Nilpotence and duality in the complete cohomology of a module

Suppose that G is a finite group and k is a field of characteristic p > 0 . We consider the complete cohomology ring E M ∗ = ∑ n ∈ Z Ext ^ kG n ( M , M ) . We show that the ring has two distinguished ideals I ∗ ⊆ J ∗ ⊆ E M ∗ such that I ∗ is bounded above in degrees, E M ∗ / J ∗ is bounded below...

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Bibliographic Details
Published in:Beiträge zur Algebra und Geometrie 2022-09, Vol.63 (3), p.647-660
Main Author: Carlson, Jon F.
Format: Article
Language:English
Subjects:
Algebra
Algebraic Geometry
Convex and Discrete Geometry
Geometry
Homology
Mathematics
Mathematics and Statistics
Original Paper
Rings (mathematics)
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Summary:Suppose that G is a finite group and k is a field of characteristic p > 0 . We consider the complete cohomology ring E M ∗ = ∑ n ∈ Z Ext ^ kG n ( M , M ) . We show that the ring has two distinguished ideals I ∗ ⊆ J ∗ ⊆ E M ∗ such that I ∗ is bounded above in degrees, E M ∗ / J ∗ is bounded below in degree and J ∗ / I ∗ is eventually periodic with terms of bounded dimension. We prove that if M is neither projective nor periodic, then the subring of all elements in negative degrees in E M ∗ is a nilpotent algebra.
ISSN:0138-4821
2191-0383
DOI:10.1007/s13366-021-00595-y