Document Zbl 1092.55013 - zbMATH Open

The mod 2 universal Steenrod algebra \(Q\) was introduced by J. P. May [Steenrod Algebra Appl., Lect. Notes Math. 168, 153–231 (1970; Zbl 0242.55023)] and is isomorphic to the algebra of cohomology operations in the category of \(H_{\infty}\)-ring spectra. It has generators \(y_i\) for \(i\in {\mathbb Z}\), subject to relations similar in form to the classical Adem relations. The main result in this paper is that the cohomology of \(Q\) is diagonal, i.e., \(\text{Ext}_{Q}^{s,t}({\mathbb F}_2,{\mathbb F}_2) = 0\) if \(s \neq t\). Consequently, by a result of L. Lomonaco [J. Pure Appl. Algebra 121, 315–323 (1997; Zbl 0891.55023)], the cohomology of \(Q\) is the completion of \(Q\) with respect to a certain chain of two-sided ideals.

MSC:

55S10	Steenrod algebra
16S37	Quadratic and Koszul algebras
18G15	Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)

References:

[1]	Bousfield, A.K., Curtis, E.B., Kan, D.M., Quillen, D.G., Rector, D.L., Schlesinger, J.W.: The mod p lower central series and the Adams spectral sequence. Topology 5, 331–342 (1966) · Zbl 0158.20502 · doi:10.1016/0040-9383(66)90024-3
[2]	Braverman, A., Gaitsgory, D.: Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type. J. Algebra 181 (2), 315–328 (1996) · Zbl 0860.17002 · doi:10.1006/jabr.1996.0122
[3]	Bruner, R.R., May, J.P., McClure, J.E., Steinberger, M.: H ring spectra and their applications. Lecture Notes in Math. 1176, Springer, Berlin, 1986 · Zbl 0585.55016
[4]	Bruner, R.: An example in the cohomology of augmented algebras. J. Pure Appl. Algebra 55 (1-2), 81–84 (1988) · Zbl 0664.16021
[5]	Löfwall, C.: On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra. Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture Notes in Math. 1183, Springer, Berlin, 1986 pp. 291–338
[6]	Lomonaco, L.A.: A basis of admissible monomials for the universal Steenrod algebra. Ricerche Mat. 40 (1), 137–147 (1991) · Zbl 0747.55012
[7]	Lomonaco, L.A.: The diagonal cohomology of the universal Steenrod algebra. J. Pure Appl. Algebra 121 (3), 315–323 (1997) · Zbl 0891.55023 · doi:10.1016/S0022-4049(96)00066-7
[8]	May, J.P.: A general algebraic approach to Steenrod operations. The Steenrod algebra and its applications. Lecture Notes in Math. 168, Springer, Berlin, 1970 pp. 153–231 · Zbl 0242.55023
[9]	Priddy, S.B.: Koszul resolutions. Trans. Amer. Math. Soc. 152, 39–60 (1970) · Zbl 0261.18016 · doi:10.1090/S0002-9947-1970-0265437-8
[10]	Steenrod, N.E., Epstein, D.B.A.: Cohomology operations. Annals of Mathematics Studies, No. 50 Princeton University Press, Princeton, N.J. 1962 · Zbl 0102.38104

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.