A Cohomology Theory for Commutative Algebras. I1
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چکیده
1. Introduction. D. K. Harrison has recently developed a co-homology theory for commutative algebras over a field [2]. A few key theorems are proved and the results applied to the theory of local rings and eventually to algebraic geometry. The main problem is that both his definitions and proofs require involved calculations. In this paper we define a cohomology theory which (a) relies on more or less straightforward techniques of homological algebra and (b) defines a cohomology theory for an algebra over any commutative ring K, whose H2 group is the group of all singular extensions, whether X-split or not. Of course, a suitable specialization of the theory gives relative case (see §4, below). The definitions are based on an idea of
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