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A note on the realization of graded complete intersection algebras by the cohomology of a space. (English) Zbl 0546.55009
The purpose of this note is to record a simple consequence of the theory developed in N. Bourbaki [Groupes et algèbres de Lie, Chap. 5 (1968; Zbl 0186.330)], C. Wilkerson [Topology 16, 227-237 (1977; Zbl 0404.55019)], and J. F. Adams and C. W. Wilkerson [Ann. Math., II. Ser. 111, 95-143 (1980; Zbl 0404.55020)] to construct and classify polynomial algebras that occur as the cohomology of a space, and a full reading of Bourbaki [loc. cit., § 5, Th. 4].
Theorem 1. Let p be a prime and \(\Lambda =P[x_ 1,...,x_ n]/(\phi_ 1,...,\phi_ n)\) where: \(1\circ.\quad (p,\deg x_ i)=1=(p,\deg \phi_ i): i=1,...,n\). \(2\circ.\quad\phi_ 1,...,\phi_ n\in P[x_ 1,...,x_ n]\) are algebraically independent. \(3\circ.\quad {\mathbb{Z}}/p[x_ 1,...,x_ n]\) is an unstable algebra over the mod p Steenrod algebra and \({\mathbb{Z}}/p[\phi_ 1,...,\phi_ n]\) is closed under this action. Then there exists a space X such that \(H^*(X;{\mathbb{Z}}/p)\cong\Lambda \) as algebras over the Steenrod algebra.