Smooth projective $\mathbb{G}_m$-varieties with isolated rational fixed points admit Tate Milnor-Witt motives. Over Euclidean fields, we give a splitting formula of such motives, which reduces the computation of their Chow-Witt groups to that of their Chow groups and cohomologies of Witt sheaf.
We study the Milnor-Witt motives which are a finite direct sum of $\mathbb{Z}(q)[p]$ and $\mathbb{Z}/η(q)[p]$. We show that for MW-motives of this type, we could determine an MW-motivic cohomology class in terms of a motivic cohomology class and a Witt cohomology class. We define the motivic Bockstein cohomology and show that it corresponds to subgroup of Witt cohomology, if the MW-motive splits as above. As an application, we give the splitting formula of Milnor-Witt motives of Grassmannian bundles and complete flag bundles. This in particular shows that the integral cohomology of real comple…
We show that finite Milnor-Witt correspondences satisfy a cancellation theorem with respect to the pointed multiplicative group scheme. This has several notable applications in the theory of Milnor-Witt motives and Milnor-Witt motivic cohomology.
We give a new proof of the fact that Milnor-Witt K-theory has geometric transfers. The proof yields to a simplification of Morel's conjecture about transfers on contracted homotopy sheaves.
We develop the theory of Milnor-Witt motives and motivic cohomology. Compared to Voevodsky's theory of motives and his motivic cohomology, the first difference appears in our definition of Milnor-Witt finite correspondences, where our cycles come equipped with quadratic forms. This yields a weaker notion of transfers and a derived category of motives that is closer to the stable homotopy theory of schemes. We prove a cancellation theorem when tensoring with the Tate object, we compare the diagonal part of our Milnor-Witt motivic cohomology to Minor-Witt K-theory and we provide spectra represen…
In the paper M. Somekawa, {\it{On Milnor $K$-groups attached at semi-Abelian varieties}}, K-theory, \textbf{4} (1990) p.105, Somekawa conjectures that his Milnor K-group $K(k,G_1,...,G_r)$ attached to semi-abelian varieties $G_1$,...,$G_r$ over a field $k$ is isomorphic to ${\rm Ext}_{\mathcal{M}_k}^r(\mathbb{Z},G_1[-1] \otimes ... \otimes G_r[-1])$ where $\mathcal{M}_k$ is a certain category of motives over $k$. The purpose of this note is to give remarks on this conjecture, when we take $\mathcal{M}_k$ as Voevodsky's category of motives ${\rm DM}^{eff}_{-}(k)$ .
We build a ring spectrum representing Milnor-Witt motivic cohomology, as well as its étale local version and show how to deduce out of it three other theories: Borel-Moore homology, cohomology with compact support and homology. These theories, as well as the usual cohomology, are defined for singular schemes and satisfy the properties of their motivic analog (and more), up to considering more general twists. In fact, the whole formalism of these four theories can be functorially attached to any ring spectrum, giving finally maps between the Milnor-Witt motivic ones to the classical motivic one…
The category of effective Grothendieck-Witt-motives $\mathbf{DM}^{GW}_{\mathrm{eff},-}(k)$ (and Witt-motives $\mathbf{DM}^W_{\mathrm{eff},-}(k)$) by Voevodsky-Suslin method starting with some category of GW-correspondences (and Witt-correspondences) over a perfect field $k$, $char\,k\neq 2$, is defined. The functor $M^{GW}_{eff}\colon Sm_k\to \mathbf{DM}^{GW}_{\mathrm{eff},-}(k)$ of Grothendieck-Witt-motives of smooth varieties is computed and it is proved that for any smooth scheme $X$ and homotopy invariant sheave with GW-transfers $F$ $$ Hom_{\mathbf{DM}^{GW}_{\mathrm{eff},-}(k)}(M^{GW}_{ef…
We study the cohomology theory and the canonical Milnor-Witt cycle module associated to a motivic spectrum. We prove that the heart of Morel-Voevodsky stable homotopy category over a perfect field (equipped with its homotopy t-structure) is equivalent to the category of Milnor-Witt cycle modules, thus generalising Déglise's thesis. As a corollary, we recover a theorem of Ananyevskiy and Neshitov and we prove that the Milnor-Witt K-theory groups are birational invariants.
The category of finite Milnor-Witt correspondences, introduced by Calmès and Fasel, provides a new type of correspondences closer to the motivic homotopy theoretic framework than Suslin-Voevodsky's correspondences. A fundamental result of the theory of ordinary correspondences concerns homotopy invariance of sheaves with transfers, and in the present paper we address this question in the setting of Milnor-Witt correspondences. Employing techniques due to Druzhinin, Fasel-Østvær and Garkusha-Panin, we show that homotopy invariance of presheaves with Milnor-Witt transfers is preserved under Nisn…
Relying on recent advances in the theory of motives we develope a general formalism for derived categories of motives with Q-coefficients on perfect (ind-)schemes. As an application we give a motivic refinement of Zhu's geometric Satake equivalence for Witt vector affine Grassmannians in this set-up.
