Cech Cohomology

Having established the basic foundations of sheaf cohomology, Ogus preceeded to prove the following:

  1. For a topological space X and a flasque sheaf F, the higher cohomology vanishes: H^q(X, F) = 0, q> 0.
  2. For X = \mbox{Spec} A, an M and A module, I have H^q(X, \widetilde{M}) = 0, q>0.

There was some face collecting along the way, for example proving that for a ringed space (X, \mathcal{O}_X) , injective \mathcal{O}_X modules are flasque.  The proof is the same as in Hartshorne, basically use the property of injective objects to dualize an injection to give the required surjective property of flasque sheaves.  Also in a s.e.s F' \to F \to F'', if the first is flasque then the global sections functor preserves exactness (ex. II.2.16).

The following result is more general than what is found in Hartshorne, but the proof uses the same ideas (the long exact sequence of cohomology and induction)

Thm: Let \mathscr{A} by an abelian category with enough injectives.  Let F \colon \mathscr{A} \to \mathscr{B} be an additive left exact functor.  Let \mathscr{I} \subset \mbox{objects} \mathscr{A} such any object of \mathscr{A} injects into an object of \mathscr{I}, if I' \to I \to I'' is a s.e.s. with the first two in \mathscr{I} then so is the third, and F(I') \to F(I) \to F(I'') still s.e.s.  Then R^qF(I) = 0, q>0, i.e. \mathscr{I} is a class of F-acyclic objects.

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This theorem gives 1.  Also it gives 2 when A is noetherian (with noetherian hypothesis can prove if I is injective A-module, then \widetilde{I} is flasque).

After this Ogus started Cech Cohomology, his development was not remarkably different except for his discussion of \check{H}^1(\mathscr{U}, E) as classifying certain E torsors trivialized on \mathscr{U}.

But here’s a fun fact, in Cech cohomology, for the cover \mathscr{U} = (U_i)_{i \in I}, if there is some U_i = X, then (Thm) all the higher cohomology vanishes for all sheafs of abel. groups E.

proof: Fix it so that U_o = X.  Simply show the complex

C^\bullet (\mathscr{U}, E) = C^0(\mathscr{U}, E) \to C^1(\mathscr{U}, E) \to C^2(\mathscr{U}, E) \to ...

Is homotopically trivial, that is, the identity C^\bullet \to C^\bullet is homotopic to the zero map.  It suffices to produce maps R^q \colon C^q \to C^{q-1} and R^0\colon C^0 \to E(X) such that d^{q-1} \circ R^q + R^{q+1} \circ d^q = id = id - 0.  Represent e \in C^q as e = (e_I)_{I = (i_0, ... , i_q)}.  Then 

R^q(e) = f = (f_J)_{J = (j_o , ... , j_{q-1})}

f_J = e_{0, j_0, ..., j_{q_1}}

Note that U_{0, j_o, ..., j_{q-1}} = X \cap U_{j_o, ..., j_{q_1}} = U_{j_o, ..., j_{q-1}}, so this is a well defined map.  And its not hard to check (but a bit of a notational nightmare to write down, so I wont) that indeed d^{q-1} \circ R^q + R^{q+1} \circ d^q = id = id - 0.

QED

Its easy to see that \check{H}^0 gives the global sections.  For \check{H}^1 there is the statement about torsors, and in addition to the result above, here are two more results on Cech Cohomology:

 

  • The natural map E \to C^\bullet (\mathscr{U},E) is a quasi iso.  This says that \underline{C}^\bullet (\mathscr{U}, E) (i.e. sheaf version) is exact on stalks in positive degree.  The sheaf version is defined as follows: V \subset X, get C^\bullet(\mathscr{U}, E)(V) = C^bullet (\mathscr{U} \cap V, E), i.e. C^q(\mathscr{U}, E)(V) = \prod_I E(U_I\cap V).
  • There are natural maps \check{H}^q(\mathscr{U}, E) \to H^q(X, E) which are isos for all q if C^q(\mathscr{U}, E) are \Gamma -acyclic or E is flasque.

 

Thm: For a ringed space X, let \mathscr{B} be a basis which is closed under finite intersections.  Suppose all U \in \mathscr{B} are quasi compact.  Let E be a sheaf.  If for all V \subset \mathscr{B} and for all finite \mathscr{B} coverings of V , the higher Cech cohomology vanishes, then the higher sheaf cohomology vanishes.  That is, \check{H}^q(\mathscr{U}, E) = 0, q>0 then H^q(X,E) = 0, q>0.

proof: Let \mathscr{F} = \{ E | \check{H}^q(\mathscr{U}, E) = 0, q>0 \mbox{ implies } H^q(X,E) = 0, q>0 \}.  Then by the first thm above it suffices to check

  1.  Any object injects into an object in \mathscr{F}
  2. F' \to F \to F'' s.e.s, first two in \mathscr{F}, then so is third
  3. global sections exact on F' \to F \to F''

All the assumptions of the thm remain true if we restrict to V \subset X, so can reduce to the case X \in \mathscr{B}.  Injective sheaves are flasque, and flasque sheaves are in \mathscr{F}, so this takes care of 1.  For 3, in fact it suffices that F' \in \mathscr{F} because in this case the long exact sequence of sheaf cohomology says there is a surjection F(U) \to F''(U) \to 0.  In light of 3, there is an exact sequence of complexes 

C^\bullet (\mathscr{U}, F') \to C^\bullet (\mathscr{U}, F) \to C^\bullet (\mathscr{U}, F'')

and general theory says this gives rise to an exact sequence of Cech cohomology groups, then the vanishing of the higher cohomology groups for F,F' show they also vanish for F''.  This is should do it.

QED

From this theorem has the following corollary (Note, no Noetherian hypothesis!)

Thm: Let X by affine and E qcoh, then H^q(X,E) = 0, q>0.

proof: Let \mathscr{B} by the basis consisting of distinguished affines.  Then by the previous result it suffices to show the higher Cech cohomology vanishes for any finite distinguished affine cover.  For such a cover \mathscr{U} write (note the underline means sheaf version)

\underline{C}^q(\mathscr{U}, E) = \prod_I E(U_I \cap V)

= \prod_I (j_{I*}j_I^*E)(V) = \oplus_I (j_{I*}j_I^*E)(V) := \oplus_I E_I(V)

where V \in \mathscr{B} and j_I \colon U_I \to X is the natural inclusion, as this is an affine morphism it is in particular quasi separated and quasi compact so a previous result of sheaves of modules says E_I is still qcoh.  By one of the bullet points above, the sheaf version the Cech complex is a resolution of E, i.e. there is a quasi iso

E \to \underline{C}^\bullet (\mathscr{U}, E)

But this is a resolution of finite direct sums of qcoh sheaves, and finite direct sums preserve qcoh, hence applying global sections preserves exactness, but this says exactly that the higher Cech cohomology of E for this covering vanishes.

QED

Yup.

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