Related papers: The localization theorem for framed motivic spaces

We reconcile the multiplications on the homotopy rings of motivic ring spectra used by Voevodsky and Dugger. While the connection is elementary and similar phenomena have been observed in situations like supersymmetry, neither we nor other…

We construct and study a motivic lift of a spectral sequence associated to a stratified scheme, recently discovered by Petersen in the context of mixed Hodge theory and $\ell$-adic Galois representations. The original spectral sequence…

The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. These are Nisnivich sheaves of…

We generalize the notion of a bornology by omitting the condition that a one-point-subset is bounded and obtain a complete and co-complete generalization of the category of bornological coarse spaces. Then we imitate the construction of…

We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent…

Let $S$ be a finite dimensional noetherian scheme. For any proper morphism between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher algebraic $K$-theory and motivic cohomology, thus with no projective hypothesis neither…

We present a general theorem which computes the cohomology of a homological vector field on global sections of vector bundles over smooth affine supervarieties. The hypotheses and results have the clear flavor of a localization theorem.

We prove a Verdier Hypercovering Theorem for cohomology theories arising from motivic spectra. This allows us to construct for smooth quasi-projective complex varieties a natural morphism from etale algebraic to Hodge filtered complex…

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…

We prove that some of the classical homological stability results for configuration spaces of points in manifolds can be lifted to motivic cohomology.

We construct and study a theory of bivariant cobordism of derived schemes. Our theory provides a vast generalization of the algebraic bordism theory of characteristic 0 algebraic schemes, constructed earlier by Levine and Morel, and a…

For any motivic $\mathbb{E}_\infty$-ring spectrum $A$ we construct an equivalence $\rho$ between the $\infty$-category of cellular motivic $A$-module spectra and modules over an $\mathbb{E}_1$-algebra $\Theta$ in $\mathbb{Z} $-graded…

We improve some foundational connectivity results and the relative Hurewicz theorem in motivic homotopy theory, study functorial central series in motivic local group theory, establish the existence of functorial Moore--Postnikov…

We build a ring spectrum representing Milnor-Witt motivic cohomology, as well as its \'etale local version and show how to deduce out of it three other theories: Borel-Moore homology, cohomology with compact support and homology. These…

We establish a purely geometric form of the concentration theorem (also called localization theorem) for actions of a linearly reductive group $G$ on an affine scheme $X$ over an affine base scheme $S$. It asserts the existence of a…

Let S be an essentially smooth scheme over a field and l a prime number invertible on S. We show that the algebra of bistable operations in the mod l motivic cohomology of smooth S-schemes is generated by the motivic Steenrod operations.…

We investigate several interrelated foundational questions pertaining to the study of motivic dga's of Dan-Cohen--Schlank [8] and Iwanari [13]. In particular, we note that morphisms of motivic dga's can reasonably be thought of as a…

We analyze stabilization with respect to ${\mathbb P}^1$ in the Morel--Voevodsky unstable motivic homotopy theory. We introduce a refined notion of cellularity (a.k.a., biconnectivity) in various motivic homotopy categories taking into…

We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in…

We prove that the $\infty$-category of $\mathrm{MGL}$-modules over any scheme is equivalent to the $\infty$-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbb{P}^1$-loop spaces,…