We generalize Rost's theory of cycle modules using 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 usual constructions are developed: proper pushfoward, (essentially) smooth pullback, long exact sequences, (coniveau) spectral sequences and products, as well as the homotopy invariance property; in addition, Gysin morphisms for lci maps are constructed. We prove an adjunction theorem linking our theory to Rost's.
The definition of Milnor-Witt cycle modules in [Feld, N., Milnor-Witt cycle modules, Journal of Pure and Applied Algebra 224 (2020) 106298] can easily be adapted over general regular base schemes. However, there are simple examples to show that Gersten complex fails to be exact for cycle modules in general if the base is not a field. The goal of this article is to show that, for a restricted class of Milnor-Witt cycle modules over an excellent DVR satisfying an extra axiom, called here as R5, the expected properties of exactness of Gersten complex and $\mathbb{A}^1$-invariance hold. Moreover R…
We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor's conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology.
The category of effective $Witt$-motives $DWM^-(k)$ with functor $WM\colon Sm_k\to DWM^-(k)$ defining motives of smooth affine varieties for perfect field $k$, $char k\neq 2$ is constructed. In the construction Voevodsky-Suslin method is applyed to a category of $Witt$-correspondence between affine smooth varieties $WCor_k$ that morphisms are defined by class in $Witt$-group of quadratic space $(P,q_P)$ with $P$ being $k[X\times Y]$-module finitely generated projective over $k[X]$ and $q_P\colon P\to Hom_{k[X]}(P,k[X])$ being $k[X\times Y]$-liner isomorphism. And the natural isomorphism $$Hom_…
The article is to construct a graded ring isomorphism between $H_0(ZF(Δ^{\bullet}_k,\mathbb{G}_m^{\wedge *}))$ and the Milnor-Witt K-theory ring $K^{MW}_{*\geqslant 0}(k)$, where $k$ is a field of characteristic zero and $ZF_*(k)$ is the category of linear framed correspondences of algebraic k-varieties, introduced by Garkusha and Panin. As it was shown by Garkusha and Panin, this partially recovers the computation of the motivic cohomotopy groups $π^{n,n}(Σ^{\infty}_{S^1}Σ^{\infty}_{\mathbb{G}_m}S^0)(k)$, which is originally due to Morel.
In these lectures, we provide a toolkit to work with Chow-Witt groups, and more generally with the homology and cohomology of the Rost-Schmid complex associated to Milnor-Witt $K$-theory.
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 $\tilde{I}$ and a sheaf of Witt rings $\tilde{W}$ in the obvious way. The Milnor conjecture then relates the associated graded of $\tilde{W}$ 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.
Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It extend the morphism sending the class of an algebraic variety over $K$ to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdan's motivic integration and Ayoub's equivalence between motives of rigid analytic varieties over $K$ and quasi-unipotent motives over $k$ ; we also…
We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, whose Milnor range is isomorphic to the Zariski sheaf of big de Rham-Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham-Witt sheaves. Several applications are derived.
We present a version of projective bundle theorem in MW-motives (resp. Chow-Witt rings), which says that $\widetilde{CH}^*(\mathbb{P}(E))$ is determined by $\widetilde{CH}^*(X)$, $\widetilde{CH}^*(X,det(E)^{\vee})$, $CH^*(X)$ and $Sq^2$ for smooth quasi-projective schemes $X$ and vector bundles $E$ over $X$ with $e(E^{\vee})=0\in H^n(X,W(det(E)))$, provided that $_2CH^*(X)=0$. As an application, we compute the MW-motives of blow-ups with smooth centers. Moreover, we discuss the invariance of Chow-Witt cycles of projective bundles under automorphisms of vector bundles.
We introduce in this work the notion of the category of pure $\mathbf{E}$-Motives, where $\mathbf{E}$ is a motivic strict ring spectrum and construct twisted $\mathbf{E}$-cohomology by using six functors formalism of J. Ayoub. In particular, we construct the category of pure Chow-Witt motives $CHW(k)_{\mathbb{Q}}$ over a field $k$ and show that this category admits a fully faithful embedding into the geometric stable $\mathbb{A}^1$-derived category $D_{\mathbb{A}^1,gm}(k)_{\mathbb{Q}}$.
In the paper we use the theory of framed correpondences to construct Milnor-Witt transfers on homotopy modules. As a consequence we identify the zeroth stable $\mathbb{A}^1$-homotopy sheaves of smooth varieties with the zeroth homology of corresponding MW-motivic complexes and prove that the hearts of homotopy $t$-structures on the stable $\mathbb{A}^1$-derived category and the category of Milnor-Witt motives are equivalent.
