Related papers: Motivic Landweber Exactness

We show that Lubin-Tate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is by application of the Goerss-Hopkins-Miller theorem to algebras with involution. For each height $n$, we compute the entire homotopy…

Given an algebraic torus $T$ over a field $F$, its lattice of characters $\Lambda$ gives rise to a topological torus $\mathfrak{T}(T)=\Lambda_{\mathbb R}/\Lambda$ with a continuous action of the absolute Galois group $G$. We construct a…

Grothendieck first defined the notion of a "motif" as a way of finding a universal cohomology theory for algebraic varieties. Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric…

Bivariant (equivariant) K-theory is the standard setting for non-commutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from…

We analyze the spectrum of the tensor-triangulated category of Artin-Tate motives over the base field R of real numbers, with integral coefficients. Away from 2, we obtain the same spectrum as for complex Tate motives, previously studied by…

In this paper we discuss a general notion of Weil cohomology theories, both in algebraic geometry and in rigid analytic geometry. We allow our Weil cohomology theories to have coefficients in arbitrary commutative ring spectra. Using the…

We prove that the map on Balmer spectra induced by a fully faithful geometric functor is a quotient map whose fibers are connected. This is an analogue of the Zariski Connectedness Theorem in algebraic geometry and it can be applied to a…

Let K be a field of characteristic 0 and A be a rigid tensor K-linear category. Let M be a finite-dimensional object of A in the sense of Kimura-O'Sullivan. We prove that the "motivic" zeta function of M with coefficients in K\_0(A) has a…

We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an \'etale topological realization of the stable motivic homotopy theory of smooth schemes over a base…

Let $\mathbb{k}$ be a field of characteristic $p$. We introduce a formalism of mixed sheaves with coefficients in $\mathbb{k}$ and showcase its use in representation theory. More precisely, we construct for all quasi-projective schemes $X$…

We provide a universal characterization of the construction taking a scheme $X$ to its stable $\infty$-category $\text{Mot}(X)$ of noncommutative motives, patterned after the universal characterization of algebraic K-theory due to…

In this article we continue the development of a theory of noncommutative motives. We construct categories of A1-homotopy noncommutative motives, described their universal properties, and compute their spectra of morphisms in terms of…

For an oriented cohomology theory A and a relative cellular space X, we decompose the A-motive of X into a direct sum of twisted motives of the base spaces. We also obtain respective decompositions of the A-cohomology of X. Applying them,…

We apply a version of the Chas-Sullivan-Cohen-Jones product on the higher loop homology of a manifold in order to compute the homology of the spaces of continuous and holomorphic maps of the Riemann sphere into a complex projective space.…

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…

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…

Over the past century, cohomology operations have played a crucial role in homotopy theory and its applications. A powerful framework for constructing such operations is the theory of commutative algebras in spectra. In this article, we…

We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential…

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements.…

We provide a proof in the language of model categories and symmetric spectra of Lurie's theorem that topological complex $K$-theory represents orientations of the derived multiplicative group. Then we generalize this result to the motivic…