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We provide a factorization model for the continuous internal Hom, in thehomotopy category of $k$-linear dg-categories, between dg-categories ofequivariant factorizations. This motivates a notion, similar to that ofKuznetsov, which we call the extended Hochschild cohomology algebra of thecategory of equivariant factorizations. In some cases of geometric interest,extended Hochschild cohomology contains Hochschild cohomology as a subalgebraand Hochschild homology as a homogeneous component. We use our factorizationmodel for the internal Hom to calculate the extended Hochschild cohomology forequivariant factorizations on affine space.  Combining the computation of extended Hochschild cohomology with theHochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov recovers andextends Griffiths' classical description of the primitive cohomology of asmooth, complex projective hypersurface in terms of homogeneous pieces of theJacobian algebra. In the process, the primitive cohomology is identified withthe fixed subspace of the cohomological endomorphism associated to aninteresting endofunctor of the bounded derived category of coherent sheaves onthe hypersurface. We also demonstrate how to understand the whole Jacobianalgebra as morphisms between kernels of endofunctors of the derived category.  Finally, we present a bootstrap method for producing algebraic cycles incategories of equivariant factorizations. As proof of concept, we show how thisreproves the Hodge conjecture for all self-products of a particular K3 surfaceclosely related to the Fermat cubic fourfold.

A category of kernels for equivariant factorizations and its implications for Hodge theory