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motivic homotopy theory

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Motivic cohomology

Contents

Idea

Motivic homotopy theory or A 1\mathbf{A}^1-homotopy theory is the homotopy theory of smooth schemes, where the affine line A 1\mathbf{A}^1 plays the role of the interval. Hence what is called the motivic homotopy category or the 𝔸 1\mathbb{A}^1-homotopy category bears the same relation to smooth varieties that the ordinary homotopy category Ho(Top)Ho(Top) bears to smooth manifolds.

Both are special cases of a homotopy theory induced by any sufficiently well-behaved interval object II in a site CC via localization at that object. Ordinary homotopy theory is obtained by taking CC to be the site of smooth manifolds and II to be the real line \mathbb{R}, and 𝔸 1\mathbb{A}^1-homotopy theory over a Noetherian scheme SS is obtained when CC is the Nisnevich site of smooth schemes of finite type over SS and

(1)I𝔸 1 I \coloneqq \mathbb{A}^1

is the standard affine line in CC.

As for the standard homotopy theory, one can furthermore pass to spectrum objects and consider the stable homotopy category. In the following we first discuss

and then

The unstable motivic homotopy category

Let SS be a fixed Noetherian base scheme, and let Sm/SSm/S be the category of smooth schemes of finite type over SS.

Definition

The motivic homotopy category H(S)\mathrm{H}(S) over SS is the homotopy localization at the affine line 𝔸 1\mathbb{A}^1 (1) of the (∞,1)-topos of (∞,1)-sheaves on the Nisnevich site Sm/SSm/S. Objects of H(S)\mathrm{H}(S) are called motivic spaces.

Thus, a motivic space over SS is an (∞,1)-presheaf FF on Sm/SSm/S such that

  • FF is an (∞,1)-sheaf for the Nisnevich topology
  • FF is 𝔸 1\mathbb{A}^1-homotopy invariant: for every XSm/SX\in Sm/S, the projection X×𝔸 1XX\times\mathbb{A}^1\to X induces an equivalence F(X)F(X×𝔸 1F(X)\simeq F(X\times\mathbb{A}^1).

As for any homotopy localization, the inclusion H(S)PSh(Sm/S)\mathrm{H}(S)\subset PSh(Sm/S) admits a left adjoint localization functor, and one can show that it preserves finite (∞,1)-products.

The (∞,1)-category H(S)\mathrm{H}(S) is a locally presentable and locally cartesian closed (∞,1)-category. However, it is not an (∞,1)-topos (see Remark 3.5 in Spitzweck-Østvær, Motivic twisted K-theory, pdf)

Motivic spheres

The Tate sphere is the pointed object of H(S)\mathrm{H}(S) defined by

T:=𝔸 1/𝔾 m, T {:=} \mathbb{A}^1/\mathbb{G}_m,

that is, TT is the homotopy cofiber of the inclusion 𝔾 m𝔸 1\mathbb{G}_m\hookrightarrow \mathbb{A}^1. More generally, any algebraic vector bundle VV on SS (or, equivalently, any locally free sheaf of finite rank on SS) has an associated motivic sphere given by its Thom space:

S V:=V/V ×. S^V {:=} V/V^\times.

Thus, T=S 𝔸 1T=S^{\mathbb{A}^1}.

A crucial observation is that TT is the suspension of 𝔾 m\mathbb{G}_m (pointed at 11):

TS 1𝔾 m. T \simeq S^1\wedge\mathbb{G}_m.

Indeed, this follows from the definition of TT and the fact that (𝔸 1,1)(\mathbb{A}^1,1) is contractible as a pointed motivic space. It is common to write, for pq0p\geq q\geq 0,

S p,q:=S pq𝔾 m q. S^{p,q} {:=} S^{p-q}\wedge \mathbb{G}_m^{\wedge q}.
Proposition

There is a canonical equivalence of pointed motivic spaces T( 1,1)T\simeq (\mathbb{P}^1,1).

