Reduction of derived Hochschild functors over commutative algebras and schemes

TL;DR

This paper investigates the structure of derived Hochschild cohomology for commutative algebras and schemes, providing explicit complexes and isomorphisms that simplify the understanding of these functors in algebraic geometry.

Contribution

It constructs a specific complex D that yields natural reduction isomorphisms for derived Hochschild cohomology over commutative algebras and schemes, extending duality theory.

Findings

Constructed a complex D for derived Hochschild cohomology

Established reduction isomorphisms for algebraic and scheme cases

Extended duality results to schemes with finite type flat maps

Abstract

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.