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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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