Milnor-Witt cycle modules

Feld, Niels

doi:10.1016/j.jpaa.2019.106298

Cited by 15 publications

(57 citation statements)

References 20 publications

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“…The compatibility of the exact sequences with change of coefficients follows from the discussion in Section 2.4. Essentially, changing the coefficients is a map of Milnor-Witt cycle modules [23], which are an oriented version of Rost's cycle modules [34]. An immediate application of the localization sequences in Chow theory is that Chern classes are uniquely determined by their compatibility with stabilization and the identification of the top Chern class as Thom class of the universal bundle.…”

Section: There Is a Long Exact Sequence Of Chow-witt Groups Of Classimentioning

confidence: 99%

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Chow–Witt rings of classifying spaces for symplectic and special linear groups

Hornbostel

Wendt

2019

Journal of Topology

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We compute the Chow-Witt rings of the classifying spaces for the symplectic and special linear groups. In the structural description we give, contributions from real and complex realization are clearly visible. In particular, the computation of cohomology with I j -coefficients is done closely along the lines of Brown's computation of integral cohomology for special orthogonal groups. The computations for the symplectic groups show that Chow-Witt groups are a symplectically oriented ring cohomology theory. Using our computations for special linear groups, we also discuss the question when an oriented vector bundle of odd-rank splits off a trivial summand. Contents

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Section: There Is a Long Exact Sequence Of Chow-witt Groups Of Classimentioning

confidence: 99%

“…The compatibility of the exact sequences with change of coefficients follows from the discussion in Section 2.4. Essentially, changing the coefficients is a map of Milnor–Witt cycle modules , which are an oriented version of Rost's cycle modules .

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…”

Section: Characteristic Classes For (Oriented) Vector Bundlesmentioning

confidence: 99%

Chow–Witt rings of classifying spaces for symplectic and special linear groups

Hornbostel

Wendt

2019

Journal of Topology

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“…The details of such a theorem have not yet been written up, but see [Dég14a]. In the homotopical case, a generalization of Rost's theory is in the work (see [Fel18]).…”

Section: 11mentioning

confidence: 99%

The homotopy Leray spectral sequence

Asok¹,

Déglise²,

Nagel³

2020

Motivic Homotopy Theory and Refined Enumerative Geometry

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In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its E 2 -page. Our description of the E 2 -page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost's cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work. Contents 2 All objects in T (S) are obtained by taking extensions of arbitrary coproducts. Equivalently, an object K of T (S) is zero if and only if: ∀X/S smooth,(n, i) ∈ Z 2 , Hom T (S) M S (X)(i)[n], K = 0. 3 Recall that this expresses the compatibility of T with projective limits. 4 This condition is automatically verified (see [BD17, 3.2.13]) if T satisfies absolute purity and the following vanishing statement holds: ∀fields E, ∀n > m, H n,m (Spec(E), T ) = 0. 5Under two geometric assumptions, this means that:• In case (S1), we will require that, for p the characteristic exponent of k, T is Z[1/p]linear.• In case (S2), we require that T is Q-linear.

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“…In [Fel18], the author introduced the theory of Milnor-Witt cycle modules, generalising the work of Rost [Ros96] on Milnor cycle modules and Schmid's thesis [Sch98]. Indeed, we have studied general cycle complexes C * (X, M, V X ) and their (co)homology groups A * (X, M, V X ) (called Chow-Witt groups with coefficients) in a quadratic setting over a perfect base field of any characteristic.…”

Section: Current and Future Workmentioning

confidence: 99%

“…

This article is a sequel of [Fel18]. We study the cohomology theory and the canonical Milnor-Witt cycle module associated to a motivic spectrum.

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confidence: 99%

Morel Homotopy Modules and Milnor-Witt Cycle Modules

Feld¹

2019

Preprint

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This article is a sequel of [Fel18]. 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 [Dé11]. As a corollary, we recover a theorem of Ananyevskiy and Neshitov [AN18] and we prove that Chow-Witt groups in degree zero are birational invariants.

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Milnor-Witt cycle modules

Cited by 15 publications

References 20 publications

Chow–Witt rings of classifying spaces for symplectic and special linear groups

Chow–Witt rings of classifying spaces for symplectic and special linear groups

The homotopy Leray spectral sequence

Morel Homotopy Modules and Milnor-Witt Cycle Modules

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