Proof

The cartesian square

𝔾 m 𝔸 1 𝔸 1 1 \array{ \mathbb{G}_m &\stackrel{}{\to}& \mathbb{A}^1 \\ \downarrow && \downarrow \\ \mathbb{A}^1 &\stackrel{}{\to}& \mathbb{P}^1 }

becomes homotopy cocartesian in the Nisnevich (even Zariski) (∞,1)-topos. By pointing all schemes at 11 and using that (𝔸 1,1)(\mathbb{A}^1,1) is contractible, we deduce that ( 1,1)S 1𝔾 mT.(\mathbb{P}^1,1)\simeq S^1\wedge \mathbb{G}_m\simeq T.

The stable motivic homotopy category

Definition

The stable motivic homotopy category SH(S)SH(S) over SS is the inverse limit of the tower of (∞,1)-categories

Ω TH *(S)Ω TH *(S)Ω TH *(S), \dots \stackrel{\Omega_T}{\to} \mathrm{H}_*(S) \stackrel{\Omega_T}{\to} \mathrm{H}_*(S) \stackrel{\Omega_T}{\to} \mathrm{H}_*(S),

where H(S)H(S) is the ordinary motivic homotopy category from def. , and where Ω T:=Hom(T,)\Omega_T {:=}Hom(T, -). An object of the stable motivic homotopy category is called a motivic spectrum (or TT-spectrum).

Thus, a motivic spectrum EE is a sequence of pointed motivic spaces (E 0,E 1,E 2)(E_0,E_1,E_2\dots) together with equivalences

Ω TE i+1E i. \Omega_T E_{i+1}\simeq E_i.

Since T 1T\simeq \mathbb{P}^1, we could equivalently use 1\mathbb{P}^1 instead of TT in the above definition.

Since TS 1𝔾 mT\simeq S^1\wedge \mathbb{G}_m, SH(S)SH(S) is indeed a stable (∞,1)-category.

Symmetric monoidal structure and universal property

The functor Ω T :SH(S)H *(S)\Omega^\infty_T\colon SH(S)\to \mathrm{H}_*(S) sending a motivic spectrum EE to its first component E 0E_0 admits a left adjoint Σ T \Sigma_T^\infty. One can then equip the category SH(S)SH(S) with the structure of a symmetric monoidal (∞,1)-category in such a way that the (∞,1)-functor Σ T \Sigma_T^\infty can be promoted to a symmetric monoidal (∞,1)-functor. As such, SH(S)SH(S) is characterized by a universal property:

Proposition

Let 𝒞\mathcal{C} be a locally presentable symmetric monoidal (∞,1)-category. The (∞,1)-functor

Fun ,L(SH(S),𝒞)Fun (Sm/S,𝒞),FFΣ T () + Fun^{\otimes, L}(SH(S),\mathcal{C}) \to Fun^{\otimes} (Sm/S, \mathcal{C}), \quad F\mapsto F\circ \Sigma_T^\infty(-)_+

(where \otimes means “symmetric monoidal” and LL means “colimit-preserving”) is fully faithful and its essential image consists of the symmetric monoidal (∞,1)-functors F:Sm/S𝒞F\colon Sm/S\to \mathcal{C} satisfying:

  • Nisnevich excision
  • 𝔸 1\mathbb{A}^1-homotopy invariance
  • TT-stability: the homotopy cofiber of F(𝔾 m)F(𝔸 1)F(\mathbb{G}_m)\to F(\mathbb{A}^1) is \otimes-invertible.

This is (Robalo, Corollary 5.11).

Stable motivic spheres

Because TS 2,1T\simeq S^{2,1}, the stable motivic spheres S p,qS^{p,q} are defined for all p,qp,q\in\mathbb{Z}.

All the other motivic spheres S VS^V, for VV a vector bundle on SS, also become invertible in SH(S)SH(S). In fact, the Picard ∞-groupoid of SH(S)SH(S) receives a map from the algebraic K-theory of SS. These invertible objects are all exotic in the sense that they are not equivalent to any of the “categorical” spheres S n=Σ nΣ T S +S^n = \Sigma^n\Sigma^\infty_T S_+. These exotic spheres play an important rôle in the formalism of six operations in stable motivic homotopy theory (see Ayoub).

Cohomology theories

Any motivic spectrum ESH(S)E\in SH(S) gives rise to a bigraded cohomology theory for smooth SS-schemes and more generally for motivic spaces:

E p,q(X):=[Σ T X +,Σ p,qE], E^{p,q}(X) {:=} [\Sigma^\infty_T X_+, \Sigma^{p,q} E],

as well as a bigraded homology theory:

E p,q(X):=[S p,q,Σ T X +E]. E_{p,q}(X) {:=} [S^{p,q}, \Sigma^\infty_T X_+ \wedge E].

