Let $R$ be the ring $M_n(k)$ of matrices of order $n$ over the field $k$. Compute $Tor_n^R(M,N)$ and $Ext_R^n(P, Q)$ for any $n \geq 0$ and every $M, N,P,Q$ - $R$ modules(left or right such that $Tor$ and $Ext$ have sense. My idea is to find a projective resolution in every case but I can't...
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Do you know that the category of modules over $M_n(k)$ is equivalent to the category of $k$ vector spaces? – Hanno Jun 10 '17 at 10:43
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The matrix ring $M_n(k)$ is semisimple (simple even). – Angina Seng Jun 10 '17 at 10:45
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@Hanno I don't know that. Why is useful? – rafa Jun 10 '17 at 10:51
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@Hanno Is this true?I do not know this,where can I find it?I think this is amazing ' – Jian Jun 10 '17 at 10:58
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@Sky That is useful for the problem? – rafa Jun 10 '17 at 11:06
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The ring $R=M_n(k)$ is semisimple (Wedderburn’s theorem or direct proof), so every module in projective and injective.
More generally, if $S$ is a ring, then the functor $F=\operatorname{Hom}(P_S,-)$ is a category equivalence between $\text{Mod-}S$ and $\text{Mod-}R$, where $R=M_n(S)$ and $P=S^n$. The inverse equivalence is $G=-\otimes_RP$ (seeing $P$ as a left module over $R$, its endomorphism ring as right $S$-module).
It's easy to prove, using exactness of $F$ and $G$, that for every pair of modules $M,N\in\text{Mod-}R$, $$ \operatorname{Ext}_R^n(M,N)\cong\operatorname{Ext}_S^n(GM,GN) $$ (as abelian groups). Similarly for Tor.
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