Project members, sometimes in collaboration with external experts, obtained the following results during the first reporting period:
A theory of motivic cohomology for quasi-compact quasi-separated equicharacteristic schemes was introduced and shown to satisfy many of the desired properties, including an Atiyah--Hirzebruch spectral sequence converging to algebraic K-theory, a theory of Chern classes such that the projective bundle formula holds, and a Nesterenko--Suslin comparison to Milnor K-theory. The motivic cohomology satisfies pro cdh descent and has a vanishing bound which refines both Weibel's conjecture and a theorem of Soulé in algebraic K-theory. The theory agrees with the classical motivic cohomology of Bloch and Voevodsky in the case of smooth algebraic varieties, and captures the Levine--Weibel Chow group of zero cycles on singular surfaces. This work is available at arXiv:2309.08463.
A new approach to A1-invariant motivic cohomology was developed, which has the advantage of being more compatible with non-A1-invariant techniques such as trace methods. A key new input was syntomic cohomology. This new approach led to the resolution of several outstanding conjectures of Voevodsky about slice filtrations (in the sense of the motivic homotopy theory which he introduced with Morel). In particular, the zero slice of the motivic sphere was shown to be stable under pullback, thereby establishing a 6-functor formalism for a candidate derived category of motives over arbitrary quasi-compact quasi-separated schemes.
A mixed characteristic, relative version of Cartier smoothness was introduced and related to the concept of F-smoothness. It was shown that valuation rings over perfectoid valuation rings are Cartier smooth, and their prismatic and syntomic cohomologies were calculated. This work is available at arXiv:2211.16371.
Non-A1-invariant motivic homotopy theory was developed further, and a Conner--Floyd isomorphism was proved for the algebraic K-theory of quasi-compact quasi-separated schemes. This work is available at arXiv:2303.02051.
The algebraic K-theory, topological cyclic homology, and syntomic cohomology of Cartier smooth Fp-algebras were calculated and shown to look the same as that of smooth algebras over fields of characteristic p. This work is available at arXiv:2306.01063.
A new approach to Thomason's filtration on K(1)-localised algebraic K-theory was developed, by identifying it with K(1)-localised topological cyclic homology and analysing the latter using perfectoid and prismatic methods. This work is available at arXiv:2305.07576.
