Perfectoid rings and THH Perfectoid rings and THH

Cyclic homology offers, as shown by Connes, a way to refine de Rham cohomology to the non-commutative setting. To unpack what this means, we will delve into the classical theory of Hochschild cohomology with an emphasis on its relation to perfect rings. The modern view point on this theory, afforded by recent developments in homotopy theory, is to consider topological variants, where the theory can be generalised to more homotopically enriched categories. It is possible, in this setting, to perform analogous computations for perfect rings in characteristic p, like in the classical case. If one is to take the analogy that perfectoid rings are characteristic 0 analogues of perfect rings in characteristic p seriously, then one would expect that similar computations can be undertaken for perfectoid rings that would yield similar results. We will attempt to survey such results, and in particular, outline how perfectoid rings can arise as certain Thom spectra.

Syntomic cohomology and ${\mathrm{\Sigma }}^{″}$ Syntomic cohomology and ${\mathrm{\Sigma }}^{″}$

Motivated by proving certain comparison theorems between crystalline and etale cohomology in p-adic hodge theory, Fontain and Messing defined their "syntomic cohomology" as a way to transfer between both worlds. As we've understood throughout this seminar, prismatic cohomology serves as a "universal" p-adic cohomology theory refining all known p-adic cohomology theories. One may ask whether there is a notion of syntomic cohomology that may compute the prismatic cohomology, in the spirit of Fontain-Messing's approach to p-adic Hodge theory. We will discuss a few interpretations of this question, and hopefully be able to compare the modern treatments with the classical treatments satisfactorily.