Main features

The six operations

The categories SH(S)SH(S) for varying base scheme SS support a formalism of six operations. This means that to every morphism of schemes f:XYf: X\to Y is associated an (inverse image \dashv direct image)-adjunction

f *:SH(Y)SH(X):f * f^* : SH(Y) \to SH(X) : f_*

and, if ff is separated of finite type, a (direct image with compact support \dashv exceptional inverse image)-adjunction

f !:SH(X)SH(Y):f !, f_! : SH(X) \to SH(Y): f^!,

satisfying the properties listed at six operations. For more details see Ayoub.

Stable homotopy functors. To construct the six operations for SHSH, Voevodsky introduced an axiomatic setting which also subsumes the classical case of étale cohomology. A stable homotopy functor is a contravariant (∞,1)-functor

D:Schemes opPrStab(,1)Cat, D\colon Schemes^{op} \to PrStab (\infty,1) Cat,

from some category of schemes to the (∞,1)-category PrStab(,1)CatPrStab (\infty,1) Cat of locally presentable stable (∞,1)-categories and colimit-preserving exact functors, satisfying the following axioms (for ff a morphism of schemes, we denote D(f)D(f) by f *f^* and its right adjoint by f *f_*):

  1. D()=0D(\emptyset)=0.

  2. If i:XYi: X\to Y is an immersion of schemes, then i *:D(X)D(Y)i_*\colon D(X)\to D(Y) is fully faithful.

  3. (Smooth base change/Beck-Chevalley condition) If ff is smooth, then f *f^* admits a left adjoint f f_\sharp. Moreover, given a cartesian square

    Y k X h f Y g X \array{ Y' &\stackrel{k}{\to}& X' \\ _h\downarrow && \downarrow \;_f \\ Y &\stackrel{g}{\to}& X }

    with ff smooth, there is a canonical equivalence h k *g *f h_\sharp k^*\simeq g^* f_\sharp.

  4. (Locality) If i:ZXi: Z\hookrightarrow X is a closed immersion with open complement j:UXj: U\hookrightarrow X, then the pair (i *,j *)(i^*,j^*) is conservative.

  5. (Homotopy invariance) If p:X×𝔸 1Xp: X\times\mathbb{A}^1\to X is the projection, then p *p^* is fully faithful.

  6. (TT-stability) If pp is as above and ss is the zero section of pp, then p s *:D(X)D(X)p_\sharp s_*: D(X)\to D(X) is an equivalence of (∞,1)-categories.

Theorem

(Ayoub, Voevodsky) Every stable homotopy functor admits a formalism of four operations f *f^\ast, f *f_\ast, f !f_!, and f !f^!.

For a more precise statement, see Ayoub, Scholie 1.4.2.

Theorem

SHSH is a stable homotopy functor.

This is essentially proved in Morel-Voevodsky. In fact, something stronger is expected to be true:

Expected Theorem

SHSH is the initial object in the (∞,2)-category of stable homotopy functors.

This theorem has not been proved yet; however, Ayoub’s thesis shows that every stable homotopy functor factors through SHSH.

Realization functors

Complex realization. The functor

Sm/SmoothMfd,XX() Sm/\mathbb{C}\to SmoothMfd,\quad X\mapsto X(\mathbb{C})

associating to a smooth \mathbb{C}-scheme XX its set of \mathbb{C}-points with its structure of smooth manifold induces a functor from H()\mathrm{H}(\mathbb{C}) to the homotopy localization of the smooth (∞,1)-topos at the interval object 𝔸 1() 2\mathbb{A}^1(\mathbb{C})\simeq \mathbb{R}^2. As this localization is equivalent to the (∞,1)-topos Grpd\infty Grpd of discrete ∞-groupoids, we obtain the complex realization functor

H()Grpd. \mathrm{H}(\mathbb{C}) \to \infty Grpd.

After TT-stabilization, we obtain a functor from SH()SH(\mathbb{C}) to the (∞,1)-category of spectra.

