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Elden Elmanto

Publications and source records attributed to Elden Elmanto.

36 records · Page 2Linked to original sources
arXiv2021

Milnor excision for motivic spectra

We prove that the $\infty$-category of motivic spectra satisfies Milnor excision: if $A\to B$ is a morphism of commutative rings sending an ideal $I\subset A$ isomorphically onto an ideal of $B$, then a motivic spectrum over $A$ is equivalent to a pair of motivic spectra over $B$ and $A/I$ that are identified over $B/IB$. Consequently, any cohomology theory represented by a motivic spectrum satisfies Milnor excision. We also prove Milnor excision for Ayoub's étale motives over schemes of finite virtual cohomological dimension.

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arXiv2021

Motivic infinite loop spaces

We prove a recognition principle for motivic infinite P1-loop spaces over a perfect field. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of E-infinity-spaces. A framed motivic space is a motivic space equipped with transfers along finite syntomic morphisms with trivialized cotangent complex in K-theory. Our main result is that grouplike framed motivic spaces are equivalent to the full subcategory of motivic spectra generated under colimits by suspension spectra. As a consequence, we deduce some representability results for suspension spectra of smooth varieties, and in particular for the motivic sphere spectrum, in terms of Hilbert schemes of points in affine spaces.

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arXiv2021

Voevodsky's slice conjectures via Hilbert schemes

Using recent development in motivic infinite loop space theory, we offer short and conceptual reproofs of some conjectures of Voevodsky's on the slice filtration using the birational geometry of Hilbert schemes. The original proofs were due to Marc Levine using very different methods, namely, the homotopy coniveau tower.

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arXiv2021

On nilpotent extensions of $\infty$-categories and the cyclotomic trace

We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty$-categories) for additive $\infty$-categories, (2) define the notion of nilpotent extensions for suitable $\infty$-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas-Goodwillie-McCarthy theorem for stable $\infty$-categories which are not monogenically generated (such as the stable $\infty$-category of Voevodsky's motives or the stable $\infty$-category of perfect complexes on some algebraic stacks). The key input in our paper is Bondarko's notion of weight structures which provides a "ring-with-many-objects" analog of a connective $\mathbb{E}_1$-ring spectrum. As applications, we prove cdh descent results for truncating invariants of stacks extending the work of Hoyois-Krishna for homotopy $K$-theory, and establish new cases of Blanc's lattice conjecture.

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arXiv2021

On the infinite loop spaces of algebraic cobordism and the motivic sphere

We obtain geometric models for the infinite loop spaces of the motivic spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are motivically equivalent to $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{lci}(\mathbb{A}^\infty)^+$, $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{or}(\mathbb{A}^\infty)^+$, and $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{fr}(\mathbb{A}^\infty)^+$, respectively, where $\mathrm{Hilb}_d^\mathrm{lci}(\mathbb{A}^n)$ (resp. $\mathrm{Hilb}_d^\mathrm{or}(\mathbb{A}^n)$, $\mathrm{Hilb}_d^\mathrm{fr}(\mathbb{A}^n)$) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$, and $+$ is Quillen's plus construction. Moreover, we show that the plus construction is redundant in positive characteristic.

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arXiv2021

$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors

In this paper, we study the Nisnevich sheafification $\mathcal{H}^1_{\acute{e}t}(G)$ 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 $\mathcal{H}^1_{\acute{e}t}(G)$ 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 $\mathcal{H}^1_{\acute{e}t}(G)$ with the sheaf of $\mathbb{A}^1$-connected components of the classifying space ${\rm B}_{\acute{e}t}G$. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of $\mathbb{A}^1$-connected components in terms of unramified $G$-torsors over function fields whenever Nisnevich-local purity holds for $G$-torsors.

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arXiv2021

Scheiderer motives and equivariant higher topos theory

We give an algebro-geometric interpretation of $C_2$-equivariant stable homotopy theory by means of the $b$-topology introduced by Claus Scheiderer in his study of $2$-torsion phenomena in étale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a $\infty$-topos $\mathscr{X}$ with $G$-action a presentable stable $\infty$-category $\mathrm{Sp}^G(\mathscr{X})$, which recovers the $\infty$-category $\mathrm{Sp}^G$ of genuine $G$-spectra when $\mathscr{X}$ is the terminal $G$-$\infty$-topos. Given a scheme $X$ with $1/2 \in \mathcal{O}_X$, our construction then specializes to produce an $\infty$-category $\mathrm{Sp}^{C_2}_b(X)$ of "$b$-sheaves with transfers" as $b$-sheaves of spectra on the small étale site of $X$ equipped with certain transfers along the extension $X[i] \rightarrow X$; if $X$ is the spectrum of a real closed field, then $\mathrm{Sp}^{C_2}_b(X)$ recovers $\mathrm{Sp}^{C_2}$. On a large class of schemes, we prove that, after $p$-completion, our construction assembles into a premotivic functor satisfying the full six functors formalism. We then introduce the $b$-variant $\mathrm{SH}_b(X)$ of the $\infty$-category $\mathrm{SH}(X)$ of motivic spectra over $X$ (in the sense of Morel-Voevodsky), and produce a natural equivalence of $\infty$-categories $\mathrm{SH}_b(X)^{\wedge}_p \simeq \mathrm{Sp}^{C_2}_b(X)^{\wedge}_p$ through amalgamating the étale and real étale motivic rigidity theorems of Tom Bachmann. This involves a purely algebro-geometric construction of the $C_2$-Tate construction, which may be of independent interest. Finally, as applications, we deduce a "$b$-rigidity" theorem, use the Segal conjecture to show étale descent of the $2$-complete $b$-motivic sphere spectrum, and construct a parametrized version of the $C_2$-Betti realization functor of Heller-Ormsby.

