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- W4387163920 abstract "AbstractWe compute the Hochschild homology and cohomology of A(1), the subalgebra of the 2-primary Steenrod algebra generated by the first two Steenrod squares, Sq1,Sq2. The computation is accomplished using several May-type spectral sequences.KEYWORDS: Hochschild homologyHochschild cohomologySteenrod algebradeformation theoryMay spectral sequence2020 MATHEMATICS SUBJECT CLASSIFICATION: 16E4055S10 Notes1 That is, the differentials in the spectral sequence obey the graded Leibniz rule. The ring structures on the input HH*(E0A,E0A) and abutment HH*(A,A) are given by the usual shuffle product on the Hochschild homology of a commutative ring.2 The notation x¯ is traditional for the conjugate of x in a Hopf algebra. In A(1)* , we have ξ¯1=ξ1 and ξ¯2=ξ2+ξ13 . Another notational point: the symbol ξ¯i,j is used to denote the image of ξ¯i2j∈A(1) in AugE0(A(1)).3 Here it is essential that we are using AugE0A(1) and not A(1), since (Sq˜2)2=0 in AugE0A(1) but (Sq2)2≠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." @default.
- W4387163920 created "2023-09-30" @default.
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- W4387163920 date "2023-09-29" @default.
- W4387163920 modified "2023-09-30" @default.
- W4387163920 title "The Hochschild homology and cohomology of <i>A</i> (1)" @default.
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- W4387163920 doi "https://doi.org/10.1080/00927872.2023.2261042" @default.
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