Milnor K-theory Research Articles

On the relative Gersten conjecture for Milnor K-theory in the smooth case

We show that the Gersten complex for the (improved) Milnor K-sheaf on a smooth scheme over an excellent discrete valuation ring is exact except at the first place and that exactness at the first place may be checked at the discrete valuation ring associated to the generic point of the special fibre. This complements results of Gillet-Levine for K-theory, Geisser for motivic cohomology and Schmidt-Strunk and the author for étale cohomology.

A motivic construction of the de Rham-Witt complex

The theory of reciprocity sheaves due to Kahn-Saito-Yamazaki is a powerful framework to study invariants of smooth varieties via invariants of pairs (X,D) of a variety X and a divisor D. We develop a generalization of this theory where D can be a Q-divisor. As an application, we provide a motivic construction of the de Rham-Witt complex, which is analogous to the motivic construction of the Milnor K-theory due to Suslin-Voevodsky.

MILNOR K-THEORY, F-ISOCRYSTALS AND SYNTOMIC REGULATORS

AbstractWe introduce a category of filtered F-isocrystals and construct a symbol map from Milnor K-theory to the group of 1-extensions of filtered F-isocrystals. We show that our symbol map is compatible with the syntomic symbol map to the log syntomic cohomology by Kato and Tsuji. These are fundamental materials in our computations of syntomic regulators which we work in other papers.

Endomorphisms of the projective plane and the image of the Suslin–Hurewicz map

The endomorphism ring of the projective plane over a field F of characteristic neither two nor three is slightly more complicated in the Morel–Voevodsky motivic stable homotopy category than in Voevodsky’s derived category of motives. In particular, it is not commutative precisely if there exists a square in F which does not admit a sixth root. A byproduct of these computations is a proof of Suslin’s conjecture on the Suslin–Hurewicz homomorphism from Quillen to Milnor K-theory in degree four, based on work of Asok et al. (Invent Math 219:39-73, 2020).

Reconstructing function fields from Milnor K-theory

Reconstructing function fields from Milnor K-theory

Cohomological Invariants in Positive Characteristic

Abstract We determine the mod $p$ cohomological invariants for several affine group schemes $G$ in characteristic $p$. These are invariants of $G$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $p$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $p$ Milnor K-theory of fields.

Witt and cohomological invariants of Witt classes

We classify all invariants of the functor $I^n$ (powers of the fundamental ideal of the Witt ring) with values in $A$, that it to say functions $I^n(K)\rightarrow A(K)$ compatible with field extensions, in the cases where $A(K)=W(K)$ is the Witt ring and $A(K)=H^*(K,\mu_2)$ is mod 2 Galois cohomology. This is done in terms of some invariants $f_n^d$ that behave like divided powers with respect to sums of Pfister forms, and we show that any invariant of $I^n$ can be written uniquely as a (possibly infinite) combination of those $f_n^d$. This in particular allows to lift operations defined on mod 2 Milnor K-theory (or equivalently mod 2 Galois cohomology) to the level of $I^n$. We also study various properties of these invariants, including behaviour under products, similitudes, residues for discrete valuations, and restriction from $I^n$ to $I^{n+1}$. The goal is to use this to study invariants of algebras with involutions in future articles.

Milnor-Witt cycle modules

We generalize Rost's theory of cycle modules [20] using the Milnor-Witt K-theory instead of the classical Milnor K-theory. We obtain a (quadratic) setting to study general cycle complexes and their (co)homology groups. The standard constructions are developed: proper pushfoward, (essentially) smooth pullback, long exact sequences, spectral sequences and products, as well as the homotopy invariance property; in addition, Gysin morphisms for lci morphisms are constructed. We prove an adjunction theorem linking our theory to Rost's. This work extends Schmid's thesis [22].

Motivic spheres and the image of the Suslin–Hurewicz map

We show that an old conjecture of A. A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree n to Milnor K-theory in degree n admits an interpretation in terms of unstable $${\mathbb {A}}^1$$-homotopy sheaves of the general linear group. Using this identification, we establish Suslin’s conjecture in degree 5 for infinite fields having characteristic unequal to 2 or 3. We do this by linking the relevant unstable $${\mathbb {A}}^1$$-homotopy sheaf of the general linear group to the stable $${\mathbb {A}}^1$$-homotopy of motivic spheres.

Algebraization for zero-cycles and the $p$-adic cycle class map

Using an idelic argument and assuming the Gersten conjecture for Milnor K-theory, we show that the restriction map from one-cycles on a smooth projective scheme over a henselian local ring to a pro-system of thickened zero-cycles is surjective. We relate this restriction map to the $p$-adic cycle class map.

