If $f:S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes: \mathcal H_*(S') \to\mathcal H_*(S)$, where $\mathcal H_*(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite etale, we show that it stabilizes to a functor $f_\otimes: \mathcal{SH}(S') \to \mathcal{SH}(S)$, where $\mathcal{SH}(S)$ is the $\mathbb P^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb Z$, the homotopy K-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb Z$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.

Norms in motivic homotopy theory

We develop various aspects of classical enumerative geometry, including Euler characteristics and formulas for counting degenerate fibres in a pencil, with the classical numerical formulas being replaced by identitites in the Grothendieck-Witt group of quadratic forms with coefficients in the base-field.

Aspects of enumerative geometry with quadratic forms

We compute the generalized slices (as defined by Spitzweck–Ostvaer) of the motivic spectrum KO (representing Hermitian K-theory) in terms of motivic cohomology and (a version of) generalized motivic cohomology, obtaining good agreement with the situation in classical topology and the results predicted by Markett–Schlichting. As an application, we compute the homotopy sheaves of (this version of) generalized motivic cohomology, which establishes a version of a conjecture of Morel.

The generalized slices of Hermitian K‐theory

Smooth Models of Motivic Spheres and the Clutching Construction

We equate various Euler classes of algebraic vector bundles, including those of [BM, KW, DJK], and one suggested by M.J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class, and give formulas for local indices at isolated zeros, both in terms of 6-functor formalism of coherent sheaves and as an explicit recipe in commutative algebra of Scheja and Storch. As an application, we compute the Euler classes associated to arithmetic counts of d-planes on complete intersections in P^n in terms of topological Euler numbers over R and C.

A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections

We show that the higher Grothendieck–Witt groups, a.k.a. algebraic hermitian $K$-groups, are represented by an infinite orthogonal Grassmannian in the $\mathbb {A}^1$-homotopy category of smooth schemes over a regular base for which $2$ is a unit in the ring of regular functions. We also give geometric models for various $\mathbb {P}^1$- and $S^1$-loop spaces of hermitian $K$-theory.

https://homepages.warwick.ac.uk/~masiap/research/GWreps4.pdf

Geometric models for higher Grothendieck–Witt groups in $\mathbb {A}^1$-homotopy theory

We show that the higher Grothendieck-Witt groups, a.k.a. algebraic hermitian K-groups, are represented by an infinite orthogonal Grassmannian in the A1-homotopy category of smooth schemes over a regular base for which 2 is a unit in the ring of regular functions. We also give geometric models for various P1- and S1-loop spaces of hermitian K-theory.

Geometric models for higher Grothendieck-Witt groups in A1-homotopy theory

In this paper we study equivariant algebraic $K$-theory for an action of a finite constant group scheme in the equivariant motivic homotopy theory. We prove that the equivariant algebraic $K$-theory is represented by an equivariant ind-scheme defined by Grassmannians. Using this result we show that in the category of equivariant motivic spaces the set of endomorphisms of the motivic space defined by $K_0(G,-)$ coincides with the set of endomorphisms of infinite Grassmannians in the equivariant motivic homotopy category. To this end we explicitly recall the folklore computation of equivariant $K$-theory of Grassmannians.

Endomorphisms of Equivariant Algebraic $K$-theory

In this paper we define the constructible Witt theory of sheaves of $\Lambda$-modules for a scheme $X$ for coefficient rings $\Lambda$ having finite characteristic not equal to 2 and prime to the residue characteristics of the scheme $X$. Our construction is based on the recent advances by Cisinski and Deglise on six-functor formalism for derived categories of \'etale motives and offers a background for the of study of constructible Witt theory as a cohomological invariant for schemes.

Girja Shanker Tripathi

Papers

Geometric models for higher Grothendieck–Witt groups in \(\mathbb {A}^1\)-homotopy theory

Geometric models for higher Grothendieck-Witt groups in A1-homotopy theory

Endomorphisms of Equivariant Algebraic $K$-theory

Constructible Witt theory of schemes