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Depth and detection for Noetherian unstable algebras. (English) Zbl 07254285
The author uses the theory of unstable algebras over the mod \(p\) Steenrod algebra to generalize known algebro-geometric properties of group cohomology. Recall the starting point: the mod \(p\) cohomology \(H^*_G\) of a compact Lie group \(G\) is a finitely-generated graded commutative ring with Krull dimension equal to the \(p\)-rank of \(G\), the rank of a maximal elementary abelian \(p\)-subgroup.
For finite \(G\), J. Duflot [Comment. Math. Helv. 56, 627–637 (1981; Zbl 0493.55003)] showed that the depth of \(H^* _G\) is at least the rank of the maximal central elementary abelian \(p\)-subgroup of \(G\). To generalize this, replacing \(H^*_G\) by a Noetherian unstable algebra \(R\), the author uses an algebraic avatar of the centre introduced by W. G. Dwyer and C. W. Wilkerson [Topology 31, No. 2, 433–443 (1992; Zbl 0756.55012)] that is defined using Lannes’ \(T\)-functor technology. A central morphism of \(R\) is a pair \((E,f)\) where \(E\) is an elementary abelian \(p\)-group and \(f\) is a finite morphism of unstable algebras \(R \rightarrow H^*_E\), such that \(R\) is canonically isomorphic to the associated component \(T_E(R;f)\) of \(T_E R\). (In subsequent work, the author has shown that one may consider the centre of \(R\), which is unique up to isomorphism.)
Using this algebraic notion of centrality, the author proves the following elegant algebraic extension of Duflot’s result: the depth of \(R\) is bounded below by the supremum of the rank of \(E\), for central morphisms \((E,f)\) as above.
He then establishes an analogue of Carlson’s detection theorem for the cohomology of finite groups. If \(R\) is a Noetherian unstable algebra of depth at least \(s\), he shows that the set of finite morphisms \(g : R \rightarrow H^*_E\) with \(E\) elementary abelian of rank \(s\) induces an embedding \[R\hookrightarrow\prod T_E (R;g). \] (The result holds more generally for finitely-generated unstable modules over \(R\).) The proof uses results from the theory of unstable modules over \(R\) that relate a version of transcendence degree defined using Lannes’ \(T\)-functor to depth.
These results are illustrated by applications to examples of significant interest not covered by the earlier results. For instance, the author considers the cohomology of \(p\)-local compact groups as well as unstable algebras from modular invariant theory.
MSC:
55S10 Steenrod algebra
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
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