Milnor K-theory of complete discrete valuation rings with finite residue fields

Consider a complete discrete valuation ring O with quotient field F and finite residue field. Then the inclusion map O↪F induces a map Kˆ⁎MO→Kˆ⁎MF on improved Milnor K-theory. We show that this map is an isomorphism in degrees bigger or equal to 3. This implies the Gersten conjecture for improved Milnor K-theory for O. This result is new in the p-adic case.

Principal ideals in mod-$$\ell $$ Milnor K-theory

Fix a symbol \(\underline{a}\) in the mod-\(\ell \) Milnor K-theory of a field k, and a norm variety X for \(\underline{a}\). We show that the ideal generated by \(\underline{a}\) is the kernel of the K-theory map induced by \(k\subset k(X)\) and give generators for the annihilator of the ideal. When \(\ell =2\), this was done by Orlov, Vishik and Voevodsky.

The Witt ring of a curve with good reduction over a non-dyadic local field

In this work, we present a generalization to varieties and sheaves of the fundamental ideal of the Witt ring of a field by defining a sheaf of fundamental ideals I˜ and a sheaf of Witt rings W˜ in the obvious way. The Milnor conjecture then relates the associated graded of W˜ to Milnor K-theory and so allows the classical invariants of a bilinear space over a field to be extended to our setting using étale cohomology. As an application of these results, we calculate the Witt ring of a smooth curve with good reduction over a non-dyadic local field.

Partial Fractions and Milnor K-Theory

This paper uses partial fraction decompositions to give a direct computation of the logarithmic derivative of the norm in Milnor K-theory for a finite separable extension. This result is useful for computations involving the relative Brauer group in finite characteristic and Witt kernels for function fields in characteristic two. Kato's result that the norm is compatible with the trace under logarithmic differentiation also follows from these tools. When F(x) is rational over F in finite characteristic ℓ, the unramified part of is computed to be .

Cycle modules and the intersection A ∞-algebra

In his paper Chow Group with Coefficients, M. Rost has developed a generalization of the classical Chow groups based on Milnor K-theory and an axiomatic generalization of it called cycles modules. For a cycle module M with a ring structure and a smooth scheme X of finite type over a field, we show that Rost’s cycle complex with coefficients \({C^*(X, M)_\mathbb{Q}}\) has a structure of an A∞-algebra. In the case of Milnor K-theory it provides an homotopy model for the classical intersection theory of cycles.

Katoʼs residue homomorphisms and reciprocity laws on arithmetic surfaces

We explicitly study Katoʼs residue homomorphisms in Milnor K-theory, which are closely related to Contou-Carrère symbols. As applications we establish several reciprocity laws for certain locally defined maps on K-groups that are associated to arithmetic surfaces.

Comparison of dualizing complexes

Abstract We prove that there is a map from Bloch's cycle complex to Kato's complex of Milnor K-theory, which induces a quasi-isomorphism from cycle complex mod pr to Moser's complex of logarithmic de Rham–Witt sheaves. Next we show that the truncation of Bloch's cycle complex at -3 is quasi-isomorphic to Spiess' dualizing complex. In the end, we prove that a weak form of the Gersten conjecture implies that Sato's dualizing complex is quasi-isomorphic to Bloch's complex.

Algebraic BP-theory and norm varieties

Let p be an odd prime and BP*(pt) ≅ $¥mathbb Z$(p)[v1,v2,…] the coefficient ring of the Brown-Peterson cohomology theory BP*(−) with |vi| = −2pi + 2. We study ABP*,*'(−) theory, which is the counter part in algebraic geometry of the BP*(−) theory. Let k be a field with k ⊂ $¥mathbb C$ and K*M(k) the Milnor K-theory. For a nonzero symbol a ∈ Kn+1M(k)/p, a norm variety Va is a smooth variety such that a|k(Va) = 0 ∈ Kn+1M(k(Va))/p and V a($¥mathbb C$) = vn. In particular, we compute ABP*,*'(Ma) for the Rost motive Ma which is a direct summand of the motive M(Va) of some norm variety Va.

Cohomologie non ramifiée et conjecture de Hodge entière

Building upon the Bloch–Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with Q/Z coefficients with a group which measures the failure of the integral Hodge conjecture in degree 4. As a first consequence, a geometric theorem of Voisin implies that the third unramified cohomology group with Q/Z coefficients vanishes on all uniruled threefolds. As a second consequence, a 1989 example by Colliot-Thélène and Ojanguren implies that the integral Hodge conjecture in degree 4 fails for unirational varieties of dimension at least 6. For certain classes of threefolds fibered over a curve, we establish a relation between the integral Hodge conjecture and the computation of the index of the generic fiber.

Hilbert 90 for Galois Cohomology

We determine precise conditions under which Hilbert 90 is valid for Milnor k-theory and Galois cohomology. In particular, Hilbert 90 holds for degree n when the cohomological dimension of the Galois group of the maximal p-extension of F is at most n.