Motivic Homotopy Research Articles

The multiplicative structures on motivic homotopy groups

The multiplicative structures on motivic homotopy groups

Motivic homotopy theory of algebraic stacks

Motivic homotopy theory of algebraic stacks

Tensor-Triangular Geometry and Interactions

The workshop brought together experts in a rapidly growing field of tensor triangular geometry highlighting applications to and techniques coming from homotopy theory, algebraic geometry, modular representation theory, motivic homotopy theory and noncommutative algebra.

Trace maps in motivic homotopy and local terms

We define a trace map for every cohomological correspondence in the motivic stable homotopy category over a general base scheme, which takes values in the twisted bivariant groups. Local contributions to the trace map give rise to quadratic refinements of the classical local terms, and some A 1 \mathbb {A}^1 -enumerative invariants, such as the local A 1 \mathbb {A}^1 -Brouwer degree and the Euler class with support, can be interpreted as local terms. We prove an analogue of a theorem of Varshavsky, which states that for a contracting correspondence, the local terms agree with the naive local terms.

Triangulated categories of framed bispectra and framed motives

An alternative approach to the classical Morel–Voevodsky stable motivic homotopy theory S H ( k ) SH(k) is suggested. The triangulated category of framed bispectra S H nis fr ( k ) SH_{\operatorname {nis}}^{\operatorname {fr}}(k) and effective framed bispectra S H nis fr , eff ( k ) SH_{\operatorname {nis}}^{\operatorname {fr},\operatorname {eff}}(k) are introduced in the paper. Both triangulated categories only involve Nisnevich local equivalences and have nothing to do with any kind of motivic equivalences. It is shown that S H nis fr ( k ) SH_{\operatorname {nis}}^{\operatorname {fr}}(k) and S H nis fr , eff ( k ) SH_{\operatorname {nis}}^{\operatorname {fr},\operatorname {eff}}(k) recover classical Morel–Voevodsky triangulated categories of bispectra S H ( k ) SH(k) and effective bispectra S H eff ( k ) SH^{\operatorname {eff}}(k) respectively. Also, S H ( k ) SH(k) and S H eff ( k ) SH^{\operatorname {eff}}(k) are recovered as the triangulated category of framed motivic spectral functors S H S 1 fr [ F r 0 ( k ) ] SH_{S^1}^{\operatorname {fr}}[\mathcal {F}r_0(k)] and the triangulated category of framed motives S H fr ( k ) \mathcal {SH}^{\operatorname {fr}}(k) constructed in the paper.

Periodic self maps and thick ideals in the stable motivic homotopy category over $${\mathbb {C}}$$ at odd primes

Periodic self maps and thick ideals in the stable motivic homotopy category over $${\mathbb {C}}$$ at odd primes

The cdh-local motivic homotopy category

We construct a cdh-local motivic homotopy category SHcdh(S) over an arbitrary base scheme S, and show that there is a canonical equivalence SHcdh(S)≃SH(S). We learned this result from D.-C. Cisinski.

Correspondences and stable homotopy theory

AbstractA general method of producing correspondences and spectral categories out of symmetric ring objects in general categories is given. As an application, stable homotopy theory of spectra is recovered from modules over a commutative symmetric ring spectrum defined in terms of framed correspondences over an algebraically closed field. Another application recovers stable motivic homotopy theory from spectral modules over associated spectral categories.

Topologie

The lectures in the workshop covered various topics in modern topology, including algebraic and geometric topology, homotopy theory, geometric group theory, and manifold topology, as well as connections to neighboring areas, most prominently symplectic topology/geometry. The following current research topics received more attention during the workshop: manifolds and K-theory, symplectic topology and Floer homology, generalizations of hyperbolic techniques in geometric group theory, and equivariant and motivic homotopy theory. The aim of the various topics was to foster communication and provide chances for participants to see and experience driving questions and important methods in nearby fields within the realm of topology.

Geometric Models for Algebraic Suspensions

Abstract We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $X$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the ${\mathbb P}^{1}$ suspension of $X$; we also analyze a host of variations on this observation. Our approach yields many examples of ${\mathbb A}^{1}$-$(n-1)$-connected smooth affine $2n$-folds and strictly quasi-affine ${\mathbb A}^{1}$-contractible smooth schemes.

Conics meeting eight lines over perfect fields

Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler number and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This provides a weighted count of the number of plane conics meeting 8 general lines, where the weight of each conic is determined the geometry of its intersections with the 8 given lines. As a corollary, real conics meeting 8 general lines come in two families of equal size.

Stable homotopy groups of spheres: from dimension 0 to 90

Using techniques in motivic homotopy theory, especially the theorem of Gheorghe, the second and the third author on the isomorphism between motivic Adams spectral sequence for Ctau and the algebraic Novikov spectral sequence for BP_{*}, we compute the classical and motivic stable homotopy groups of spheres from dimension 0 to 90, except for some carefully enumerated uncertainties.

Transfers on Milnor-Witt K-theory

We study the existence of transfers on a generalization of Milnor K-theory called Milnor-Witt K-theory. We give a new proof of the fact that Milnor-Witt K-theory has geometric transfers. Moreover, we explain how our proof yields a simplification of Morel's conjecture about Bass-Tate-Kato transfers on contracted homotopy sheaves in the context of motivic homotopy theory.

