Topological Hochschild homology of algebras in characteristic \(p\) (Q1963989)

Topological Hochschild homology (\(THH\)) is notoriously harder to calculate than its algebraic cousin Hochschild homology (\(HH\)). However, there are spectral sequences relating them, and occasionally these give simple answers. In this paper, the authors focus on commutative \(\mathbb{F}_p\)-algebras \(A\), and say that \(THH_*(A)\) splits weakly if there is an isomorphism of graded \(A\)-modules \[ THH_*(A)\cong HH_*(A)\otimes THH_*(\mathbb{F}_p) \] This almost amounts to a calculation, since \(THH_*(\mathbb{F}_p)\) is known to be the ring of polynomials on an element of degree \(2\). The authors present a sufficient condition on \(A\) for \(THH_*(A)\) to split weakly. This hinges on the two results: ``If \(THH\) splits weakly for the henselizations of every local ring \(A_{\mathfrak p}\) of \(A\) then \(THH\) splits weakly for \(A\)'' and ``If \(\Pi\) is a finitely generated monoid with unit and \(K/\mathbb{F}_p\) is a finitely generated field over \(\mathbb{F}_p\), then \(THH(K[\Pi])\) splits weakly''. Unfortunately the methods used in the paper does not shed light on the question of whether the weak splitting is canonical, and the answer to this question would be very welcome. The paper contains many example.