February | 2013 | Random Math

In this post, I will prove that a Noetherian and Hausdorff topological space is finite (and therefore has the discrete topology, being Hausdorff). The proof is very short and pleasant.

Proof: Let ${X}$ be such a space, and suppose that it is infinite. Let ${\Sigma}$ be the collection of infinite closed subsets of ${X}$ . It is nonempty since ${X \in \Sigma}$ , and therefore has a minimal member ${Z}$ by the Noetherian assumption. Let ${p,q}$ be points of ${Z}$ , and ${U,V}$ be disjoint open neighborhoods of ${p}$ and ${q}$ respectively (such ${U}$ and ${V}$ exist by the Hausdorff assumption). Then ${X = (X-U) \cup (X-V)}$ since ${U}$ and ${V}$ are disjoint, so ${Z = (Z \cap (X-U)) \cup (Z \cap (X-V))}$ . Now each of ${Z \cap (X-U)}$ and ${Z \cap (X-V)}$ is closed in ${X}$ , and is properly contained in ${Z}$ (the first one doesn’t contain ${p}$ , and the second one doesn’t contain ${q}$ ). Therefore, by minimality of ${Z}$ , each must be finite, and therefore ${Z}$ is also finite, which is a contradiction. $\Box$

Corollary: in any infinite Hausdorff space, there exists a strictly descending infinite chain of closed subsets $Z_1 \supset Z_2 \supset Z_3 \dots$ . The proof above can be easily adapted to construct such a sequence.

In this post, I will prove the Mayer-Vietoris Sequence for sheaf cohomology.

In what follows, ${X}$ is a topological space and ${\mathcal F, \mathcal G, \mathcal H}$ are sheaves of abelian groups on ${X}$ . Let ${Z}$ be a closed subset of ${X}$ . We let ${\Gamma_Z(X,\mathcal F)}$ denote the global sections of ${\mathcal F}$ with support in ${Z}$ . The functor ${\Gamma_Z(X, -)}$ is a left-exact additive functor from sheaves on ${X}$ to abelian groups, and its right derived functors, denoted ${H^i_Z(X, -)}$ , is the ${i}$ -th cohomology of ${X}$ with support in ${Z}$ . If ${\mathcal F}$ is a sheaf, the presheaf ${U \mapsto \Gamma_{Y \cap U}(U, \mathcal F)}$ is also a sheaf on ${X}$ , denoted ${\mathcal H^0_Y(\mathcal F)}$ and called the “subsheaf of ${\mathcal F}$ with support in ${Y}$ “.

The Mayer-Vietoris sequence, for a sheaf ${\mathcal F}$ and for a pair of closed subsets ${Y,Z \subseteq X}$ , is the long exact sequence of cohomology with supports

$\displaystyle \dots \rightarrow H^i_{Y \cap Z}(X, \mathcal F) \rightarrow H^i_Y(X, \mathcal F) \oplus H^i_Z(X, \mathcal F) \rightarrow H^i_{Y \cup Z}(X, \mathcal F) \rightarrow H^{i+1}_{Y \cap Z}(X, \mathcal F) \rightarrow \dots$

We will prove the existence of this sequence in several steps.

Lemma 1 Let ${\mathcal E}$ be a flasque sheaf, ${Y}$ a closed subset of ${X}$ , and ${U=X-Y}$ . Then the sequence
$\displaystyle 0 \rightarrow \Gamma_Y(X, \mathcal E) \rightarrow \Gamma(X, \mathcal E) \rightarrow \Gamma(U, \mathcal E) \rightarrow 0$

is exact.

Proof: Trivial. $\Box$

Lemma 2 Let ${\mathcal E}$ be a flasque sheaf, and let ${Y, Z}$ be closed subsets of ${X}$ . Then the sequence
$\displaystyle 0 \rightarrow \Gamma_{Y \cap Z}(X, \mathcal E) \rightarrow \Gamma_Y(X, \mathcal E) \oplus \Gamma_Z(X, \mathcal E) \rightarrow \Gamma_{Y \cup Z}(X, \mathcal E) \rightarrow 0$

is exact (where the first map is the diagonal embedding, and the second map is ${(s, t) \mapsto s-t}$ ).

Proof: Exactness is clear except possibly on the right. Let $U=X-Y, V=X-Z$ , and Let ${D, E}$ be the short exact sequences

$\displaystyle 0 \rightarrow \Gamma(X, \mathcal E) \rightarrow \Gamma(X, \mathcal E) \oplus \Gamma(X, \mathcal E) \rightarrow \Gamma(X, \mathcal E) \rightarrow 0$

and

$\displaystyle 0 \rightarrow \Gamma(U \cup V, \mathcal E) \rightarrow \Gamma(U, \mathcal E) \oplus \Gamma(V, \mathcal E) \rightarrow \Gamma(U \cap V, \mathcal E) \rightarrow 0$

where the maps are defined similarily as in the statement of the Lemma. There is an obvious morphism of short exact sequences ${D \rightarrow E}$ . Since ${\mathcal E}$ is flasque, this morphism is surjective onto each term of ${E}$ . By the snake lemma, and using Lemma 1, we get the desired short exact sequence. $\Box$

Now we are ready to prove the existence of the Mayer-Vietoris sequence for ${\mathcal F}$ . Let

$\displaystyle 0 \rightarrow \mathcal F \rightarrow \mathcal E^0 \rightarrow \mathcal E^1 \rightarrow \dots$

be a flasque resolution of ${\mathcal F}$ . By the lemma, we have a short exact sequence of complexes

$\displaystyle 0 \rightarrow \Gamma_{Y \cap Z}(X, \mathcal E^\bullet) \rightarrow \Gamma_Y(X, \mathcal E^\bullet) \oplus \Gamma_Z(X, \mathcal E^\bullet) \rightarrow \Gamma_{Y \cup Z}(X, \mathcal E^\bullet) \rightarrow 0.$

The long exact sequence of cohomology associated to this short exact sequences of complexes is precisely the Mayer-Vietoris sequence.

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A Noetherian and Hausdorff space is finite

The Mayer-Vietoris sequence in sheaf cohomology

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