Related papers: The stable Galois correspondence for real closed f…

For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that…

Let k be an algebraically closed field of characteristic zero. Let SH(k) denote the motivic stable homotopy category of T-spectra over k and SH the classical stable homotopy category. Let c:SH -> SH(k) be the functor induced by sending a…

We reconstruct (appropriately completed) categories of cellular motivic spectra over fields of small cohomological dimension in terms of only their absolute Galois groups. As our main application, we determine the motivic stable stems (away…

Consider a reflection from a finitely-complete category $\mathbb{C}$ into its full subcategory $\mathbb{M}$, with unit $\eta :1_\mathbb{C}\rightarrow HI$. Suppose there is a left-exact functor $U$ into the category of sets, such that $UH$…

Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely there exists a finite field extension $F/k(t)$,…

We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents…

Let k be an infinite perfect field. We provide a general criterion for a spectrum in the stable homotopy category over k to be effective, i.e. to be in the localizing subcategory generated by the suspension spectra of smooth schemes. As a…

Over any field of characteristic not 2, we establish a 2-term resolution of the $\eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the…

We establish axiomatic characterizations of $K$-theory and $KK$-theory for real C*-algebras. In particular, let $F$ be an abelian group-valued functor on separable real C*-algebras. We prove that if $F$ is homotopy invariant, stable, and…

In this article we introduce and study a motivic category in the arithmetic of function fields, namely the category of motives over an algebraic closure $L$ of a finite field with coefficients in a global function field over this finite…

This paper deals with the Weak Inverse Galois Problem which, for a given field $k$, states that, for every finite group $G$, there exists a finite separable extension $L/k$ such that ${\rm{Aut}}(L/k)=G$. One of its goals is to explain how…

The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. In the case of a Galois extension of a number field F/L with rings of integers A/B respectively, this K-theory of the "norm functor" is an extension of…

In this article, we give a construction of the (un-)stable motivic homotopy category of an algebraic stack in the spirit of Morel-Voevodsky. We prove that this new construction agrees with the stable motivic homotopy category defined by…

We show that Klemenc's stable envelope of exact $\infty$-categories induces an equivalence between stable $\infty$-categories with a bounded heart structure and weakly idempotent complete exact $\infty$-categories. Moreover, we generalise…

Given a 0-connective motivic spectrum $E \in SH(k)$ over a perfect field k, we determine $h_0$ of the associated motive $M E \in DM(k)$ in terms of $\pi_0 (E)$. Using this we show that if k has finite 2-\'etale cohomological dimension, then…

Given a field $k$ and a finite group $G$, the Beckmann--Black problem asks whether every Galois field extension $F/k$ with group $G$ is the specialization at some $t_0 \in k$ of some Galois field extension $E/k(T)$ with group $G$ and $E…

Let $k$ be an algebraically closed field of exponential characteristic $p$. Given any prime $\ell\neq p$, we construct a stable \'etale realization functor $$\underline{\text{\'Et}}_{\ell}:\text{Spt}(k)\rightarrow…

For a countable group $G$ we construct a small, idempotent complete, symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}_{\mathrm{sep}}$ whose homotopy category recovers the triangulated equivariant Kasparov category of separable…

We study the question of the surjectivity of the Galois correspondence from subHopf algebras to subfields given by the Fundamental Theorem of Galois Theory for abelian Hopf Galois structures on a Galois extension of fields with Galois group…

We show that there exists a Galois correspondence between subalgebras of an H-comodule algebra A over a base ring R and generalised quotients of a Hopf algebra H if both A and H are flat Mittag--Leffler modules. We also provide new criteria…