The extension algebra of some cohomological Mackey functors

Let $k$ be a field of characteristic $p$. We construct a new inflation functor for cohomological Mackey functors for finite groups over $k$. Using this inflation functor, we give an explicit presentation of the graded algebra of self extensions of the simple functor $S_\un^G$, when $p$ is odd and $G$ is an elementary abelian $p$-group.

💡 Research Summary

The paper tackles two intertwined problems in the theory of Mackey functors: (1) how to define an inflation (or “inflation‑up”) operation that behaves well for cohomological Mackey functors, and (2) how to give an explicit presentation of the self‑extension algebra of the simplest such functor. Working over a field $k$ of characteristic $p$, the authors first restrict attention to the category $\mathsf{Mack}_k^{\mathrm{coh}}$ of cohomological Mackey functors. In this setting each subgroup $H\le G$ is assigned the cohomology ring $H^*(H,k)$, and the usual Mackey axioms (restriction, induction, conjugation) are compatible with the cup product.

The classical inflation functor, defined via bisets, does not preserve exactness when one passes from ordinary Mackey functors to the cohomological subcategory. To overcome this, the authors introduce a “normalized biset” construction. For a normal subgroup $N\trianglelefteq G$ they consider a $G$–$G/N$ biset $U$ equipped with a $G$‑equivariant map to the trivial $G/N$‑set. By carefully arranging the left and right actions, they obtain a $k$‑linear functor
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