Abstract
The \(\mod 2\) universal Steenrod algebra \(Q\) is a non-locally finite homogeneous quadratic algebra closely related to the ordinary \(\mod 2\) Steenrod algebra and the Lambda algebra. The algebra \(Q\) provides an example of a Koszul algebra which is a direct limit of a family of certain non-Koszul algebras \(R_k\)’s. In this paper we see how far the several \(R_k\)’s are to be Koszul by chasing in their cohomology non-trivial cocycles of minimal homological degree.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger, R.: Koszulity for nonquadratic algebras. J. Algebra 239(2), 705–734 (2001)
Brunetti, M., Ciampella, A.: A Priddy-type koszulness criterion for non-locally finite algebras. Colloq. Math. 109(2), 179–192 (2007)
Brunetti, M., Ciampella, A., Lomonaco, L.A.: An embedding for the \(E_2\)-term of the Adams spectral sequence at odd primes. Acta Math. Sin. (Engl. Ser.) 22(6), 1657–1666 (2006)
Brunetti, M., Ciampella, A., Lomonaco, L.A.: Homology and cohomology operations in terms of differential operators. Bull. Lond. Math. Soc. 42(1), 53–63 (2010)
Brunetti, M., Ciampella, A., Lomonaco, L.A.: The cohomology of the universal Steenrod algebra. Manuscripta Math. 118(3), 271–282 (2005)
Brunetti, M., Lomonaco, L.A.: An embedding for the \(E_2\)-term of the Adams spectral sequence. Ricerche Mat. 54(1), 185200 (2005)
Bruner, R.R., May, J.P., McClure, J.E., Steinberger, M.: \(H_\infty \) Ring Spectra and Their Applications. Lecture Notes in Mathematics, vol. 1176. Springer, Berlin (1986)
Cassidy, T., Shelton, B.: Generalizing the notion of Koszul algebra. Math. Z. 260(1), 93–114 (2008)
Ciampella, A., Lomonaco, L.A.: Homological computations in the universal Steenrod algebra. Fund. Math. 183(3), 245–252 (2004)
Ciampella, A., Lomonaco, L.A.: The universal Steenrod algebra at odd primes. Comm. Algebra 32(7), 2589–2607 (2004)
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 Mathematics, vol. 1183, pp. 291–338. Springer, Berlin (1986)
Lomonaco, L.A.: A basis of admissible monomials for the universal Steenrod algebra. Ricerche Mat. 40, 137–147 (1991)
Lomonaco, L.A.: A phenomenon of reciprocity in the universal Steenrod algebra. Trans. Amer. Math. Soc. 330(2), 813–821 (1992)
Lomonaco, L.A.: The diagonal cohomology of the universal Steenrod algebra. J. Pure Appl. Algebra 121(3), 315–323 (1997)
May, J.P.: The cohomology of augmented algebras and generalized Massey products for DGA-algebras. Trans. Amer. Math. Soc. 122, 334–340 (1966)
May, J.P.: A General Algebraic Approach to Steenrod Operations. The Steenrod Algebra and Its Applications. Lecture Notes in Mathematics, vol. 168, pp. 153–231. Springer, Berlin (1970)
Polishchuk, A., Positselski, L.: Quadratic Algebras. University Lecture Series, vol. 37. American Mathematical Society, Providence (2005)
Priddy, S.B.: Koszul resolutions. Trans. Amer. Math. Soc. 152, 39–60 (1970)
Steenrod, N.E., Epstein, D.B.A.: Cohomology Operations. Annals of Mathematics Studies, vol. 50. Princeton University Press, Princeton (1962)
Acknowledgments
This research has been carried out as part of “Programma STAR”, financially supported by UniNA and Compagnia di San Paolo.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Salvatore Rionero.
Rights and permissions
About this article
Cite this article
Brunetti, M., Lomonaco, L.A. Chasing non-diagonal cycles in a certain system of algebras of operations. Ricerche mat. 63 (Suppl 1), 57–68 (2014). https://doi.org/10.1007/s11587-014-0190-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-014-0190-z