اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدHermitian K-theory via oriented Gorenstein algebras
135
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We show that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along finite Gorenstein morphisms with trivialized dualizing sheaf. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith-Winograd tensor.
قيم البحث
اقرأ أيضاً
We study generically split octonion algebras over schemes using techniques of ${mathbb A}^1$-homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classification results over smooth affine s
chemes of small dimension. In particular, for smooth affine schemes over algebraically closed fields, we show that generically split octonion algebras may be classified by characteristic classes including the second Chern class and another mod $3$ invariant. We review Zorns vector matrix construction of octonion algebras, generalized to rings by various authors, and show that generically split octonion algebras are always obtained from this construction over smooth affine schemes of low dimension. Finally, generalizing P. Gilles analysis of octonion algebras with trivial norm form, we observe that generically split octonion algebras with trivial associated spinor bundle are automatically split in low dimensions.
In this paper, we construct a refined, relative version of the etale realization functor of motivic spaces, first studied by Isaksen and Schmidt. Their functor goes from the $infty$-category of motivic spaces over a base scheme $S$ to the $infty$-cat
egory of $p$-profinite spaces, where $p$ is a prime which is invertible in all residue fields of $S$. In the first part of this paper, we refine the target of this functor to an $infty$-category where $p$-profinite spaces is a further completion. Roughly speaking, this $infty$-category is generated under cofiltered limits by those spaces whose associated local system on $S$ is $A^1$-invariant. We then construct a new, relative version of their etale realization functor which takes into account the geometry and arithmetic of the base scheme $S$. For example, when $S$ is the spectrum of a field $k$, our functor lands in a certain $infty$-category equivariant for the absolute Galois group. Our construction relies on a relative version of etale homotopy types in the sense of Artin-Mazur-Friedlander, which we also develop in some detail, expanding on previous work of Barnea-Harpaz-Schlank. We then stabilize our functor, in the $S^1$-direction, to produce an etale realization functor for motivic $S^1$-spectra (in other words, Nisnevich sheaves of spectra which are $A^1$-invariant). To this end, we also develop an $infty$-categorical version of the theory of profinite spectra, first explored by Quick. As an application, we refine the construction of the etale $K$-theory of Dwyer and Friedlander, and define its non-commutative extension. This latter invariant should be seen as an $ell$-adic analog of Blancs theory of semi-topological $K$-theory of non-commutative schemes. We then formulate and prove an analog of Blancs conjecture on the torsion part of this theory, generalizing the work of Antieau and Heller.
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 finit
e 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 give an algebro-geometric interpretation of $C_2$-equivariant stable homotopy theory by means of the $b$-topology introduced by Claus Scheiderer in his study of $2$-torsion phenomena in etale cohomology. To accomplish this, we first revisit and ex
tend work of Scheiderer on equivariant topos theory by functorially associating to a $infty$-topos $mathscr{X}$ with $G$-action a presentable stable $infty$-category $mathrm{Sp}^G(mathscr{X})$, which recovers the $infty$-category $mathrm{Sp}^G$ of genuine $G$-spectra when $mathscr{X}$ is the terminal $G$-$infty$-topos. Given a scheme $X$ with $1/2 in mathcal{O}_X$, our construction then specializes to produce an $infty$-category $mathrm{Sp}^{C_2}_b(X)$ of $b$-sheaves with transfers as $b$-sheaves of spectra on the small etale site of $X$ equipped with certain transfers along the extension $X[i] rightarrow X$; if $X$ is the spectrum of a real closed field, then $mathrm{Sp}^{C_2}_b(X)$ recovers $mathrm{Sp}^{C_2}$. On a large class of schemes, we prove that, after $p$-completion, our construction assembles into a premotivic functor satisfying the full six functors formalism. We then introduce the $b$-variant $mathrm{SH}_b(X)$ of the $infty$-category $mathrm{SH}(X)$ of motivic spectra over $X$ (in the sense of Morel-Voevodsky), and produce a natural equivalence of $infty$-categories $mathrm{SH}_b(X)^{wedge}_p simeq mathrm{Sp}^{C_2}_b(X)^{wedge}_p$ through amalgamating the etale and real etale motivic rigidity theorems of Tom Bachmann. This involves a purely algebro-geometric construction of the $C_2$-Tate construction, which may be of independent interest. Finally, as applications, we deduce a $b$-rigidity theorem, use the Segal conjecture to show etale descent of the $2$-complete $b$-motivic sphere spectrum, and construct a parametrized version of the $C_2$-Betti realization functor of Heller-Ormsby.
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 hom
otopy 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.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
