A note on measurings and higher order Hochschild homology of algebras

TL;DR

The paper studies whether coalgebra measurings between commutative algebras induce morphisms on higher order Hochschild homology as defined by Pirashvili. It introduces $C$-comodule measurings $\phi:D\to \mathrm{Hom}_k(M,N)$ and constructs, for each $t\in D$, a natural transformation $\mathcal{L}^\phi(t)$ that yields maps $\phi^Y_\bullet(t):HH_\bullet^Y(A,M)\to HH_\bullet^Y(B,N)$ on Hochschild homology of order $Y$, functorial in pointed simplicial maps $g:Y\to Z$. These maps are compatible with simplicial morphisms and specialize to the familiar Hochschild maps $\psi^Y(s):HH_\bullet^Y(A)\to HH_\bullet^Y(B)$ when $M=A$, $N=B$, $D=C$, and $\phi=\psi$. The results provide a general, functorial bridge from coalgebra measurings to higher order Hochschild invariants, enriching the interaction between algebraic measurings and (co)homology theories.

Abstract

We know that coalgebra measurings behave like generalized maps between algebras. In this note, we show that coalgebra measurings between commutative algebras induce morphisms between higher order Hochschild homology groups of algebras. By higher order Hochschild homology, we mean the the Hochschild homology groups of a commutative algebra with respect to a simplicial set as introduced by Pirashvili.

Theorems & Definitions (5)

• Definition 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof