06/02/2024, 10:00 — 11:00 — Room P3.10, Mathematics Building
Clover May, Norwegian University of Science and Technology
Classifying modules of equivariant Eilenberg-MacLane spectra
Cohomology with $\mathbb{Z}/p$-coefficients is represented by a stable object, an Eilenberg-MacLane spectrum $H\mathbb{Z}/p$. Classically, since $\mathbb{Z}/p$ is a field, any module over $H\mathbb{Z}/p$ splits as a wedge of suspensions of $H\mathbb{Z}/p$ itself. Equivariantly, cohomology and the module theory of $G$-equivariant Eilenberg-MacLane spectra are much more complicated.
For the cyclic group $G=C_p$ and the constant Mackey functor $\underline{\mathbb{Z}}/p$, there are infinitely many indecomposable $H\underline{\mathbb{Z}}/p$-modules. Previous work together with Dugger and Hazel classified all indecomposable $H\underline{\mathbb{Z}}/2$-modules for the group $G=C_2$. The isomorphism classes of indecomposables fit into just three families. By contrast, we show for $G=C_p$ with $p$ an odd prime, the classification of indecomposable $H\underline{\mathbb{Z}}/p$-modules is wild. This is joint work in progress with Grevstad.
