stacks/stacks-project

catenary+local+Noetherian+S_2 => equidimensional

aisejohan committed Jul 24, 2019
1 parent 16a99fa commit c53944e3e6d45d95ebdfc1c1face139f740f4fbd
Showing with 76 additions and 37 deletions.
1. +65 −1 local-cohomology.tex
2. +10 −35 pione.tex
3. +1 −1 tags/tags
 @@ -292,6 +292,70 @@ \section{Generalities} \section{Hartshorne's connectedness lemma} \label{section-hartshorne-connectedness} \noindent The title of this section refers to the following result. \begin{lemma} \label{lemma-depth-2-connected-punctured-spectrum} \begin{reference} \cite[Proposition 2.1]{Hartshorne-connectedness} \end{reference} \begin{slogan} Hartshorne's connectedness \end{slogan} Let $A$ be a Noetherian local ring of depth $\geq 2$. Then the punctured spectra of $A$, $A^h$, and $A^{sh}$ are connected. \end{lemma} \begin{proof} Let $U$ be the punctured spectrum of $A$. If $U$ is disconnected then we see that $\Gamma(U, \mathcal{O}_U)$ has a nontrivial idempotent. But $A$, being local, does not have a nontrivial idempotent. Hence $A \to \Gamma(U, \mathcal{O}_U)$ is not an isomorphism. By Lemma \ref{lemma-local-cohomology} we conclude that either $H^0_\mathfrak m(A)$ or $H^1_\mathfrak m(A)$ is nonzero. Thus $\text{depth}(A) \leq 1$ by Dualizing Complexes, Lemma \ref{dualizing-lemma-depth}. To see the result for $A^h$ and $A^{sh}$ use More on Algebra, Lemma \ref{more-algebra-lemma-henselization-depth}. \end{proof} \begin{lemma} \label{lemma-catenary-S2-equidimensional} \begin{reference} \cite[Corollary 5.10.9]{EGA} \end{reference} Let $A$ be a Noetherian local ring which is catenary and $(S_2)$. Then $\Spec(A)$ is equidimensional. \end{lemma} \begin{proof} Set $X = \Spec(A)$. Say $d = \dim(A) = \dim(X)$. Inside $X$ consider the union $X_1$ of the irreducible components of dimension $d$ and the union $X_2$ of the irreducible components of dimension $< d$. Of course $X = X_1 \cup X_2$. If $X_2 = \emptyset$, then the lemma holds. If not, then $Z = X_1 \cap X_2$ is a nonempty closed subset of $X$ because it contains at least the closed point of $X$. Hence we can choose a generic point $z \in Z$ of an irreducible component of $Z$. Recall that the spectrum of $\mathcal{O}_{Z, z}$ is the set of points of $X$ specializing to $z$. Since $z$ is both contained in an irreducible component of dimension $d$ and in an irreducible component of dimension $< d$ we obtain nontrivial specializations $x_1 \leadsto z$ and $x_2 \leadsto z$ such that the closures of $x_1$ and $x_2$ have different dimensions. Since $X$ is catenary, this can only happen if at least one of the specializations $x_1 \leadsto z$ and $x_2 \leadsto z$ is not immediate! Thus $\dim(\mathcal{O}_{Z, z}) \geq 2$. Therefore $\text{depth}(\mathcal{O}_{Z, z}) \geq 2$ because $A$ is $(S_2)$. However, the punctured spectrum $U$ of $\mathcal{O}_{Z, z}$ is disconnected because the closed subsets $U \cap X_1$ and $U \cap X_2$ are disjoint (by our choice of $z$) and cover $U$. This is a contradiction with Lemma \ref{lemma-depth-2-connected-punctured-spectrum} and the proof is complete. \end{proof} @@ -405,7 +469,7 @@ \section{Cohomological dimension} construction of local cohomology in Dualizing Complexes, Section \ref{dualizing-section-local-cohomology} combined with More on Algebra, Lemma \ref{more-algebra-lemma-local-cohomology-closed}. More on Algebra, Lemma \ref{more-algebra-lemma-local-cohomology-closed} or it follows from Lemma \ref{lemma-local-cohomology-is-local-cohomology}. To see (3) we use Lemma \ref{lemma-cd} and the vanishing result of Cohomology of Schemes, Lemma
