# Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if $\check{H}^q(U_s,\mathcal{F})=0$,

$q>0$ where $U_s$ is any finite intersection of elements of $\mathcal{U}$, then $$\check{H}^p(\mathcal{U}, \mathcal{F})\cong H^p(X,\mathcal{F})$$ where $H^*(X,\mathcal{F})$ denotes the derived functor cohomology.

Note in particular the hypothesis, under the seeingly weaker hypothesis, $H^q(U_s,\mathcal{F})=0$, I have a proof sketched below.

My questions is essentially how to prove Leray's this under the stronger hypothesis given ?

Concerning the literature, its weak on this point, many books either do not prove the theorem, or only state the theorem with the weaker hypothesis, or give references to original works where the notation leaves one in doubt as what the hypothesis are and whose proofs lack detail and clarity.

Now follows a sketch of a proof assuming the derived functor cohomology is zero, $H^q(U_s,\mathcal{F})=0$. Let $0\rightarrow \mathcal{F}\rightarrow \mathcal{I}^0\rightarrow \ldots$ be an injective resolution of $\mathcal{F}$ and $\check{C}^p(\mathcal{U},\mathcal{I}^q)$ the Godement resolution of $\mathcal{I}^q$. So we have a double complex, (which I dont know how to latex here) the columns (except the first) are exact since $\mathcal{I}^q$ is flabby and the homology of the rows gives $\check{C}^p(\mathcal{U},\mathcal{H}^q(\mathcal{F}))$ and since this latter is $\prod_s H^q(U_s, \mathcal{F})=0$ we see that the rows (except the first) are exact and one can now chase to define an isomorphism between the homologies of the first row and first column. Alternatively one can view this as taking $E_2^{p,q}$ of the double complex spectral sequence which must degenerate and so the edge homomorphism gives the isomorphism.

So how to carry this out only assuming $\check{H}^q(U_s,\mathcal{F})=0$ ?

$\bf{Edit :}$ In the meantime I have a solution. I decided not to delete the question, or post my own solution, instead I communicate here my proof and remark that if anyone wishes to post answers which clarify any point related I will upvote and eventually accept an answer. The relevant literature is Godement 5.9.2 where he attributes the stronger theorem to Cartan. The idea is to show by induction on $q$ that $\check{H}^q(U_s,\mathcal{F})\rightarrow H^q(U_s,\mathcal{F})$ is an isomorphism for all $s$. For if we have this for all $q<n$ the we apply the previous version with $U_s$ instead of $X$ and taking now those intersections of $\mathcal{U}$ contained in $U_s$. The induction hypothesis gives not that all rows are exact but only that the first $n$ are exact (or zero in $E_2$). This suffices to show the isomorphism at the next stage. Lastly remark that the previous double complex under the limit of $\mathcal{U}$ gives a double complex for Cech to derived cohomology.

• Have you looked at Lemma 01E7 in the Stacks project? – Zhen Lin Jul 14 '14 at 16:50
• @ZhenLin Thanks, I have never been on that site, but its a nice proof, using the long exact sequence. – Rene Schipperus Jul 14 '14 at 17:24
• I think this is also covered in Tamme's book on Etale cohomology. – Martin Brandenburg Jul 16 '14 at 8:01