Real realization. The functor

Sm//2SmoothMfd,XX() Sm/\mathbb{R}\to \mathbb{Z}/2-SmoothMfd,\quad X\mapsto X(\mathbb{C})

associating to a smooth \mathbb{R}-scheme XX its set of \mathbb{C}-points with its structure of smooth manifold together with the action of /2\mathbb{Z}/2 by complex conjugation induces as in the complex case the Real realization functor

H()PSh (𝒪 /2), \mathrm{H}(\mathbb{R}) \to PSh_\infty(\mathcal{O}_{\mathbb{Z}/2}),

where 𝒪 /2\mathcal{O}_{\mathbb{Z}/2} is the orbit category of /2\mathbb{Z}/2. After TT-stabilization, we obtain a functor from SH()SH(\mathbb{R}) to the (∞,1)-category of genuine /2\mathbb{Z}/2-spectra.

Étale realization. Over a separably closed field kk, we can consider the étale homotopy type functor

Sh ((Sm/k) Nis)Sh ((Sm/k) et)ΠPro(Grpd) Sh_{\infty}((Sm/k)_{Nis})\to Sh_{\infty}((Sm/k)_{et}) \stackrel{\Pi}{\to} Pro(\infty Grpd)

(see shape of an (∞,1)-topos). However, it does not descend to H(k)\mathrm{H}(k) because the étale homotopy type is not 𝔸 1\mathbb{A}^1-homotopy invariant. To rectify this, we choose a prime lchar(k)l\neq \operatorname{char}(k) and consider the reflexive localization Pro(Grpd) l Pro(\infty Grpd)^\wedge_{l} of Pro(Grpd)Pro(\infty Grpd) at the class of maps inducing isomorphisms on pro-homology groups with coefficients in /l\mathbb{Z}/l. We then obtain an étale realization functor

H(k)Pro(Grpd) l . \mathrm{H}(k) \to Pro(\infty Grpd)^\wedge_{l}.

The slice filtration

See motivic slice filtration.

The slice filtration is a filtration of H(S)\mathrm{H}(S) and of SH(S)SH(S) which is analogous to the Postnikov filtration for (∞,1)-topoi. It generalizes the coniveau filtration in algebraic K-theory, the fundamental filtration on Witt groups?, and the weight filtration on mixed Tate motives.

If SS is smooth over a field, the layers of the slice filtration of a motivic spectrum (called its slices) are modules over the motivic Eilenberg–Mac Lane spectrum H()H(\mathbb{Z}). At least if SS is a field of characteristic zero, this is the same thing as an integral motive. The spectral sequences associated to the slice filtration are analogous to the Atiyah-Hirzebruch spectral sequences in that their first page consists of motivic cohomology groups.

The 𝔸 1\mathbb{A}^1-Postnikov filtration

One can also consider the filtration on H(S)\mathrm{H}(S) induced by the Postnikov filtration in the containing Nisnevich (∞,1)-topos. A motivic space is 𝔸 1\mathbb{A}^1-n-connected if it is n-connected as a Nisnevich (∞,1)-sheaf, and the 𝔸 1\mathbb{A}^1-homotopy groups π n 𝔸 1(X,x)\pi_n^{\mathbb{A}^1}(X,x) of a motivic space are its homotopy groups as a Nisnevich (∞,1)-sheaf.

If SS has finite Krull dimension?, 𝔸 1\mathbb{A}^1-homotopy groups detect equivalences because the Nisnevich (∞,1)-topos is hypercomplete.

Remark

The usage of the 𝔸 1\mathbb{A}^1- prefix in the above definitions may seem strange since all these notions are simply inherited from the Nisnevich (∞,1)-topos. The point is that, when a smooth scheme XX is viewed as a motivic space, a localization functor is implicitly applied. The underlying Nisnevich (∞,1)-sheaf of the motivic space “XX” can thus be very different from the Nisnevich (∞,1)-sheaf represented by XX (for which these definitions would not be interesting at all!).

Intuitively, 𝔸 1\mathbb{A}^1-connectedness corresponds to the topological connectedness of the real points rather than of the complex points. For example, 𝔾 m\mathbb{G}_m is not 𝔸 1\mathbb{A}^1-connected, and 1\mathbb{P}^1 is 𝔸 1\mathbb{A}^1-connected but not 𝔸 1\mathbb{A}^1-simply connected.