 @@ -1865,35 +1865,10 @@ \section{Local connectedness} In view of Lemma \ref{lemma-same-etale-extensions} it is interesting to know when the punctured spectrum of a ring (and of its strict henselization) is connected. The following famous lemma due to Hartshorne gives a sufficient condition. \begin{lemma} \label{lemma-depth-2-connected-punctured-spectrum} \begin{reference} \cite[Proposition 2.1]{Hartshorne-connectedness} \end{reference} \begin{slogan} Hartshorne's connectedness \end{slogan} Let $A$ be a Noetherian local ring of depth $\geq 2$. Then the punctured spectra of $A$, $A^h$, and $A^{sh}$ are connected. \end{lemma} \begin{proof} Let $U$ be the punctured spectrum of $A$. If $U$ is disconnected then we see that $\Gamma(U, \mathcal{O}_U)$ has a nontrivial idempotent. But $A$, being local, does not have a nontrivial idempotent. Hence $A \to \Gamma(U, \mathcal{O}_U)$ is not an isomorphism. By Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local} we conclude that either $H^0_\mathfrak m(A)$ or $H^1_\mathfrak m(A)$ is nonzero. Thus $\text{depth}(A) \leq 1$ by Dualizing Complexes, Lemma \ref{dualizing-lemma-depth}. To see the result for $A^h$ and $A^{sh}$ use More on Algebra, Lemma \ref{more-algebra-lemma-henselization-depth}. \end{proof} is connected. There is a famous lemma due to Hartshorne which gives a sufficient condition, see Local Cohomology, Lemma \ref{local-cohomology-lemma-depth-2-connected-punctured-spectrum}. \begin{lemma} \label{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian} @@ -5729,8 +5704,8 @@ \section{Purity in local case, III} (Algebra, Lemma \ref{algebra-lemma-apply-grothendieck}). Thus the functor is fully faithful by Lemma \ref{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian} combined with Lemma \ref{lemma-depth-2-connected-punctured-spectrum}. combined with Local Cohomology, Lemma \ref{local-cohomology-lemma-depth-2-connected-punctured-spectrum}. \medskip\noindent Let $W \to V$ be a finite \'etale morphism. Let $B \to C$ be the unique finite @@ -5794,8 +5769,8 @@ \section{Purity in local case, III} \begin{proof} The functor is fully faithful by Lemma \ref{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian} combined with Lemma \ref{lemma-depth-2-connected-punctured-spectrum} combined with Local Cohomology, Lemma \ref{local-cohomology-lemma-depth-2-connected-punctured-spectrum} (plus an application of Algebra, Lemma \ref{algebra-lemma-apply-grothendieck} to check the depth condition). @@ -5875,8 +5850,8 @@ \section{Purity in local case, III} (Algebra, Lemma \ref{algebra-lemma-apply-grothendieck}). Thus the functor is fully faithful by Lemma \ref{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian} combined with Lemma \ref{lemma-depth-2-connected-punctured-spectrum}. combined with Local Cohomology, Lemma \ref{local-cohomology-lemma-depth-2-connected-punctured-spectrum}. \medskip\noindent Denote $A^\wedge$ and $B^\wedge$ the completions of $A$ and $B$
 @@ -14600,7 +14600,7 @@ 0BLN,pione-lemma-faithful-general 0BLP,pione-lemma-fully-faithful-general 0BLQ,pione-lemma-same-etale-extensions 0BLR,pione-lemma-depth-2-connected-punctured-spectrum 0BLR,local-cohomology-lemma-depth-2-connected-punctured-spectrum 0BLS,pione-lemma-lift-purity-general 0BLT,local-cohomology-lemma-finiteness-for-finite-locally-free 0BLU,pione-section-pi1-punctured-spec-II