Periodic Reporting for period 2 - MoCoS (Motivic Cohomology of Schemes)
Berichtszeitraum: 2023-03-01 bis 2024-08-31
Algebraic K-theory is a universal linear invariant appearing in pure mathematics. It is a powerful tool within arithmetic algebraic geometry, a branch of pure mathematics at the interface of number theory and geometry, but in practice is difficult to analyse. In the 1980s Beilinson and Lichtenbaum predicted that it should be possible to break up algebraic K-theory into simpler pieces which they named "motivic cohomology"; it was predicted that these pieces would be of a more geometric nature and thus be more accessible. In the special case of the K-theory of smooth algebraic varieties, their vision was realised by Bloch, Friedlander, Levine, Suslin, Voevodsky and others over a period of almost 30 years. Among the spectacular applications of the theory for smooth algebraic varieties, the most important is undoubtedly the resolution of the famous Bloch-Kato conjecture.
The overall goal of the project is to develop a theory of motivic cohomology for arbitrarily singular schemes, thereby extending the existing theory to much greater generality. This requires replacing the traditional techniques for smooth algebraic varieties (which often exploit homotopy invariance) by modern techniques such as derived de Rham cohomology, topological cyclic homology, and perfectoid methods. The resulting theory of motivic cohomology should have various applications within the fields of arithmetic algebraic geometry and K-theory, such as a new approach to intersection theory on singular varieties, special value conjectures about zeta functions of singular varieties, and new cohomological vanishing theorems.
The overall goal of the project is to develop a theory of motivic cohomology for arbitrarily singular schemes, thereby extending the existing theory to much greater generality. This requires replacing the traditional techniques for smooth algebraic varieties (which often exploit homotopy invariance) by modern techniques such as derived de Rham cohomology, topological cyclic homology, and perfectoid methods. The resulting theory of motivic cohomology should have various applications within the fields of arithmetic algebraic geometry and K-theory, such as a new approach to intersection theory on singular varieties, special value conjectures about zeta functions of singular varieties, and new cohomological vanishing theorems.
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.
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.
All results obtained in the first reporting period are beyond the state of the art. In the remainder of the project the following problems will be studied:
The new non-A1-invariant motivic cohomology of equicharacteristic schemes will be related to algebraic cycles on singular algebraic varieties. In particular, cycle classes will be constructed for cycles of arbitrary codimension, thereby yielding a novel approach to intersection theory on singular varieties. In the presence of a fixed divisor at infinity, comparisons to the theory of Chow groups with modulus will also be investigated. It will also be studied from the point of view of non-A1-invariant motivic homotopy theory, and a realisation of it will be sought as a slice of algebraic K-theory, or as a motivic delooping.
Work on the extension of the new motivic cohomology to mixed characteristic schemes will continue. The case of regular schemes in mixed characteristic will receive special attention, where the project will aim in particular to improve the Beilinson--Lichtenbaum comparison which was established in the first reporting period.
The new non-A1-invariant motivic cohomology of equicharacteristic schemes will be related to algebraic cycles on singular algebraic varieties. In particular, cycle classes will be constructed for cycles of arbitrary codimension, thereby yielding a novel approach to intersection theory on singular varieties. In the presence of a fixed divisor at infinity, comparisons to the theory of Chow groups with modulus will also be investigated. It will also be studied from the point of view of non-A1-invariant motivic homotopy theory, and a realisation of it will be sought as a slice of algebraic K-theory, or as a motivic delooping.
Work on the extension of the new motivic cohomology to mixed characteristic schemes will continue. The case of regular schemes in mixed characteristic will receive special attention, where the project will aim in particular to improve the Beilinson--Lichtenbaum comparison which was established in the first reporting period.