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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Homological algebra in topoi


Author: D. H. Van Osdol
Journal: Proc. Amer. Math. Soc. 50 (1975), 52-54
MSC: Primary 18G99; Secondary 14F20
MathSciNet review: 0369481
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Abstract: Let $ {\mathbf{A}}$ be the category of sheaves of universal algebras of some fixed type on a Grothendieck topology. There is defined a cohomology theory $ {H^\ast }$ such that if $ A$ is an object in $ {\mathbf{A}}$ and $ M$ is an $ A$-module then $ {H^1}(A,M)$ is in one-to-one correspondence with the equivalence classes of singular extensions of $ A$ by $ M$. When $ {\mathbf{A}}$ is the category of sheaves of $ R$-modules for some sheaf of rings $ R$, then $ {H^n}(A,M) \cong {\operatorname{Ext} ^n}(A,M)$ for all $ n \geq 0$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0369481-1
PII: S 0002-9939(1975)0369481-1
Keywords: Cotriple, topos, principal object, extension
Article copyright: © Copyright 1975 American Mathematical Society