vanishing theorems - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T19:45:49Z http://mathoverflow.net/feeds/question/29883 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29883/vanishing-theorems vanishing theorems Agusti Roig 2010-06-29T08:53:01Z 2010-07-06T10:39:05Z <p>I would be glad to know about possible generalizations of the following results:</p> <p>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.]</p> <p>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]</p> <p>More precisely, I'm interested in dropping the "abelian groups" hypothesis: could I take sheaves in any, say, AB5 abelian category?</p> <p>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?</p> <p>Are those generalizations trivial ones? False for trivial reasons?</p> <p>Any hints or references will be welcome.</p> http://mathoverflow.net/questions/29883/vanishing-theorems/30756#30756 Answer by Agusti Roig for vanishing theorems Agusti Roig 2010-07-06T10:39:05Z 2010-07-06T10:39:05Z <p>Well, I think I can answer my question, thanks to Boyarsky's remark.</p> <p>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, <a href="http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem</a>, 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 <em>exact</em>. That is to say, $V$ sends exact sequences to exact sequences. So $H^n(X;\cal{F})$ = $H^n(X;V(\cal{F}))$.</p> <p>Hence, both vanishing theorems are also (trivially) true for sheaves with values in any abelian category.</p>