Cyclotomic Spectra and p-Divisible Groups

The PI proposes to pursue three projects related to algebraic K-theory, motivic filtrations, and p-adic Hodge theory, all connected via the theory of cyclotomic spectra and p-divisible groups. Algebraic K-theory is a subtle invariant of rings and algebraic varieties and is the subject of many of the deepest conjectures in algebraic number theory and algebraic geometry (for instance the Hodge and Tate conjectures). However, K-theory itself is extremely difficult to control, so one attempts to either break it into pieces via the motivic filtration and understand those pieces, motivic cohomology, or to compare K-theory to a more computable theory, usually by mapping from it via the trace map K(X) → TC(X). Here, TC(X) is constructed from topological Hochschild homology THH(X), a lift of classical Hochschild homology to the world of stable homotopy theory. These two approaches to understanding algebraic K-theory, the motivic filtration and the trace approach, are so far useful for radically different types of objects. The former is best developed for smooth schemes over a field or Dedekind domain, following work of Friedlander, Geisser, Grayson, Levine, Suslin, and Voevodsky. The latter is effective for smooth formal schemes in the p-adic context or even better for the quasiregular semiperfectoid rings of Bhatt, Morrow, and Scholze. However, Bhatt, Morrow, and Scholze construct a filtration on p-complete TC with graded pieces given by syntomic cohomology, suggesting a close connection to the motivic filtration for K-theory. The proposal’s three main objectives are (1) to directly compare these approaches by showing that if R is a smooth commutative Z(p)-algebra, then the map K(R) → TC(R) respects the motivic and syntomic filtrations after p-completion, (2) to construct a theory of coefficient systems for p- adic cohomology using cyclotomic spectra and to verify the PI’s liftability conjecture, and (3) to understand the filtration on prismatic cohomology arising from the cyclotomic t-structure (more on this below). Topological Hochcshild homology THH(X) is a cyclotomic spectrum, i.e., a spectrum with an action of the circle and with certain additional ‘Frobenius’ maps. From this data one constructs TC(M ) which acts as a form of fixed points for Frobenius. If X is a scheme, set TC(X) = TC(THH(X)). Cyclotomic spectra were introduced by B¨okstedt, Hsiang, and Madsen in the late 80s and early 90s. It was not until very recently that an appropriate homotopy theory of cyclotomic spectra was constructed, first by Blumberg and Mandell and then by Nikolaus and Scholze. In order to approach the objectives above, the PI proposes to use the cyclotomic t-structure, which was discovered by the PI in joint work with Thomas Nikolaus in 2018. This t-structure gives a way of filtering TC(X) and breaking it up into smaller, more understandable pieces. In characteristic p, these pieces are essentially syntomic cohomology. However, in mixed characteristic, the connection to syntomic cohomology is more mysterious. The PI conjectures that there exists a hierarchy of cohomology theories interpolating between the prismatic theory of Bhatt, Morrow, and Scholze and the relative de Rham–Witt theory and that this interpolation is realized by the Postnikov tower with respect to the cyclotomic t-structure. Moreover, the PI conjectures that these intermediate theories admit site-theoretic descriptions analogous to prismatic cohomology. The liftability conjecture will help to explain the window-frame approach to the classification of formal groups and