arXiv subjects

Maria Yakerson

Publications and source records attributed to Maria Yakerson.

13 recordsLinked to original sources

arXiv2025

$p$-perfection and group completion of $\mathbb{E}_\infty$-monoids

We study $\mathbb{E}_\infty$-monoids on which a prime $p$ acts invertibly, which we call $p$-perfect, in the non-group-complete situation. In particular, we prove that in many examples, they almost embed in their group-completion. We further study the $p$-perfection functor, and describe it in terms of Quillen's $+$-construction, similarly to group-completion. This gives an alternative description of the $p$-inverted higher algebraic $K$-theory of a ring.

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arXiv2024

An Alternative to Spherical Witt Vectors

We give a direct construction of the ring spectrum of spherical Witt vectors of a perfect $\mathbb{F}_p$-algebra R as the completion of the spherical monoid algebra $\mathbb{S}[R]$ of the multiplicative monoid $(R,\cdot)$ at the ideal $I = \mathrm{fib}(\mathbb{S}[R] \to R)$. This generalizes a construction of Cuntz and Deninger. We also use this to give a description of the category of p-complete modules over the spherical Witt vectors and a universal property for spherical Witt vectors as an $\mathbb{E}_1$-ring.

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arXiv2023

The motivic Adams conjecture

We solve a motivic version of the Adams conjecture with the exponential characteristic of the base field inverted. In the way of the proof we obtain a motivic version of mod k Dold theorem and give a motivic version of Brown's trick studying the homogeneous variety of maximal tori in a general linear group, which turns out to be not stably A1-connected. We also show that the higher motivic stable stems are of bounded torsion.

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arXiv2022

Hermitian K-theory via oriented Gorenstein algebras

We show that the hermitian K-theory space of a commutative ring R can be identified, up to A^1-homotopy, with the group completion of the groupoid of oriented finite Gorenstein R-algebras, i.e., finite locally free R-algebras with trivialized dualizing sheaf. We deduce that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along oriented finite Gorenstein morphisms. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith-Winograd tensor.

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arXiv2022

Twisted K-theory in motivic homotopy theory

In this paper, we study twisted algebraic $K$-theory from a motivic viewpoint. For a smooth variety $X$ over a field of characteristic zero and an Azumaya algebra $\mathcal{A}$ over $X$, we construct the $\mathcal{A}$-twisted motivic spectral sequence, by computing the slices of the motivic twisted algebraic $K$-theory spectrum as a twisted form of motivic cohomology. This generalizes previous results due to Kahn-Levine where $\mathcal{A}$ is assumed to be pulled back from a base field. Our methods use interaction between the slice filtration and birational geometry. Along the way, we prove a representability result, expressing the motivic space of twisted $K$-theory as an extension of the twisted Grassmannian by the sheaf of "twisted integers". This leads to a proof of cdh descent and Milnor excision for twisted homotopy $K$-theory.

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arXiv2021

The Hilbert scheme of infinite affine space and algebraic K-theory

We study the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ from an $\mathbb{A}^1$-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ is $\mathbb{A}^1$-equivalent to the Grassmannian of $(d-1)$-planes in $\mathbb{A}^\infty$. We then describe the $\mathbb{A}^1$-homotopy type of $\mathrm{Hilb}_d(\mathbb{A}^n)$ in a range, for $n$ large compared to $d$. For example, we compute the integral cohomology of $\mathrm{Hilb}_d(\mathbb{A}^n)(\mathbb{C})$ in a range. We also deduce that the forgetful map $\mathrm{FFlat}\to\mathrm{Vect}$ from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an $\mathbb{A}^1$-equivalence after group completion. This implies that the moduli stack $\mathrm{FFlat}$, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum $\mathrm{kgl}$ representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the $\mathrm{kgl}$-homology of smooth proper schemes over a perfect field.

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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

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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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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arXiv2019

Towards conservativity of $\mathbb{G}_m$-stabilization

We study the interplay of the homotopy coniveau tower, the Rost-Schmid complex of a strictly homotopy invariant sheaf, and homotopy modules. For a strictly homotopy invariant sheaf $M$, smooth $k$-scheme $X$ and $q \geqslant 0$ we construct a novel cycle complex $C^*(X, M, q)$ and we prove that in favorable cases, $C^*(X, M, q)$ is equivalent to the homotopy coniveau tower $M^{(q)}(X)$. To do so we establish moving lemmas for the Rost-Schmid complex. As an application we deduce a cycle complex model for Milnor-Witt motivic cohomology. Furthermore we prove that if $M$ is a strictly homotopy invariant sheaf, then $M_{-2}$ is a homotopy module. Finally we conjecture that for $q>0$, $\underlineπ_0(M^{(q)})$ is a homotopy module, explain the significance of this conjecture for studying conservativity properties of the $\mathbb{G}_m$-stabilization functor $\mathcal{SH}^{S^1}\!(k) \to \mathcal{SH}(k)$, and provide some evidence for the conjecture.

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arXiv2019

The unit map of the algebraic special linear cobordism spectrum

In joint work with Elmanto, Hoyois, Khan and Sosnilo, we computed infinite $\mathbb{P}^1$-loop spaces of motivic Thom spectra, using the technique of framed correspondences. This result allows us to express non-negative $\mathbb{G}_m$-homotopy groups of motivic Thom spectra in terms of geometric generators and relations. Using this explicit description, we show that the unit map of the algebraic special linear cobordism spectrum induces an isomorphism on $\mathbb{G}_m$-homotopy sheaves.

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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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arXiv2016

Algebraic K-theory of varieties $SL_{2n}/Sp_{2n}$, $E_6/F_4$ and their twisted forms

Let $SL_{2n}$, $Sp_{2n}$, $E_6 = G^{sc}(E_6)$, $F_4 = G(F_4)$ be simply connected split algebraic groups over an arbitrary field $F$. Algebraic K-theory of affine homogeneous varieties $SL_{2n}/Sp_{2n}$ and $E_6/F_4$ is computed. Moreover, explicit elements that generate $K_*(SL_{2n}/Sp_{2n})$ and $K_*(E_6/F_4)$ as $K_*(F)$-algebras are provided. For some twisted forms of these varieties K-theory is also computed.

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