A Noetherian and Hausdorff space is finite

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.

The Mayer-Vietoris sequence in sheaf cohomology

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.