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arXiv2021

Descent for semiorthogonal decompositions

We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of derived categories and even marked filtrations, where one descends not only admissible subcategories but also preferred objects.

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arXiv2020

Modules over algebraic cobordism

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, 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 $Ω^\infty_{\mathbb{P}^1}\mathrm{MGL}$ is the $\mathbb{A}^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$, $Ω^\infty_{\mathbb{P}^1} Σ^n_{\mathbb{P}^1} \mathrm{MGL}$ is the $\mathbb{A}^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$.

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arXiv2020

THH and TC are (very) far from being homotopy functors

We compute the $\mathbb{A}^1$-localization of several invariants of schemes namely, topological Hochschild homology ($\mathrm{THH}$), topological cyclic homology ($\mathrm{TC}$) and topological periodic cyclic homology ($\mathrm{TP}$). This procedure is quite brutal and kills the completed versions of most of these invariants. The main ingredient for the vanishing statements is the vanishing of $\mathbb{A}^1$-localization of de Rham cohomology (and, eventually, crystalline cohomology) in positive characteristics.

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arXiv2020

Cdh descent, cdarc descent, and Milnor excision

We give necessary and sufficient conditions for a cdh sheaf to satisfy Milnor excision, following ideas of Bhatt and Mathew. Along the way, we show that the cdh infinity-topos of a quasi-compact quasi-separated scheme of finite valuative dimension is hypercomplete, extending a theorem of Voevodsky to nonnoetherian schemes. As an application, we show that if E is a motivic spectrum over a field k which is n-torsion for some n invertible in k, then the cohomology theory on k-schemes defined by E satisfies Milnor excision.

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arXiv2020

A descent view on Mitchell's theorem

In this short note, we given a new proof of Mitchell's theorem that $L_{T\left(n\right)} K(Z) \cong 0$ for $n \geq 2$. Instead of reducing the problem to delicate representation theory, we use recently established hyperdescent technology for chromatically-localized algebraic K-theory.

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arXiv2019

Framed transfers and motivic fundamental classes

We relate the recognition principle for infinite $\mathbf P^1$-loop spaces to the theory of motivic fundamental classes of Déglise, Jin, and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with $\mathbf A^n/(\mathbf A^n-0)$, and the Gysin transfers defined via Verdier's deformation to the normal cone. We then introduce the category of finite E-correspondences for E a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calmès and Fasel's category of finite Milnor-Witt correspondences. Using the formalism of fundamental classes, we show that the natural functor from the category of framed correspondences to the category of E-module spectra factors through the category of finite E-correspondences.

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arXiv2019

Perfection in motivic homotopy theory

We prove a topological invariance statement for the Morel-Voevodsky motivic homotopy category, up to inverting exponential characteristics of residue fields. This implies in particular that SH[1/p] of characteristic p>0 schemes is invariant under passing to perfections. Among other applications we prove Grothendieck-Verdier duality in this context.

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arXiv2019

On Modules Over Motivic Ring Spectra

In this note, we provide an axiomatic framework that characterizes the stable $\infty$-categories that are module categories over a motivic spectrum. This is done by invoking Lurie's $\infty$-categorical version of the Barr--Beck theorem. As an application, this gives an alternative approach to Röndigs and Østvær's theorem relating Voevodsky's motives with modules over motivic cohomology, and to Garkusha's extension of Röndigs and Østvær's result to general correspondence categories, including the category of Milnor-Witt correspondences in the sense of Calmès and Fasel. We also extend these comparison results to regular Noetherian schemes over a field (after inverting the residue characteristic), following the methods of Cisinski and Déglise.

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arXiv2018

Relative Étale Realizations of Motivic Spaces and Dwyer-Friedlander $K$-Theory of Noncommutative Schemes

In this paper, we construct a refined, relative version of the étale 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 the $\infty$-category of $p$-profinite spaces, where $p$ is a prime which is invertible in all residue fields of $S$. In the first part of this paper, we refine the target of this functor to an $\infty$-category where $p$-profinite spaces is a further completion. Roughly speaking, this $\infty$-category is generated under cofiltered limits by those spaces whose associated "local system" on $S$ is $A^1$-invariant. We then construct a new, relative version of their étale realization functor which takes into account the geometry and arithmetic of the base scheme $S$. For example, when $S$ is the spectrum of a field $k$, our functor lands in a certain $\infty$-category equivariant for the absolute Galois group. Our construction relies on a relative version of étale homotopy types in the sense of Artin-Mazur-Friedlander, which we also develop in some detail, expanding on previous work of Barnea-Harpaz-Schlank. We then stabilize our functor, in the $S^1$-direction, to produce an étale realization functor for motivic $S^1$-spectra (in other words, Nisnevich sheaves of spectra which are $A^1$-invariant). To this end, we also develop an $\infty$-categorical version of the theory of profinite spectra, first explored by Quick. As an application, we refine the construction of the étale $K$-theory of Dwyer and Friedlander, and define its non-commutative extension. This latter invariant should be seen as an $\ell$-adic analog of Blanc's theory of semi-topological $K$-theory of non-commutative schemes. We then formulate and prove an analog of Blanc's conjecture on the torsion part of this theory, generalizing the work of Antieau and Heller.

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arXiv2016

A primer for unstable motivic homotopy theory

In this expository article, we give the foundations, basic facts, and first examples of unstable motivic homotopy theory with a view towards the approach of Asok-Fasel to the classification of vector bundles on smooth complex affine varieties. Our focus is on making these techniques more accessible to algebraic geometers.

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