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catenary+local+Noetherian+S_2 => equidimensional

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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

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