Theorem

Let kk be a perfect field (resp. a field) and XX a Nisnevich (∞,1)-sheaf of spaces (resp. of spectra) on Sm/kSm/k. If XX is n-connected for some n, then its 𝔸 1\mathbb{A}^1-localization is also n-connected.

This is Morel’s connectivity theorem (Morel, Theorem 5.38). It follows that the 𝔸 1\mathbb{A}^1-Postnikov filtration on H(k)\mathrm{H}(k) “extends” to a t-structure on the stable (∞,1)-category SH(k)SH(k), called the homotopy t-structure.

Relation to the theory of motives

The stable motivic homotopy category SH(S)SH(S) is the basis for several definitions of the derived category of mixed motives over SS. See there for more details.

Relation to the theory of symmetric bilinear forms

Motivic homotopy theory is also related to the classical theory of symmetric bilinear forms (or quadratic forms in characteristic 2\neq 2). Invariants such as Witt groups?, oriented Chow groups, and Hermitian K-theory are representable in the motivic homotopy category.

A central theorem of Fabien Morel states that, if kk is a field, the ring of endomorphisms of the motivic sphere spectrum S 0SH(k)S^0\in SH(k) is canonically isomorphic to the Grothendieck–Witt ring GW(k)GW(k): this is the group completion of the semiring of isomorphism classes of nondegenerate symmetric bilinear forms over kk (Morel, Corollary 5.43). The case k=k=\mathbb{R} is especially enlightening: there the stable homotopy class of a pointed endomorphism of 1\mathbb{P}^1 corresponds to the nondegenerate symmetric bilinear form over \mathbb{R} whose dimension is the degree of the induced endomorphism of 1()S 2\mathbb{P}^1(\mathbb{C})\simeq S^2 and whose signature is the degree of the induced endomorphism of 1()S 1\mathbb{P}^1(\mathbb{R})\simeq S^1.

Equivariant motivic homotopy theory

A general theory of equivariant (unstable and stable) motivic homotopy theory was introduced in (Carlsson-Joshua 2014) and further developed in (Hoyois 15).

Applications and examples

𝔸 1\mathbb{A}^1-coverings and the 𝔸 1\mathbb{A}^1-fundamental groupoid

Like the 𝔸 1\mathbb{A}^1-Postnikov filtration, 𝔸 1\mathbb{A}^1-coverings and the 𝔸 1\mathbb{A}^1-fundamental groupoid Π 1 𝔸 1\Pi_1^{\mathbb{A}^1} are defined in the containing Nisnevich (∞,1)-topos. A morphism of motivic spaces f:YXf: Y\to X is an 𝔸 1\mathbb{A}^1-covering if it is 0-truncated as a morphism between Nisnevich (∞,1)-sheaves. Such a morphism is determined by its 1-truncation τ 1f\tau_{\leq 1}f, and hence there is an equivalence between the category of 𝔸 1\mathbb{A}^1-coverings of XX and that of 𝔸 1\mathbb{A}^1-invariant objects in the classifying topos of the Nisnevich sheaf of groupoids

Π 1 𝔸 1(X):=τ 1X. \Pi_1^{\mathbb{A}^1}(X):=\tau_{\leq 1}X.
Theorem

Let kk be a field and XH *(k)X\in\mathrm{H}_\ast(k) a pointed 𝔸 1\mathbb{A}^1-connected motivic space. Let X˜\tilde X be the 𝔸 1\mathbb{A}^1-localization of the 1-connected cover of XX (as a pointed Nisnevich (∞,1)-sheaf). Then:

  1. X˜\tilde X is the initial object in the category of pointed 𝔸 1\mathbb{A}^1-coverings of XX.

  2. X˜\tilde X is the unique pointed 𝔸 1\mathbb{A}^1-covering of XX which is 𝔸 1\mathbb{A}^1-simply connected.

  3. The Nisnevich sheaf of (unpointed) automorphisms Aut X(X˜)Aut_X(\tilde X) is canonically isomorphic to π 1 𝔸 1(X)\pi_1^{\mathbb{A}^1}(X).

