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Let $X$ be a topological space and $A$ a sheaf of noncommutative associative algebras over a fixed field $k$. My questions are:

1) Does the category of modules over A have enough injective?

2) If we have the affirmative answer of question (1), then suppose $M$ is a quasi-coherent sheaf over $A$ then the first cohomology of M: $H^1(X,M)$ is $0$?

I learn the cohomology of module over a scheme $X$ in Hartshorne book. But when we deform the structure sheaf $O_X$ of $X$, then does the cohomology theory still hold for quasi-coherent sheaves? Would you please tell me some books/papers about cohomology theory of module over a sheaf of (noncommutative) rings?

Many thanks!!!

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Crossposted from MSE: math.stackexchange.com/questions/403710 –  Zhen Lin May 28 '13 at 15:06
What do you mean by «does the cohomology theory still hold for quasi-coherent sheaves?»? –  Mariano Suárez-Alvarez May 28 '13 at 17:39
I wonder that does the results in sections 3&4 in chapter III in Hartshorne's book Algebraic Geometry still hold? –  vdm123 May 29 '13 at 9:11

1 Answer 1

up vote 5 down vote accepted

For the first question, the answer is yes, since the category of sheaves of $A$-modules is a Grothendieck Abelian Category. The least obvious condition one has to check is that this category has a generator - to construct it, just use

$\bigoplus_{U\subseteq X} (j_U)_!(A|_U)$,

where $U$ ranges over all open sets of $X$, $A$ is considered as sheaf of (left or right) $A$-modules, $j_U:U\rightarrow X$ is the natural inclusion, and $(j_U)_!$ is extension by zero.

For a nice introduction to Grothendieck Abelian Categories, check out section 14 of http://stacks.math.columbia.edu/download/injectives.pdf.

For the second question, if we take $X$ to be an elliptic curve, take $A = O_X$, and $M=A$, then we already have $H^1(X,M) \ne 0$.

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