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 an 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 this kind of theory.

Many thanks!!!

  • 4
    $\begingroup$ For any sheaf $A$ of rings whatsoever (commutative or not), the category of $A$-modules has enough injectives. $\endgroup$
    – Zhen Lin
    May 27, 2013 at 12:19
  • $\begingroup$ thank you. where can I find a proof for this? $\endgroup$
    – vdm123
    May 27, 2013 at 13:29
  • $\begingroup$ You can deduce it from the case where $A = \mathbb{Z}$. See the last paragraph here. $\endgroup$
    – Zhen Lin
    May 27, 2013 at 14:25
  • $\begingroup$ This has been answered at MO mathoverflow.net/questions/132107 $\endgroup$ Jun 17, 2013 at 15:23


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