  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.
