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Register a new userLectures on Chow-Witt groups
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Jean Fasel
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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.
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We compute Witt groups of maximal isotropic Grassmannians, aka. spinor varieties. They are examples of type D homogenuous varieties. Our method relies on the Blow-up setup of Balmer-Calm`es, and we investigate the connecting homomorphism via the projective bundle formula of Walter-Nenashev, the projection formula of Calm`es-Hornbostel and the excess intersection formula of Fasel. The computation in the Type D case can be presented by so called even shifted young diagrams.
We define the Chow $t$-structure on the $infty$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k)^{cheartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $SH(k)^{cell, cheartsuit}$ as the category of even graded $MU_{2*}MU$-comodules. Furthermore, we show that the $infty$-category of modules over the Chow truncated sphere spectrum is algebraic. Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: To integral results; To all base fields other than just $C$; To the entire $infty$-category of motivic spectra $SH(k)$, rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field $k$ using the Postnikov tower associated to the Chow $t$-structure and the motivic Adams spectral sequences over $k$.
We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $operatorname{Pic}(X)$ and pointed monoid of regular functions $Gamma(X, mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.
We complement our previous computation of the Chow-Witt rings of classifying spaces of special linear groups by an analogous computation for the general linear groups. This case involves discussion of non-trivial dualities. The computation proceeds along the lines of the classical computation of the integral cohomology of ${rm BO}(n)$ with local coefficients, as done by Cadek. The computations of Chow-Witt rings of classifying spaces of ${rm GL}_n$ are then used to compute the Chow-Witt rings of the finite Grassmannians. As before, the formulas are close parallels of the formulas describing integral cohomology rings of real Grassmannians.
Let $X$ be a smooth projective quadric defined over a field of characteristic 2. We prove that in the Chow group of codimension 2 or 3 of $X$ the torsion subgroup has at most two elements. In codimension 2, we determine precisely when this torsion subgroup is nontrivial. In codimension 3, we show that there is no torsion if {$dim Xge 11$.} This extends the analogous results in characteristic different from 2, obtained by Karpenko in the nineteen-nineties.
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