Milnor-Witt Cycle Modules

Feld, Niels

doi:10.48550/arxiv.1811.12163

Cited by 2 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 [Fel18], which are an oriented version of Rost's cycle modules [Ros96].…”

Section: Chmentioning

confidence: 99%

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: Chmentioning

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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Milnor-Witt Cycle Modules

Cited by 2 publications

References 0 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

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