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Can someone point me the name of this theorem or where can I read about it? It's about Morita's equivalences:

If $F$ is a functor which is an equivalence of categories between ${}_A{\rm Mod}$ and ${}_B{\rm Mod}$ and $M$ is an $A$-module then there exists a $(B,A)$-bimodule ${}_BP_{A}$, an $B-A-Mod$, which is projective finitely generated such that $F(M) \simeq {}_BP_{A} \otimes M $

Pedro
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Werner Germán Busch
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1 Answers1

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This is a particular case of the Eilenberg-Watts theorems. I have written a post about them here. They say that

If $F:\mathcal A\longrightarrow\mathcal B$ is a functor between (say left) module categories, then the following are equivalent:

$(1)$ $F$ preserves colimits

$(2)$ $F$ is a left adjoint

$(3)$ $F\simeq -\otimes M$ for some bimodule $M$

$(4)$ $F$ preserves cokernels and coproducts

Similarly

If $G:\mathcal A\longrightarrow B$ is a functor between module categories, then the following are equivalent

$(1)$ $F$ preserves limits

$(2)$ $F$ is a right adjoint

$(3)$ $F\simeq {\rm Hom}(M,-)$ for some bimodule $M$

$(4)$ $F$ preserves kernels and products

There's yet a third version for the contravariant hom, which you can probably guess by now.

ADD To prove that if $F$ is an equivalence then $P$ is projective finitely generated for $A$ (it is also the case for $B$) you need to note that $P$ is the image of $A$ under $F$, so it is in fact finitely presented since being finitely presented is invariant under category equivalences (this follows from the fact that being finitely presented can be stated purely in functorial terms involving the hom: $M$ is finitely presented if and only if ${\rm Hom}(M,-)$ preserves filtered colimits), and since $\otimes P$ is exact (for it is an equivalence) it follows that $P$ is finitely presented flat, whence it must be projective.

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Pedro
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  • @WernerGermánBusch Did you take a course on homological algebra last semester? – Pedro Sep 05 '15 at 03:00
  • Yes but I didn't understand a thing. I was re reading my notes and for the most part they are indecipherable – Werner Germán Busch Sep 05 '15 at 03:03
  • So this is not related to Morita at all? i had it in that section, and the proof of the isomorphism was not simple at all. – Werner Germán Busch Sep 05 '15 at 04:09
  • @Werner: this is one possible approach to Morita theory. I happen to think that it's cleaner than an approach avoiding the Eilenberg-Watts theorem. Incidentally, "right exact" and "left exact" above are redundant, and implied by preserving colimits and limits respectively. – Qiaochu Yuan Sep 05 '15 at 06:08
  • @QiaochuYuan I was meaning to state the theorems in a different form and ended up with an amalgamated (and redundant) version. Thanks. – Pedro Sep 05 '15 at 06:34