# vanishing theorems

I would be glad to know about possible generalizations of the following results:

1) (Grothendieck) Let $X$ be a noetherian topological space of dimension $n$. Then for all $i>n$ and all sheaves of abelian groups $\cal{F}$ on $X$, we have $H^i(X; \cal{F})=$ 0. [See Hartshorne, Algebraic Geometry, III.2.7.]

2) Let $X$ be an $n$-dimensional $C^0$-manifold. Then for all $i>n$ and all sheaves of abelian groups $\cal{F}$ on $X$, we have $H^i(X; \cal{F})=$ 0 . [See Kashiwara-Schapira, Sheaves on manifolds, III.3.2.2]

More precisely, I'm interested in dropping the "abelian groups" hypothesis: could I take sheaves in any, say, AB5 abelian category?

Apparently, in Grothendieck's theorem, the "abelian groups" hypothesis is necessary -at least in Hartshorne's proof-, because at the end you see a big constant sheaf $\mathbf{Z}$. But what happens if we talk about sheaves of $R$-modules, with $R$ any commutative ring with unit, for instance?

Are those generalizations trivial ones? False for trivial reasons?

Any hints or references will be welcome.

• For any sheaf of rings $O$, sheaf cohomology on the category of $O$-modules coincides with such cohomology on underlying abelian sheaves (due to acyclicity of flasques). So the generalizations are obvious. In (2) it isn't necessary to restrict to manifolds; any separable metric space (or disjoint union thereof) with dimension $n$ in the sense of topological dimension theory satisfies (2) (see Engelking's book "General Topology", especially the notion of "covering dimension"; recall that Cech = derived functor cohomology on paracompact Hausdorff spaces, and metric spaces are such spaces). – Boyarsky Jun 29 '10 at 10:44
• @Boyarsky. Thanks. So the generalization to sheaves of O-modules is trivial. Do you know anything about possible generalizations to sheaves with values in an (AB5?) abelian category? – Agustí Roig Jun 29 '10 at 16:14
• @Agusti: goodness, I can't even remember which one AB5 is...but is there some real reason for asking that kind of question? Like an example to motivate it? – Boyarsky Jun 29 '10 at 16:20
• @Boyarsky. Thanks for your help again, Boyarsky. Well, I have a nasty spectral sequence with this kind of guys which I would like to converge strongly. For this, I need some zeros in it. As for AB5, is just a conjecture: exactness of filtered colimits seems to me, at first sight, the less you should ask -or the less I need- to work with sheaves with values in an abelian category – Agustí Roig Jun 29 '10 at 17:28

Well, I think I can answer my question, thanks to Boyarsky's remark.

The point is that, since the theorem is also true for sheaves of $R$-modules, given a sheaf $\cal{F}$ with values in an abelian category $\cal{A}$, with the help of Mitchell's embedding theorem, http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem, we can consider it as a sheaf of $R$-modules, for some ring $R$. Moreover, the embedding $V: {\cal A} \longrightarrow \mathbf{Mod}_R$ is full, faithful, and exact. That is to say, $V$ sends exact sequences to exact sequences. So $H^n(X;\cal{F})$ = $H^n(X;V(\cal{F}))$.

Hence, both vanishing theorems are also (trivially) true for sheaves with values in any abelian category.