Motivic Homotopy Category Research Articles

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.

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.

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 + .

Norms in motivic homotopy theory

If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)$, where $\mathcal{S}\mathcal{H}(S)$ is the $\mathbb{P}^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.

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 twisted bivariant theory, extending the formalism of Fulton and MacPherson. We import the tools of Fulton’s intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess of intersection, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor–Witt spectrum recently constructed by Déglise–Fasel, we get a bivariant theory extending the Chow–Witt groups of Barge–Morel, in the same way the higher Chow groups extend the classical Chow groups. As another application we prove a motivic Gauss–Bonnet formula, computing Euler characteristics in the motivic homotopy category.

Isotropic stable motivic homotopy groups of spheres

In this paper we explore the isotropic stable motivic homotopy category constructed from the usual stable motivic homotopy category, following the work of Vishik on isotropic motives (see [28]), by killing anisotropic varieties. In particular, we focus on cohomology operations in the isotropic realm and study the structure of the isotropic Steenrod algebra. Then, we construct an isotropic version of the motivic Adams spectral sequence and apply it to find a complete description of the isotropic stable homotopy groups of the sphere spectrum, which happen to be isomorphic to the Ext-groups of the topological Steenrod algebra. At the end, we will see that this isomorphism is not only additive but respects higher products, completely identifying, as rings with higher structure, the cohomology of the classical Steenrod algebra with isotropic stable homotopy groups of spheres.

Rigidity in étale motivic stable homotopy theory

For a scheme X, denote by SH(X_et^hyp) the stabilization of the hypercompletion of its etale infty-topos, and by SH_et(X) the localization of the stable motivic homotopy category SH(X) at the (desuspensions of) etale hypercovers. For a stable infty-category C, write C_p^comp for the p-completion of C. We prove that under suitable finiteness hypotheses, and assuming that p is invertible on X, the canonical functor e_p^comp: SH(X_et^hyp)_p^comp -> SH_et(X)_p^comp is an equivalence of infty-categories. The primary novelty of our argument is that we use the pro-etale topology to construct directly an invertible object Sptw[1] in SH(X_et^hyp)_p^comp with the property that e_p^comp(Sptw[1]) = Sigma^infty Gm.

On the rational motivic homotopy category

We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of constructible objects, and Grothendieck-Verdier duality for SH_Q. Next, we prove that SH_Q is canonically SL-oriented; we compare SH_Q with the category of rational Milnor-Witt motives; and we relate the rational bivariant A^1-theory to Chow-Witt groups. These results are derived from analogous statements for the minus part of SH[1/2].

Smooth Affine Model for the Framed Correspondences Spectrum

Morel–Voevodsky’s unstable pointed motivic homotopy category H●(k) over an infinite perfect field is considered. For a smooth affine scheme Y over k, a smooth ind-scheme Fl(Y) and an open subscheme El(Y) are constructed for all l > 0, so that the motivic space Fl(Y)/El(Y) is equivalent in H●(k) to the motivic space $$ {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\sum \limits_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left(Y\times {T}^l\right),\kern0.5em T=\left({\mathbbm{A}}^1/{\mathbbm{A}}^1-0\right),\kern0.5em l>0 $$ . The construction is not functorial on the category of affine schemes but is functorial on the category of so-called framed schemes constructed for this purpose.

Künneth formulas for motives and additivity of traces

We prove several Künneth formulas in motivic homotopy categories and deduce a Verdier pairing in these categories following SGA5, which leads to the characteristic class of a constructible motive, an invariant closely related to the Euler-Poincaré characteristic. We prove an additivity property of the Verdier pairing using the language of derivators, following the approach of May and Groth-Ponto-Shulman; using such a result we show that in the presence of a Chow weight structure, the characteristic class for all constructible motives is uniquely characterized by proper covariance, additivity along distinguished triangles, refined Gysin morphisms and Euler classes. In the relative setting, we prove the relative Künneth formulas under some transversality conditions, and define the relative characteristic class.

A transfer morphism for Hermitian K-theory of schemes with involution

In this paper, we consider the Hermitian K-theory of schemes with involution, for which we construct a transfer morphism and prove a version of the dévissage theorem. This theorem is then used to compute the Hermitian K-theory of P1 with involution given by [X:Y]↦[Y:X]. We also prove the C2-equivariant A1-invariance of Hermitian K-theory, which confirms the representability of Hermitian K-theory in the C2-equivariant motivic homotopy category of Heller, Krishna and Østvær [14].

A motivic homotopy theory without $$\mathbb {A}^{1}$$-invariance

In this paper, we continue the program initiated by Kahn-Saito-Yamazaki by constructing and studying an unstable motivic homotopy category with modulus, extending the Morel-Voevodsky construction from smooth schemes over a field $k$ to certain diagrams of schemes. We present this category as a candidate environment for studying representability problems for non $\mathbb{A}^1$-invariant generalized cohomology theories.

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.

Geometrical symmetric powers in the motivic homotopy category

Symmetric powers of quasi-projective schemes can be extended via Kan extensions to geometrical symmetric powers of motivic spaces. In this work, we study geometrical symmetric powers and compare them with various symmetric powers in the unstable and stable A1-homotopy category of schemes over a field.

Affine Grassmannians in $$\mathbb A^1$$ A 1 -homotopy theory

Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* \to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence Omega_Gm G = Gr_G. If k is perfect, we use this to compute the motive M(Omega_Gm G) in DM(k, Z).

Primes and fields in stable motivic homotopy theory

Let F be a field of characteristic different than 2. We establish surjectivity of Balmer's comparison map rho^* from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor-Witt K-theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.

The homogeneous spectrum of Milnor–Witt K-theory

For any field F (of characteristic not equal to 2), we determine the Zariski spectrum of homogeneous prime ideals in K⁎MW(F), the Milnor–Witt K-theory ring of F. As a corollary, we recover Lorenz and Leicht's classical result on prime ideals in the Witt ring of F. Our computation can be seen as a first step in Balmer's program for studying the tensor triangular geometry of the stable motivic homotopy category.

Symmetric powers in abstract homotopy categories

We study symmetric powers in the homotopy categories of abstract closed symmetric monoidal model categories, in both unstable and stable settings. As an outcome, we prove that symmetric powers preserve the Nisnevich and étale homotopy type in the unstable and stable motivic homotopy theories of schemes over a base. More precisely, if f is a weak equivalence of motivic spaces, or a weak equivalence between positively cofibrant motivic spectra, with respect to the Nisnevich or étale topology, then all symmetric powers Symn(f) are weak equivalences too. This gives left derived symmetric powers in the corresponding motivic homotopy categories of schemes over a base, which aggregate into a categorical λ-structures on these categories.