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Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\\infty$-)categories of spans (or correspondences). In this paper, we study the more complicated setup where we have two pushforwards (an ‘additive’ and a ‘multiplicative’ one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\\infty,2)$-categories of bispans, characterized by a universal property: they corepresent functors out of $\\infty$-categories of spans where the pullbacks have left adjoints and certain canonical 2-morphisms (encoding base change and distributivity) are invertible. This gives a universal way to obtain functors from bispans, which amounts to upgrading ‘monoid-like’ structures to ‘ring-like’ ones. For example, symmetric monoidal $\\infty$-categories can be described as product-preserving functors from spans of finite sets, and if the tensor product is compatible with finite coproducts our universal property gives the canonical semiring structure using the coproduct and tensor product. More interestingly, we encode the additive and multiplicative transfers on equivariant spectra as a functor from bispans in finite $G$-sets, extend the norms for finite étale maps in motivic spectra to a functor from certain bispans in schemes, and make $\\mathrm {Perf}(X)$ for $X$ a spectral Deligne–Mumford stack a functor of bispans using a multiplicative pushforward for finite étale maps in addition to the usual pullback and pushforward maps. Combining this with the polynomial functoriality of $K$-theory constructed by Barwick, Glasman, Mathew, and Nikolaus, we obtain norms on algebraic $K$-theory spectra.

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 $\\mathbf{P} ^1$-loop spaces, we deduce that very effective $\\mathrm{MGL} $-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $\\Omega ^\\infty _{\\mathbf{P} ^1}\\mathrm{MGL} $ is the $\\mathbf{A} ^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$, $\\Omega ^\\infty _{\\mathbf{P} ^1} \\Sigma ^n_{\\mathbf{P} ^1} \\mathrm{MGL} $ is the $\\mathbf{A} ^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$.

We prove a new convergence result for the slice spectral sequence, following work by Levine and Voevodsky. This verifies a derived variant of Voevodsky’s conjecture on convergence of the slice spectral sequence. This is, in turn, a necessary ingredient for our main theorem: a Thomason-style étale descent result for the Bott-inverted motivic sphere spectrum, which generalizes and extends previous étale descent results for special examples of motivic cohomology theories. Combined with first author’s étale rigidity results, we obtain a complete structural description of the étale motivic stable category.

In this paper, we study the Nisnevich sheafification Heˊt1​(G) of the presheaf associating to a smooth scheme the set of isomorphism classes of G-torsors, for a reductive group G. We show that if G-torsors on affine lines are extended, then Heˊt1​(G) is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for G-torsors. We also identify the sheaf Heˊt1​(G) with the sheaf of A1-connected components of the classifying space Beˊt​G. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of A1-connected components in terms of unramified G-torsors over function fields whenever Nisnevich-local purity holds for G-torsors.

In this paper, we study the Nisnevich sheafification H^1_ét(G) of the presheaf associating to a smooth scheme the set of isomorphism classes of G -torsors, for a reductive group G . We show that if G -torsors on affine lines are extended, then H^1_ét(G) is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for G -torsors. We also identify the sheaf H^1_ét(G) with the sheaf of A^1 -connected components of the classifying space B_étG . This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of A^1 -connected components in terms of unramified G -torsors over function fields whenever Nisnevich-local purity holds for G -torsors.

The moduli stack of G-bundles on a smooth complete curve C over a field, BunG(C), is an immensely rich geometric object and is of central importance to the Geometric Langlands program. This thesis represents a contribution towards a motivic, in the sense of Voevodsky and Morel-Voevodsky, understanding of this stack. Following the strategy of Gaitsgory and Gaitsgory-Lurie we view the Beilinson-Drinfeld Grassmanian, GrG(C) as a more tractable, homological approximation to BunG( C). In the main theorem of this thesis we prove, using two different approaches, that the motive of the fiber of the approximation map Gr G(C) → BunG( C) is, in a number of different and precise ways, motivically contractible. This fiber is the space of rational maps, as introduced by Gaitsgory. One approach is to work in the context of E-modules where E is a motivic E∞-ring spectra and show that there is a motivic equivalence between the space of rational maps and a version of the Ran space. Via various realization functors, we obtain the contractibility theorems of Gaitsgory and Gaitsgory-Lurie. A second, novel approach is to study the unstable motivic homotopy type using a theorem of Suslin and a model of the space of rational maps as introduced by Barlev.

The Lichtenbaum-Quillen conjecture for smooth complex varieties states that algebraic and topological K-theory with finite coefficients become isomorphic in high degrees. We define the \"Lichtenbaum-Quillen dimension\" of a variety in terms of the point where this happens, show that it is surprisingly computable, and analyze many examples. It gives an obstruction to rationality, but one that turns out to be weaker than unramified cohomology and some related birational invariants defined by Colliot-Thélène and Voisin using Bloch-Ogus theory. Because it is compatible with semi-orthogonal decompositions, however, it allows us to prove some new cases of the integral Hodge conjecture using homological projective duality, and to compute the higher algebraic K-theory of the Kuznetsov components of the derived categories of some Fano varieties.

We construct a theory of motivic cohomology for quasi-compact, quasi-separated schemes of equal characteristic, which is related to non-connective algebraic \\(K\\)-theory via an Atiyah--Hirzebruch spectral sequence, and to étale cohomology in the range predicted by Beilinson and Lichtenbaum. On smooth varieties over a field our theory recovers classical motivic cohomology, defined for example via Bloch's cycle complex. Our construction uses trace methods and (topological) cyclic homology. As predicted by the behaviour of algebraic \\(K\\)-theory, the motivic cohomology is in general sensitive to singularities, including non-reduced structure, and is not \\(\\mathbb{A}^1\\)-invariant. It nevertheless has good geometric properties, satisfying for example the projective bundle formula and pro cdh descent. Further properties of the theory include a Nesterenko--Suslin comparison isomorphism to Milnor \\(K\\)-theory, and a vanishing range which simultaneously refines Weibel's conjecture about negative \\(K\\)-theory and a vanishing result of Soulé for the Adams eigenspaces of higher algebraic \\(K\\)-groups. We also explore the relation of the theory to algebraic cycles, showing in particular that the Levine--Weibel Chow group of zero cycles on a surface arises as a motivic cohomology group.

We construct motivic power operations on the mod-\\(p\\) motivic cohomology of \\(_p\\)-schemes using a motivic refinement of Nizioł's theorem. The key input is a purity theorem for motivic cohomology established by Levine. Our operations satisfy the expected properties (naturality, Adem relations, and the Cartan formula) for all bidegrees, generalizing previous results of Primozic which were only know along the ``Chow diagonal.'' We offer geometric applications of our construction: 1) an example of non-(quasi-)smoothable algebraic cycle at the characteristic, 2) an answer to the motivic Steenrod problem at the characteristic, 3) a counterexample to the integral version of a crystalline Tate conjecture.

We construct motivic power operations on the mod-\\(p\\) motivic cohomology of \\(_p\\)-schemes using a motivic refinement of Nizioł's theorem. The key input is a purity theorem for motivic cohomology established by Levine. Our operations satisfy the expected properties (naturality, Adem relations, and the Cartan formula) for all bidegrees, generalizing previous results of Primozic which were only know along the ``Chow diagonal.'' We offer geometric applications of our construction: 1) an example of non-(quasi-)smoothable algebraic cycle at the characteristic, 2) an answer to the motivic Steenrod problem at the characteristic, 3) a counterexample to the integral version of a crystalline Tate conjecture.