This is Morel, Theorem 6.8. The key input for this theorem is the fact that Π 1 𝔸 1(X)\Pi_1^{\mathbb{A}^1}(X) is 𝔸 1\mathbb{A}^1-invariant; this is only known for 𝔸 1\mathbb{A}^1-connected motivic spaces over fields, which explains the hypotheses of the theorem.

When XX is a smooth SS-scheme, examples of 𝔸 1\mathbb{A}^1-coverings of XX include 𝔾 m\mathbb{G}_m-torsors and finite Galois coverings of degree prime to the characteristics of XX (Morel, Lemma 6.5). This can be used to compute some 𝔸 1\mathbb{A}^1-fundamental groups, for example:

Proposition

If n2n\geq 2, π 1 𝔸 1( n)=𝔾 m\pi_1^{\mathbb{A}^1}(\mathbb{P}^n)=\mathbb{G}_m.

Proof

The projection 𝔸 n+10 n\mathbb{A}^{n+1}-0\to\mathbb{P}^n is a 𝔾 m\mathbb{G}_m-torsor and hence an 𝔸 1\mathbb{A}^1-covering. If n2n\geq 2, 𝔸 n+10\mathbb{A}^{n+1}-0 is moreover 𝔸 1\mathbb{A}^1-simply connected and hence is the universal 𝔸 1\mathbb{A}^1-covering of n\mathbb{P}^n. Thus, π 1 𝔸 1( n)=Aut n(𝔸 n+10)=𝔾 m\pi_1^{\mathbb{A}^1}(\mathbb{P}^n)=Aut_{\mathbb{P}^n}(\mathbb{A}^{n+1}-0)=\mathbb{G}_m.

𝔸 1\mathbb{A}^1-h-cobordisms and the classification of surfaces

An 𝔸 1\mathbb{A}^1-h-cobordism is a surjective proper map X𝔸 1X\to\mathbb{A}^1 in Sm/SSm/S such that, for i=0,1i=0,1, the fiber X iX_i is smooth and the inclusion X iXX_i\hookrightarrow X becomes an equivalence in H(S)\mathrm{H}(S).

Asok and Morel used 𝔸 1\mathbb{A}^1-h-cobordisms to classify rational smooth proper surfaces over algebraically closed fields up to 𝔸 1\mathbb{A}^1-homotopy. See Asok-Morel.

Euler classes and splittings of algebraic vector bundles

Let kk be a perfect field. If XX is a smooth affine kk-scheme, Morel proved that

Vect n(X)[X,BGL n], Vect_n(X) \cong [X,BGL_n],

where [,][-,-] denote homotopy classes of maps in the motivic homotopy category H(k)\mathrm{H}(k), def. . The classical problem of determining whether a rank nn vector bundle splits off a trivial line bundle is thus equivalent to determining whether the classifying map XBGL nX\to BGL_n lifts to BGL n1BGL_{n-1} in H(k)\mathrm{H}(k). If the Nisnevich cohomological dimension of XX is at most nn, we can use obstruction theory together with the fiber sequence

𝔸 n0BGL n1BGL n \mathbb{A}^n-0 \to BGL_{n-1} \to BGL_n

to obtain the following criterion:

Theorem

(Morel) Suppose that n2n\geq 2 and that XX has dimension n\leq n. Let ξ\xi be a vector bundle of rank nn over XX. Then there exists a canonical class

e(ξ)H Nis n(X,π n1 𝔸 1(𝔸 n0)(detξ)) e(\xi)\in H^n_{Nis}(X,\pi_{n-1}^{\mathbb{A}^1}(\mathbb{A}^n-0)(\mathrm{det} \xi))

which vanishes if and only if ξ\xi splits off a trivial line bundle.

The twist by the determinant detξ\mathrm{det} \xi comes from the nontrivial 𝔸 1\mathbb{A}^1-fundamental group π 1 𝔸 1(BGL n)=𝔾 m\pi_1^{\mathbb{A}^1}(BGL_n)=\mathbb{G}_m.

