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Register a new userRepresentability of the local motivic Brouwer degree
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We study which quadratic forms are representable as the local degree of a map $f : A^n to A^n$ with an isolated zero at $0$, following the work of Kass and Wickelgren who established the connection to the quadratic form of Eisenbud, Khimshiashvili, and Levine. Our main observation is that over some base fields $k$, not all quadratic forms are representable as a local degree. Empirically the local degree of a map $f : A^n to A^n$ has many hyperbolic summands, and we prove that in fact this is the case for local degrees of low rank. We establish a complete classification of the quadratic forms of rank at most $7$ that are representable as the local degree of a map over all base fields of characteristic different from $2$. The number of hyperbolic summands was also studied by Eisenbud and Levine, where they establish general bounds on the number of hyperbolic forms that must appear in a quadratic form that is representable as a local degree. Our proof method is elementary and constructive in the case of rank 5 local degrees, while the work of Eisenbud and Levine is more general. We provide further families of examples that verify that the bounds of Eisenbud and Levine are tight in several cases.
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We prove that the $infty$-category of motivic spectra satisfies Milnor excision: if $Ato B$ is a morphism of commutative rings sending an ideal $Isubset A$ isomorphically onto an ideal of $B$, then a motivic spectrum over $A$ is equivalent to a pair of motivic spectra over $B$ and $A/I$ that are identified over $B/IB$. Consequently, any cohomology theory represented by a motivic spectrum satisfies Milnor excision. We also prove Milnor excision for Ayoubs etale motives over schemes of finite virtual cohomological dimension.
If $f:S to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal norm functor $f_otimes: mathcal H_*(S) tomathcal H_*(S)$, 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 Grothendiecks Galois theory, with Betti realization, and with Voevodskys slice filtration; we prove that the norm functors categorify Rosts multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $Hmathbb Z$, the homotopy K-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $Hmathbb Z$ is a common refinement of Fulton and MacPhersons mutliplicative transfers on Chow groups and of Voevodskys power operations in motivic cohomology.
We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [X/G], where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of motivic homotopy theory: the (unstable) purity and gluing theorems of Morel and Voevodsky and the (stable) ambidexterity theorem of Ayoub. Our proof of the latter is different than Ayoubs and is of interest even when G is trivial. Using these results, we construct a formalism of six operations for equivariant motivic spectra, and we deduce that any cohomology theory for G-schemes that is represented by an absolute motivic spectrum satisfies descent for the cdh topology.
We obtain geometric models for the infinite loop spaces of the motivic spectra $mathrm{MGL}$, $mathrm{MSL}$, and $mathbf{1}$ over a field. They are motivically equivalent to $mathbb{Z}times mathrm{Hilb}_infty^mathrm{lci}(mathbb{A}^infty)^+$, $mathbb{Z}times mathrm{Hilb}_infty^mathrm{or}(mathbb{A}^infty)^+$, and $mathbb{Z}times mathrm{Hilb}_infty^mathrm{fr}(mathbb{A}^infty)^+$, respectively, where $mathrm{Hilb}_d^mathrm{lci}(mathbb{A}^n)$ (resp. $mathrm{Hilb}_d^mathrm{or}(mathbb{A}^n)$, $mathrm{Hilb}_d^mathrm{fr}(mathbb{A}^n)$) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree $d$ in $mathbb{A}^n$, and $+$ is Quillens plus construction. Moreover, we show that the plus construction is redundant in positive characteristic.
The dual motivic Steenrod algebra with mod $ell$ coefficients was computed by Voevodsky over a base field of characteristic zero, and by Hoyois, Kelly, and {O}stv{ae}r over a base field of characteristic $p eq ell$. In the case $p = ell$, we show that the conjectured answer is a retract of the actual answer. We also describe the slices of the algebraic cobordism spectrum $MGL$: we show that the conjectured form of $s_n MGL$ is a retract of the actual answer.
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