Reduction of derived Hochschild functors over commutative algebras and schemes

Details Reduction of derived Hochschild functors over commutative algebras and schemes Reduction of derived Hochschild functors over commutative algebras and schemes

2009-04-25

arxiv.org/abs/0904.4004v2

Mathematics

Commutative Algebra

32 pages. Minor changes from previous version. To appear in the Advances in Mathematics

Scientific paper

We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms Ext^*_{S\otimes^L_{K}S}(S|K;M\otimes^L_{K}N) ~ Ext^*_S(RHom_S(M,D),N) for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analogous isomorphisms for any essentially finite type flat maps f: X->Y of noetherian schemes, with f^!(O_Y) in place of D.

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