The Nisnevich sheaf π n1 𝔸 1(𝔸 n0)\pi_{n-1}^{\mathbb{A}^1}(\mathbb{A}^n-0) is the n-th Milnor–Witt K-theory sheaf K n MW\mathbf{K}^{MW}_n, so that

H Nis n(X,π n1 𝔸 1(𝔸 n0)(detξ))CH˜ n(X;detξ) H^n_{Nis}(X,\pi_{n-1}^{\mathbb{A}^1}(\mathbb{A}^n-0)(\mathrm{det} \xi)) \cong \widetilde{CH}^n(X; \mathrm{det} \xi)

is the n-th oriented Chow group of XX. When kk is algebraically closed, this is just the usual Chow group of XX and e(ξ)e(\xi) can be identified with the top Chern class of ξ\xi.

Motivic cohomology

Suppose that SS is a smooth scheme over a field. Then the motivic cohomology of smooth SS-schemes is representable in H(S)\mathrm{H}(S) and in SH(S)SH(S). If AA is an abelian group and pq0p\geq q\geq 0, there exist a pointed motivic spaces K(A(q),p)K(A(q),p), called a motivic Eilenberg–MacLane space, such that, for every XSm/SX\in Sm/S,

H pr,qs(X,A)=[Σ r,sX +,K(A(q),p)]. H^{p-r,q-s}(X,A) = [\Sigma^{r,s}X_+, K(A(q),p)].

Voevodsky’s cancellation theorem implies that Ω TK(A(q+1),p+2)K(A(q),p)\Omega_T K(A(q+1), p+2)\simeq K(A(q), p). It follows that the sequence of motivic spaces K(A(n),2n)K(A(n),2n) form a TT-spectrum H(A)H(A), called a motivic Eilenberg–MacLane spectrum, such that

H p,q(X,A)=[Σ T X +,Σ p,qH(A)]. H^{p,q}(X,A) = [\Sigma_T^\infty X_+, \Sigma^{p,q} H(A)].

If RR is a commutative ring, the motivic spectrum H(R)H(R) has a canonical structure of E E_\infty-algebra in SH(S)SH(S) which induces the ring structure in motivic cohomology.

Unlike in topology, H()H(\mathbb{Q}) is not always equivalent to the rational motivic sphere spectrum S 0S^0_{\mathbb{Q}}: this is only the case if 1-1 is a sum of squares in the base field. In general, H()H(\mathbb{Q}) is a direct summand of S 0S^0_{\mathbb{Q}}.

The stable (∞,1)-category of H()H(\mathbb{Q})-modules is equivalent to the derived category of mixed motives. See there for more details.

Algebraic K-theory

See algebraic K-theory spectrum.

Algebraic cobordism

See algebraic cobordism.

References

  • Aravind Asok?, Fabien Morel, Smooth varieties up to A1-homotopy and algebraic h-cobordisms, Adv. Math. 227 (5) (2011), pp. 1990-2058 (arXiv)

References

The original “textbook” references are

  • Vladimir Voevodsky, A 1\mathbf{A}^1-Homotopy Theory. Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math. 1998, Extra Vol. I, 579–604 (electronic). web

  • Fabien Morel, Vladimir Voevodsky, 𝔸 1\mathbb{A}^1-homotopy theory of schemes K-theory, 0305 (web pdf)

  • Fabien Morel, 𝔸 1\mathbb{A}^1-algebraic topology over a field, LNM 2052, 2012, (pdf)

Readable introductions to the subject are:

The model structure on simplicial presheaves on the Nisnevich site and its homotopy localization to A1-homotopy theory is in

For more on the general procedure see homotopy localization.

The universal property of SH(S)SH(S) is proved in

  • Marco Robalo, Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes, 2013 (pdf)

For the formalism of six operations see

The slice filtration was defined in

Important representability results are in

and

Discussion related to étale homotopy is in

  • Daniel Isaksen, Étale realization of the 𝔸 1\mathbb{A} ^1-homotopy theory of schemes, Advances in Mathematics 184 (2004)

Discussion about thick ideals is in

  • Ruth Joachimi?, Thick ideals in equivariant and motivic stable homotopy categories, arXiv:1503.08456.

Motivic homotopy theory in other contexts

equivariant motivic homotopy theory is developed in

This was vastly generalized and studied more thoroughly in

Motivic homotopy theory of noncommutative spaces (associative dg-algebras) is studied in

  • Marco Robalo, Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes, 2013 (pdf)

Motivic homotopy theory of associative nonunital rings is studied in

See also

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