Niels Feld
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.
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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.
On Tate Milnor-Witt Motives
该研究针对带孤立有理不动点的光滑射影$\u0001D706_m$-簇，证明其具有Tate Milnor-Witt motives，并在欧几里得域下给出分裂公式，简化Chow-Witt群的计算。
The equivariant Milnor-Witt motive of $\overline{\mathcal{M}}_{1,2}$
本文研究了稳定曲线模空间$\overline{\mathcal{M}}_{1,2}$的等变Milnor-Witt动机分解，采用代数几何与李代数理论方法，揭示了其结构特性与代数循环的关系。
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…
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 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.
Homological Milnor-Witt modules and Chow-Witt groups over general bases
本文提出了一种一般基上的同调Milnor-Witt模理论，扩展了Chow-Witt群，并证明了广义Bloch公式和有限性结果。
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…
Hermitian $K$-theory and Milnor-Witt motivic cohomology over $\mathbb Z$
本文利用Milnor-Witt母题上同调与切片谱序列计算Hermitian $K$-群，证明Grothendieck-Riemann-Roch定理，并给出整数环的Hermitian $K$-群与高阶Witt群。
Milnor-Witt motivic cohomology of complements of hyperplane arrangements
本文计算仿射空间中超平面排列补集的Milnor-Witt动机上同调，给出其代数结构，并通过实现到经典上同调获得推论。
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.
Milnor-Witt K-theory and Witt K-theory of a field
本文回顾了域的Milnor-Witt K-理论的基本计算，并利用Kato对Milnor二次型猜想的解答，给出了特征为2的域的Witt K-理论同构于对称双线性形式Witt环基本理想的Rees代数的初等证明。
Notes on Milnor-Witt K-theory
本文为任意特征域建立 Milnor-Witt K-理论基础，利用 Grothendieck 迹映射处理不可分转移，完善其函子性公理。
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…
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…
Operations on Milnor-Witt K-theory
本文定义了从 Milnor-Witt K-理论到同伦模的运算，证明其生成所有非加性运算的模，并显式计算了相关理论间的运算群。
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.
Milnor-Witt motivic decomposition of Stiefel varieties
本文计算了Stiefel流形的整MW-动机上同调群，并建立了其整MW-动机分解，从而证明了先前工作中的猜想。
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.
The big de Rham-Witt forms over fields and motives of non-reduced schemes
本文通过代数循环证明了大德拉姆-维它形式群与Artin局部代数的相对改进米尔诺K群同构，并解释了其在非减方案动机上同调中的应用。
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.
Milnor-Witt motivic cohomology and linear algebraic groups
本文计算了辛群及一般线性群等线性代数群的 Milnor-Witt 动机上同调，并确定了其乘法结构。
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.
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.
Let $X$ be a smooth proper variety over a field $k$ and suppose that the degree map $\mathrm{CH}_0(X \otimes_k K) \to \mathbb{Z}$ is isomorphic for any field extension $K/k$. We show that $G(\mathrm{Spec} k) \to G(X)$ is an isomorphism for any $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers $G$. This generalize a result of Binda-Rülling-Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge-Witt cohomology is a $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers.
The rigidity theorem for homotopy invariant presheaves with Witt-transfers on the category of smooth affine varieties over a field $k$ with characteristic not equal to 2 is proved. Namely for such a presheaf $\mathcal F$ the isomorphism $\mathcal F(U)\simeq \mathcal F(x)$ where $U$ is henseliation of a variety at smooth closed point with separable residue field (over $k$) is proved. The rigidity for presheaves $W^i(X\times -)$ where $X$ is smooth variety and $W^i(-)$ are derived Witt-groups ($i\in \mathbb Z/4\mathbb Z$) follows as corollary.
The strictly homotopy invariance of the associated Nisnevish sheave $\widetilde{\mathcal F}_{Nis}$ of a homotopy invariant presheave $\mathcal F$ with GW-transfers (or Witt-transfers) on the category of smooth varieties over a prefect field $k$, $char\,k \neq 2$, is proved, i.e. the isomorphism $$H^i_{Nis}(\mathbb A^1\times X,\widetilde{\mathcal F}_{Nis})\simeq H^i_{Nis}(X,\widetilde{\mathcal F}_{Nis})$$ for any $X\in Sm_k$ is obtained. This theorem is necessary for the construction of the triangulated category of GW-motives $\mathbf{DM}^{GW}(k)$ and Witt-motives $\mathbf{DM}^W(k)$ by the Voev…
Two monoidal structures on Satake category in mixed characteristic
本文解释了两种几何Satake等价性的重合，包括Satake范畴上两种对称monoidal结构的重合。
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