Related papers: Modules homotopiques (Homotopy modules)
Let X be a smooth scheme, Z a smooth closed subscheme, and U the open complement. Given any localizing and A1-homotopy invariant of dg categories E, we construct an associated Gysin triangle relating the value of E at the dg categories of…
The homotopy category of complexes of projective left-modules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite…
In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log…
Modern categories of spectra such as that of Elmendorf et al equipped with strictly symmetric monoidal smash products allows the introduction of symmetric monoids providing a new way to study highly coherent commutative ring spectra. These…
We prove a duality statement on modules over KH-theory in the stable motivic homotopy category whose dualizing object is given by G-theory, over any quasi-excellent scheme of characteristic zero.
This work is devoted to study orientation theory in arithmetic geometric within the motivic homotopy theory of Morel and Voevodsky. The main tool is a formulation of the absolute purity property for an \emph{arithmetic cohomology theory},…
Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity,…
Grothendieck-Chow motives of quadric hypersurfaces have provided many insights into the theory of quadratic forms. Subsequently, the landscape of motives of more general projective homogeneous varieties has begun to emerge. In particular,…
A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy…
We develop a $\mathbf{P}^1$-unstable non-$\mathbf{A}^1$-invariant theory of motivic spaces and spectra, and construct the Gysin map therein for regular immersions. This in particular gives the Gysin map in the Annala--Hoyois--Iwasa…
We compare three different ways of defining group cohomology with coefficients in a crossed-module: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where…
We prove that the Gysin map is compatible with mixed Hodge Structures.
We develop a theory of motives with compact support for logarithmic schemes over a field. Starting from the notion of finite logarithmic correspondences with compact support, we define the logarithmic motive with compact support analogous…
We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank 3 and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a…
Motivic cohomology is powerful tool in algebraic geometry with associated realization maps giving important information about the relations between cohomological invariants of schemes and their classifying spaces. The problem of computing…
The moduli spaces of stable quasimaps unify various moduli appearing in the study of Gromov-Witten Theory. This note is a survey article on the moduli of stable quasimaps, based on joint papers with Ciocan-Fontanine and Maulik as well as…
The Hochschild and (cotriple) cyclic homologies of crossed modules of (not-necessarily-unital) associative algebras are investigated. Wodzicki's excision theorem is extended for inclusion crossed modules in the category of crossed modules…
This an extended version of the previous preprint dated by February 2005. We prove that the Chow motive of an anisotropic projective homogeneous variety of type F4 is isomorphic to the direct sum of twisted copies of a generalized Rost…
We develop a motivic cohomology theory, representable in the Voevodsky's triangulated category of motives, for smooth separated Deligne-Mumford stacks and show that the resulting higher Chow groups are canonically isomorphic to the higher…
Consider the formal power series $\sum [C_{p, \alpha}(X)]t^{\alpha}$ (called Motivic Chow Series), where $C_p(X)=\disjoint C_{p, \alpha}(X)$ is the Chow variety of $X$ parametrizing the $p$-dimensional effective cycles on $X$ with $C_{p,…