The Hochschild homology and cohomology of A(1)

TL;DR

This paper calculates the Hochschild homology and cohomology of a specific subalgebra of the Steenrod algebra, using spectral sequences, to deepen understanding of algebraic structures in algebraic topology.

Contribution

It provides the first detailed computation of Hochschild homology and cohomology for the algebra A(1), employing May-type spectral sequences.

Findings

Explicit Hochschild homology groups of A(1)

Explicit Hochschild cohomology groups of A(1)

Methodology using spectral sequences for algebraic computations

Abstract

We compute the Hochschild homology and cohomology of $A(1)$, the subalgebra of the $2$-primary Steenrod algebra generated by the first two Steenrod squares, $Sq^1$ and $Sq^2$. The computation is accomplished using several May-type spectral sequences.