Motivic infinite loop spaces

@article{Elmanto2021MotivicIL,
  title={Motivic infinite loop spaces},
  author={Elden Elmanto and Marc Hoyois and Adeel A. Khan and Vladimir A. Sosnilo and Maria Yakerson},
  journal={Cambridge Journal of Mathematics},
  year={2021}
}
We prove a recognition principle for motivic infinite P1-loop spaces over an infinite 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… 
Fundamental classes in motivic homotopy theory
We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of
Framed motivic $\Gamma$-spaces
We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach
Bivariant theories in motivic stable homotopy
The purpose of this work is to study the notion of bivariant theory introduced by Fulton and MacPherson in the context of motivic stable homotopy theory, and more generally in the broader framework
Framed transfers and motivic fundamental classes
We relate the recognition principle for infinite P1 ‐loop spaces to the theory of motivic fundamental classes of Déglise, Jin and Khan. We first compare two kinds of transfers that are naturally
Generalized cohomology theories for algebraic stacks
We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendieck’s six operations. Objects in this
Stable motivic invariants are eventually étale local
In this paper we prove a Thomason-style descent theorem for the $\rho$-complete sphere spectrum. In particular, we deduce a very general etale descent result for torsion, $\rho$-complete motivic
η$\eta$‐Periodic motivic stable homotopy theory over Dedekind domains
We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer--Witt K-theory (among others) to mixed characteristic Dedekind schemes on
On the effectivity of spectra representing motivic cohomology theories
Let k be an infinite perfect field. We provide a general criterion for a spectrum in the stable homotopy category over k to be effective, i.e. to be in the localizing subcategory generated by the
Cancellation theorem for motivic spaces with finite flat transfers
We show that the category of motivic spaces with transfers along finite flat morphisms, over a perfect field, satisfies all the properties we have come to expect of good categories of motives. In
Notes on motivic infinite loop space theory
In fall of 2019, the Thursday Seminar at Harvard University studied motivic infinite loop space theory. As part of this, the authors gave a series of talks outlining the main theorems of the theory,
...
...

References

SHOWING 1-10 OF 80 REFERENCES
Fundamental classes in motivic homotopy theory
We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of
The intrinsic stable normal cone
We construct an analog of the intrinsic normal cone of Behrend-Fantechi in the equivariant motivic stable homotopy category over a base-scheme B and construct a fundament class in E-cohomology for
Motivic twisted K-theory
This paper sets out basic properties of motivic twisted K-theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K-theory is defined in terms of such motivic
Framed transfers and motivic fundamental classes
We relate the recognition principle for infinite P1 ‐loop spaces to the theory of motivic fundamental classes of Déglise, Jin and Khan. We first compare two kinds of transfers that are naturally
Framed and MW-transfers for homotopy modules
In the paper we use the theory of framed correspondences to construct Milnor–Witt transfers on homotopy modules. As a consequence we identify the zeroth stable $${\mathbb {A}}^1$$A1-homotopy sheaf of
Cdh descent in equivariant homotopy K-theory
We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the
A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the "Euler characteristic integral" of a
Framed correspondences and the zeroth stable motivic homotopy group in odd characteristic
We extend the results of G.~Garkusha and I.~Panin on framed motives of algebraic varieties [4] to the case of a finite base field, and extend the computation of the zeroth cohomology group
...
...