Nil-localization of unstable algebras over the Steenrod algebra
Let p be an odd prime, \(A\) \(*_ p\) the mod p Steenrod algebra, K the category of unstable \(A\) \(*_ p\)-algebras and K' the full subcategory of K of those algebras which are concentrated in even degrees. We say that an unstable \(A\) \(*_ p\)-algebra is Nil-closed if the underlying \(A\) \(*_ p\)-module is Nil-closed in the sense of \textit{J. Lannes} and the second author [Ann. Sci. Éc. Norm. Sup. IV, Sér. 19, 303-333 (1986; Zbl 0608.18006), 6.2.1]. In the first part of the note we introduce the concept of Nil- localization of unstable \(A\) \(*_ p\)-algebras and we prove: Theorem. For any unstable \(A\) \(*_ p\)-algebra K, there exists a unique (up to isomorphism) nil-closed unstable \(A\) \(*_ p\)-algebra, denoted \(N_ K^{-1}(K)\), and a natural map of \(A\) \(*_ p\)-algebras \(\mu _ K: K\to N_ K^{-1}(K)\) which is initial among maps of unstable \(A\) \(*_ p\)- algebras from K to nil-closed unstable \(A\) \(*_ p\)-algebras. In the second part we show that some results of Adams-Wilkerson and of Rector, which occur in K' can be extended to K. This allows us to determine the Nil-localization of certain unstable \(A\) \(*_ p\)-algebras.
category of unstable algebras over the Steenrod algebra
scientific article; zbMATH DE number 4042848
scientific article