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When I say the cohomology of a monoid, I mean that of its classifying space (considering the monoid as a category with a single object).

Let $M_n(R)$ be the monoid of matrices with matrix multiplication over some (commutative) ring $R$, sometimes called the full linear monoid. Is there any literature about the cohomology of $M_n(R)$ for certain classes (say, finite fields) of $R$?

I thought there should be at least something out there, considering the attention the general linear group gets. But I can't find anything. Perhaps I am not searching with the right keywords.

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    $\begingroup$ The classifying space is contractible: there exists an element $0$ such that $0A=0=A0$, which allows you to construct a homotopy between the identity map of the classifying space and a constant map. $\endgroup$
    – Charles Rezk
    Commented Apr 12, 2018 at 23:18
  • $\begingroup$ Ah, sure. (Fulfilling the character quota with humility) $\endgroup$
    – Cihan
    Commented Apr 12, 2018 at 23:22
  • $\begingroup$ You would probably do better looking at the Hochschild-Mitchell cohomology. $\endgroup$
    – Benjamin Steinberg
    Commented Apr 13, 2018 at 2:20

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