Abstract
We compute the Hochschild homology and cohomology of , the subalgebra of the 2-primary Steenrod algebra generated by the first two Steenrod squares, . The computation is accomplished using several May-type spectral sequences.
Notes
1 That is, the differentials in the spectral sequence obey the graded Leibniz rule. The ring structures on the input and abutment are given by the usual shuffle product on the Hochschild homology of a commutative ring.
2 The notation is traditional for the conjugate of x in a Hopf algebra. In , we have and . Another notational point: the symbol is used to denote the image of in .
3 Here it is essential that we are using and not , since in but in .
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 has infinite Hochschild dimension.