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 combines $\Gamma$-spaces and Voevodsky's framed correspondences into the concept of framed motivic $\Gamma$-spaces; these are continuous or enriched functors of two variables that take values in framed motivic spaces. We craft proofs of our main results by imposing further axioms on framed motivic $\Gamma$-spaces such as a Segal condition for simplicial Nisnevich sheaves, cancellation, $\mathbb{A}^1$- and $\sigma$-invariance, Nisnevich excision, Suslin contractibility, and grouplikeness. This adds to the discussion in the literature on coexisting points of view on the $\mathbb{A}^1$-homotopy theory of algebraic varieties.

Stable 𝔸1-connectivity over a base

Abstract Morel’s stable connectivity theorem states the vanishing of the sheaves of the negative motivic homotopy groups π ¯ i s ⁢ ( Y ) {\underline{\pi}^{s}_{i}(Y)} and π ¯ i + j , j s ⁢ ( Y ) {\underline{\pi}^{s}_{i+j,j}(Y)} , i < 0 {i<0} , in the stable motivic homotopy categories 𝐒𝐇 S 1 ⁢ ( k ) {\mathbf{SH}^{S^{1}}(k)} and 𝐒𝐇 ⁢ ( k ) {\mathbf{SH}(k)} for an arbitrary smooth scheme Y over a field k. Originally the same property was conjectured in the relative case over a base scheme S. In view of Ayoub’s conterexamples the modified version of the conjecture states the vanishing of stable motivic homotopy groups π ¯ i s ⁢ ( Y ) {\underline{\pi}^{s}_{i}(Y)} (and π ¯ i + j , j s ⁢ ( Y ) {\underline{\pi}^{s}_{i+j,j}(Y)} ) for i < - d {i<-d} , where d = dim ⁡ S {d=\dim S} is the Krull dimension. The latter version of the conjecture is proven over noetherian domains of finite Krull dimension under the assumption that residue fields of the base scheme are infinite. This is the result by J. Schmidt and F. Strunk for Dedekind schemes case, and the result by N. Deshmukh, A. Hogadi, G. Kulkarni and S. Yadavand for the case of noetherian domains of an arbitrary dimension. In the article, we prove the result for any locally noetharian base scheme of finite Krull dimension without the assumption on the residue fields, in particular for 𝐒𝐇 S 1 ⁢ ( ℤ ) {\mathbf{SH}^{S^{1}}(\mathbb{Z})} and 𝐒𝐇 ⁢ ( ℤ ) {\mathbf{SH}(\mathbb{Z})} . In the appendix, we modify the arguments used for the main result to obtain the independent proof of Gabber’s Presentation Lemma over finite fields.

Motivic Rigidity for Smooth Affine Henselian Pairs over a Field

Abstract Let $Z\to X$ be a closed immersion of smooth affine schemes over a field $k$, and let $X^h_Z$ denote the henselisation of $X$ along $Z$. We prove that $E(X^h_Z)\simeq E(Z)$ for every additive presheaf $E\colon \textbf {SH}(k)^{\textrm {op}}\to \textrm {Ab}$ on the stable motivic homotopy category over $k$ that is $l_\varepsilon $-torsion or $l$-torsion, where $l\in \mathbb Z$ is coprime to $\operatorname {char} k$, and $l_\varepsilon =\sum _{i=1}^n \langle (-1)^i \rangle $. More generally, the isomorphism holds for any homotopy invariant $l_\varepsilon $-torsion or $l$-torsion linear $\sigma $-quasi-stable framed additive presheaf on $\textrm {Sm}_{k}$. This generalises the result known earlier for local schemes. We prove the above isomorphism by constructing (stable) ${\mathbb {A}}^1$-homotopies of motivic spaces via algebraic geometry. To achieve this, we replace Quillen’s trick with an alternative and more general construction that provides relative curves required in our setting.

Motives and homotopy theory in logarithmic geometry

This document is a short user’s guide to the theory of motives and homotopy theory in the setting of logarithmic geometry. We review some of the basic ideas and results in relation to other works on motives with modulus, motivic homotopy theory, and reciprocity sheaves.

Detecting motivic equivalences with motivic homology

Let k k be a field, let R R be a commutative ring, and assume the exponential characteristic of k k is invertible in R R . In this note, we prove that isomorphisms in Voevodsky’s triangulated category of motives D M ( k ; R ) \mathcal {DM}(k;R) are detected by motivic homology groups of base changes to all separable finitely generated field extensions of k k . It then follows from previous conservativity results that these motivic homology groups detect isomorphisms between certain spaces in the pointed motivic homotopy category H ( k ) ∗ \mathcal {H}(k)_* .

On some finiteness results in real étale cohomology

We show that for quasi-compact quasi-separated schemes of finite dimension, the constructibility condition in real \'etale cohomology agrees with a notion of constructibility arising naturally from topology. As application we prove that the derived direct image functor preserves constructibility under some assumptions, and compute the Grothendieck group of the constructible rational stable motivic homotopy category for reasonable schemes. We prove the generic base change property for constructible real \'etale sheaves, and deduce the same property for rational motivic spectra and $b$-sheaves.

Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category SH ( k ) in terms of Voevodsky's framed correspondences. In particular, the motivically fibrant Ω-resolution in positive degrees of the motivic suspension spectrum Σ P 1 ∞ X + , where X + = X ⨿ ⁎ , for a smooth scheme X ∈ Sm k over an infinite perfect field k , is computed. The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres ( A l × X ) / ( ( A l − 0 ) × X ) , X ∈ Sm k , is one of ingredients in the theory. In the article we extend this result to the case of a pair ( X , U ) given by a smooth affine variety X over k and an open subscheme U ⊂ X . The result gives an explicit motivically fibrant Ω-resolution in positive degrees for the motivic suspension spectrum Σ P 1 ∞ ( X + / U + ) of the quotient-sheaf X + / U + .