Isomorphisms between endomorphism rings of progenerators.(English)Zbl 0554.16009

Wedderburn proved that if $$\Phi:\text{End}_{D_ 1}(V_ 1)\to\text{End}_{D_ 2}(V_ 2)$$ is a ring isomorphism, where each $$V_ i$$ is a finite dimensional vector space over a division ring $$D_ i$$, then there exists an isomorphism $$\alpha: D_ 1\to D_ 2$$ and an $$\alpha$$-semilinear isomorphism $$g: V_ 1\to V_ 2$$ such that $$\Phi (f)=gfg^{- 1}$$ for all $$f\in\text{End}_{D_ 1}(V_ 1)$$. This result has been generalized by Wolfson, Morita, and Jategaonkar.
The author considers the same problem from a categorical point of view and proves this Theorem: Let $$\Lambda$$ and $$\Delta$$ be rings with progenerators $$P$$ and $$Q$$ in the module categories $${\mathcal M}_{\Lambda}$$ and $${\mathcal M}_{\Delta}$$ respectively. If $$\Phi:\Lambda'\to\Delta'$$ is a ring isomorphism (where $$\Lambda'=\text{End}_{\Lambda}P)$$ and $$\Delta'=\text{End}_{\Delta}(Q))$$, then there exists a category equivalence $$F_{\Phi}: {\mathcal M}_{\Lambda}\to {\mathcal M}_{\Delta}$$ unique up to natural isomorphism such that $$F_{\Phi}(P)=Q$$ and $$F_{\Phi}(f)=\Phi(f)$$ for all $$f\in\Lambda'$$. Since free modules are progenerators, this theorem includes isomorphisms between matrix rings. From Morita theory there is an exact sequence relating the automorphisms of $$\text{End}_{\Lambda}(P)$$, where $$P$$ is a $$\Lambda$$-progenerator to the autoequivalences of the category $${\mathcal M}_{\Lambda}$$. Using this exact sequence and the preceding theorem, the author shows how semilinear maps are related to category equivalences and generalizes the results of Wedderburn, Wolfson, Morita, and Jategaonkar with a single method of proof. In cases where the semilinear description fails he shows that it is often possible to extend the isomorphism to a larger setting where it is induced by a semilinear.
Reviewer: T.W.Hungerford

MSC:

 16S50 Endomorphism rings; matrix rings 16D90 Module categories in associative algebras 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16W20 Automorphisms and endomorphisms 18G99 Homological algebra in category theory, derived categories and functors
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References:

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