# stacks/stacks-project

Hartshorne-Lichtenbaum vanishing

aisejohan committed Oct 29, 2017
1 parent e062c42 commit 9d8d796075d30cfd9584b949ee3c7f5544a9515e
Showing with 255 additions and 2 deletions.
1. +245 −2 local-cohomology.tex
2. +10 −0 my.bib
 @@ -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}
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},