Chasing non-diagonal cycles in a certain system of algebras of operations
Let \(p\) be a prime and \(\mathcal{Q}_{p}\) be the mod \(p\) universal Steenrod algebra (see [\textit{J. P. May}, Lect. Notes Math. 168, 153--231 (1970; Zbl 0242.55023)] for \(p=2\), [\textit{A. Ciampella} and \textit{L. A. Lomonaco}, Commun. Algebra 32, No. 7, 2589--2607 (2004; Zbl 1061.55014)] for \(p>2\)). It has been proved that \(\mathcal{Q}_{p}\) is a Koszul algebra (see [\textit{M. Brunetti} and \textit{A. Ciampella}, Colloq. Math. 109, No. 2, 179--192 (2007; Zbl 1151.16027)], [Manuscr. Math. 118, No. 3, 271--282 (2005; Zbl 1092.55013)]). The method used, which is adapted for non-locally finite algebras (like \(\mathcal{Q}_{2}\)), is to prove that: \(\mathcal{Q}_{2}= \lim_{\rightarrow} \{R_{k}, \; \Phi_{k} \}\) where \(\{R_{k}, \; \Phi_{k} \}\) is a system of locally finite quadratic algebras. The aim of this paper is to prove that the algebras \(R_{k}\)'s are not Koszul for \(k \geq 3\); this shows that \(\mathcal{Q}_{2}\) is an example of a Koszul algebra which is a direct limit of non-Koszul algebras.
scientific article; zbMATH DE number 6378233
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