The Hochschild homology and cohomology of A(1)

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, ${\text{Sq}}^1,{\text{Sq}}^2$. The computation is accomplished using several May-type spectral sequences.

KEYWORDS:

• Hochschild homology
• Hochschild cohomology
• Steenrod algebra
• deformation theory
• May spectral sequence

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16E40
• 55S10

Notes

1 That is, the differentials in the spectral sequence obey the graded Leibniz rule. The ring structures on the input $H{H}_{*}(E_{0}A,E_{0}A)$ and abutment $H{H}_{*}(A,A)$ are given by the usual shuffle product on the Hochschild homology of a commutative ring.

2 The notation $\overline{x}$ is traditional for the conjugate of x in a Hopf algebra. In $A{(1)}_{*}$ , we have ${\overline{\xi }}_1={\xi }_1$ and ${\overline{\xi }}_2={\xi }_2+{\xi }_1^3$ . Another notational point: the symbol ${\overline{\xi }}_{i,j}$ is used to denote the image of ${\overline{\xi }}_i^{2^j}\in A(1)$ in ${}^{\mathit{\text{Aug}}}{E}_0(A(1))$.

3 Here it is essential that we are using ${}^{\mathit{\text{Aug}}}{E}_{0}A(1)$ and not $A(1)$, since ${({\widetilde{\text{Sq}}}^2)}^2=0$ in ${}^{\mathit{\text{Aug}}}{E}_{0}A(1)$ but ${({\text{Sq}}^2)}^2\ne 0$ in $A(1)$.

4 A good reference for this classical May spectral sequence is Example 3.2.7 of [19].

5 The author does not know where this duality originally appeared in the literature, but a nice account of the duality appears in [2] and in section 3.1 of [11]. See sections 1.1 and 1.2 of [7] and section 3 of [20] for good accounts of the most fundamental properties of graded Frobenius algebras. The duality does not seem to be a special case of van den Bergh’s Poincaré duality for Hochschild (co)homology [22]: as van den Bergh remarks in [23], the duality of [22] requires the ring to be of finite Hochschild dimension, but the calculation of Hochschild homology we have just made in Theorem 4.10 demonstrates that $A(1)$ has infinite Hochschild dimension.