arXiv subjects
Publications and source records attributed to Federico Ernesto Mocchetti.
We use the motivic Greenlees spectral sequence from arXiv:2408.00338 to compute Hochschild homology in the stable motivic homotopy category over an algebraically closed field. Our target is $MHH(M\mathbb{Z}/p)/τ^{p-1}$, where $M\mathbb{Z}/p$ is modulo $p$ motivic cohomology, $p$ a prime number different from the characteristic of the base.
We define a motivic Greenlees spectral sequence by characterising an associated $t$-structure. We then examine a motivic version of topological Hochschild homology for the motivic cohomology spectrum modulo a prime number $p$. Finally, we use the motivic Greenlees spectral sequence to determine the homotopy ring of a related spectrum, given that the base field is algebraically closed with a characteristic that is coprime to $p$.
In arXiv:1712.00555, H. Heine shows that given a symmetric monoidal $\infty$-category $\mathcal{V}$ and a weakly $\mathcal{V}$-enriched monad $T$ over an $\infty$-category $\mathcal{C}$, then there is an induced action of $\mathcal{V}$ on $LMod_T(\mathcal{C})$. Moreover, properties like tensoring or enrichment can be transferred from the action on $\mathcal{C}$ to that on $LMod_T(\mathcal{C})$. We see that the action of an internal operad $O \in Alg(sSeq(\mathcal{C}))$ can be interpreted as the action of a monad $T_O$, such that $Alg_O(\mathcal{C})\cong LMod_{T_O}(\mathcal{C})$. We can then prove that, under a presentability assumption, if the category $\mathcal{C}$ admits cotensors with respect to the action of $\mathcal{V}$, then so does $Alg_O(\mathcal{C})\cong LMod_{T_O}(\mathcal{C})$. This is used to show that the coproduct-idempotent algebras are fixed by the induced tensoring action. We apply this to the stable motivic homotopy category and prove that the tensor of any motivic sphere with rational motivic cohomology is equivalent to the latter.