Extensions of sheaves of commutative algebras by nontrivial kernels

Donovan Van Osdol
1974 Pacific Journal of Mathematics  
~~~r2£~!2n· Let X be a topological space, R a sheaf of commutative algebras on X , and A a sheaf of R-modules considered as an algebra with trivial EUltiplication. It was shown in [5] that the group of equivalence classes of commutative algebra extensions of R with A as kernel is isomorphic to H 1 (R,A) , the first bicohomology group of R with coefficients in A • In this paper we will not assune·that A has trivial multiplication; we will find that, if ZA is the center of A , then H 2 (R,ZA)
more » ... then H 2 (R,ZA) contains all of the obstructions to the existence of extensions of R by A which "realize" a given morphism. This will generalize the results of [1] to the category of sheaves, and of [4] in that no assumptions need be made on X or R • In order to keep this paper as short as possible, we shall follow the format of [1] • We shall not, however, generalize section 4 of [1] • There are two reasons for this: first, we do not know how to globalize Barr's theory, although we can do his section 4 locally using only tripletheoretic techniques (and then the underlying set of A is ZXK where K is the kernel of R's structure morphism); secondly, the correct setting for completely characterizing the bicohomology ~. n>1, will not be known until Duskin writes up his results [3] • *
doi:10.2140/pjm.1974.55.531 fatcat:5bxkavx6ejguzm2f5y5sy646ga