In this work, we compute the $0$th cohomology group of a complex of groups of cobordism-framed correspondences, and prove the isomorphism to Milnor $K$-groups. An analogous result for common framed correspondences has been proved by A. Neshitov in his paper "Framed correspondences and the Milnor---Witt $K$-theory". Neshitov's result is, at the same time, a computation of the homotopy groups $π_{i,i}(S^0)(Spec(k)).$ This work could be used in the future as basis for computing homotopy groups $π_{i,i}(MGL_{\bullet})(Spec(k))$ of the spectrum $MGL_{\bullet}.$
In this paper we construct new derived invariants with integral coefficients using the theory of motifs, and give several applications. Specifically, we obtain the following results: For complex algebraic surfaces, we prove that certain torsion in the abelianized fundamental group is a derived invariant. We prove that the collection of Hodge-Witt cohomology groups is a derived invariant. In particular, Hodge-Witt reduction and ordinary reduction are preserved by derived equivalence when the characteristic is sufficiently large. Finally, using the techniques of non-commutative algebraic geometr…
We compute the perverse delta-homotopy heart of the motivic stable homotopy category over a base scheme with a dimension function delta, rationally or after inverting the exponential characteristic in the equicharacteristic case. In order to do that, we define the notion of homological Milnor-Witt cycle modules and construct a homotopy-invariant Rost-Schmid cycle complex. Moreover, we define the category of cohomological Milnor-Witt cycle modules and show a duality result in the smooth case.
The goal of this paper is to give an explicit description of the triangulated categories of Tate and Artin-Tate motives with finite coefficients Z/m over a field K containing a primitive m-root of unity as the derived categories of exact categories of filtered modules over the absolute Galois group of K with certain restrictions on the successive quotients. This description is conditional upon (and its validity is equivalent to) certain Koszulity hypotheses about the Milnor K-theory/Galois cohomology of K. This paper also purports to explain what it means for an arbitrary nonnegatively graded …
Ananyevsky has recently computed the stable operations and cooperations of rational Witt theory. These computations enable us to show a motivic analog of Serre's finiteness result: Theorem: Let $k$ be a field. Then $π^{\mathbb{A}^1}_ n(\mathbb{S}^- _k )_*$ is torsion for $n > 0$. As an application we define a category of Witt motives over $k$ and show that rationally this category is equivalent to the minus part of $SH(k)_\mathbb{Q}$.
In the present article we discuss an approach to cohomological invariants of algebraic groups over fields of characteristic zero based on the Morava $K$-theories, which are generalized oriented cohomology theories in the sense of Levine--Morel. We show that the second Morava $K$-theory detects the triviality of the Rost invariant and, more generally, relate the triviality of cohomological invariants and the splitting of Morava motives. We describe the Morava $K$-theory of generalized Rost motives, compute the Morava $K$-theory of some affine varieties, and characterize the powers of the fundam…
We compute the Chow-Witt rings of split quadrics over a field of characteristic not two. We even determine the full bigraded I-cohomology and Milnor-Witt cohomology rings, including twists by line bundles. The results on I-cohomology corroborate the general philosophy that I-cohomology is an algebro-geometric version of singular cohomology of real varieties: our explicit calculations confirm that the I-cohomology ring of a split quadric over the reals is isomorphic to the singular cohomology ring of the space of its real points.
We study intersection theoretic problems in the setting of Chow-Witt groups with coefficients in a fixed Milnor-Witt cycle algebra over a perfect field. We prove that the product maps on such groups satisfy the following property: given two points in a regular local scheme with supports which do not intersect properly, their product vanishes. This gives an analogue of Serre's vanishing result for intersection multiplicities.
Let $\mathrm{k}_{0}$ be a field and $W$ a finite orthogonal reflection group over $\mathrm{k}_{0}$. We prove Serre's splitting principle for cohomological invariants of $W$ with values in Rost's cycle modules (over $\mathrm{k}_{0}$) if the characteristic of $\mathrm{k}_{0}$ is coprime to $|W|$. We then show that this principle for such groups holds also for Witt- and Milnor-Witt $K$-theory invariants.
We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of Fulton-MacPherson. We import the tools of Fulton's intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess intersections, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor-Witt spectrum recently constructed by Déglise-Fasel, we get a bivariant theory extending the Chow-Witt groups of Bar…
The main purpose of this article is to define a quadratic analog of the Chern character, the so-called Borel character, which identifies rational higher Grothendieck-Witt groups with a sum of rational MW-motivic cohomologies and rational motivic cohomologies. We also discuss the notion of ternary laws due to Walter, a quadratic analog of formal group laws, and compute what we call the additive ternary laws, associated with MW-motivic cohomology. Finally, we provide an application of the Borel character by showing that the Milnor-Witt K-theory of a field F embeds into suitable higher Grothendie…
We compute the invariants so of Weyl groups in mod 2 Milnor K-theory and more general cycle modules, which are annihilated by 2. Over a base field of characteristic coprime to the group order, the invariants decompose as direct sums of the coefficient module. All basis elements are induced either by Stiefel-Whitney classes or specific invariants in the Witt ring. The proof is based on Serre's splitting principle that guarantees detection of invariants on elementary abelian 2-subgroups generated by reflections.
We provide a cohomological interpretation of the zeroth stable $\mathbb{A}^1$-homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a certain sheaf closely related to the first Milnor--Witt $\mathrm{K}$-theory sheaf. This cohomology group can be computed using an explicit Gersten-type complex. We show that if the base field is algebraically closed then the zeroth stable $\mathbb{A}^1$-homotopy group of a smooth curve coincides with the zeroth Suslin homology group that was identified by Sus…
