, Lemma
\ref{more-algebra-lemma-completion-normal-local-ring}
the completion $A^\wedge$ is normal and hence a domain.
Thus the assumption of
Proposition \ref{proposition-Hartshorne-Lichtenbaum-vanishing}
holds and we conclude. The statement on affineness
follows from Lemma \ref{lemma-cd-is-one}.
\end{proof}
\section{Formal functions for a principal ideal}
\label{section-formal-functions-principal}
View
10 my.bib
@@ -2490,6 +2490,16 @@ @BOOK{RD
VOLUME = {20},
}
@ARTICLE{CD,
AUTHOR = {Hartshorne, Robin},
TITLE = {Cohomological dimension of algebraic varieties},
JOURNAL = {Ann. of Math. (2)},
VOLUME = {88},
YEAR = {1968},
PAGES = {403--450},
URL = {https://doi.org/10.2307/1970720},
}
@ARTICLE{Has,
AUTHOR = {Hassett, Brendan},
TITLE = {Special Cubic Fourfolds},

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