Motivic Homotopy: Foundations¶
Job: construct Morel–Voevodsky \(\mathbb{A}^1\)-homotopy theory, the stable category \(\mathrm{SH}(S)\), motivic spheres and connectivity, and point to algebraic \(K\)-theory and motivic cohomology.
Core¶
Unstable \(\mathbb{A}^1\)-homotopy¶
Fix a Noetherian base \(S\) of finite dimension (often \(S=\mathrm{Spec}\,k\)). Let \(\mathbf{Sm}/S\) be smooth \(S\)-schemes of finite type.
- Simplicial Nisnevich sheaves. Form the category of simplicial sheaves on \(\mathbf{Sm}/S\) with the Nisnevich topology (elementary distinguished squares: Cartesian with \(U\to X\) open, \(V\to X\) étale, bijective on residual points over \(X\setminus U\)).
- Model / localization. Start from the injective (or projective) local model structure for Nisnevich-local weak equivalences; then \(\mathbb{A}^1\)-localize: force \(X\times\mathbb{A}^1\to X\) to be a weak equivalence (Bousfield localization).
- Result. Unstable pointed/unpointed homotopy categories \(\mathcal{H}_\bullet(S)\), \(\mathcal{H}(S)\). Objects: motivic spaces; morphisms: \(\mathbb{A}^1\)-homotopy classes after fibrant replacement.
Elementary distinguished squares become homotopy pushouts in \(\mathcal{H}\) — the Nisnevich descent property that makes schemes buildable from covers like CW cells.
| Topology | Why used / avoided |
|---|---|
| Zariski | too coarse for excision in \(K\)-theory |
| Nisnevich | descent for \(K\), \(\mathbb{A}^1\)-homotopy works |
| Étale | finer; \(\mathbb{A}^1\)-local étale homotopy is a different theory (Artin–Mazur / étale topological type) |
Motivic spheres¶
Pointed motivic spaces include:
| Sphere | Model | Role |
|---|---|---|
| \(S^1=S^{1,0}\) | simplicial circle | suspension (simplicial) |
| \(\mathbb{G}_m=S^{1,1}\) | \((\mathbb{A}^1\setminus 0,1)\) | weight / multiplicative sphere |
| \(\mathbb{P}^1\simeq S^{2,1}\) | \((\mathbb{P}^1,\infty)\) | stable suspension coordinate |
| \(S^{p,q}\) | \(S^{p-q}\wedge\mathbb{G}_m^{\wedge q}\) | bigraded spheres (\(p\ge q\ge 0\) unstable range) |
Smash product of pointed motivic spaces induces the monoidal structure underlying stabilization.
Stable motivic homotopy \(\mathrm{SH}(S)\)¶
Stabilization. Invert smashing with \(\mathbb{P}^1\) (equivalently with \(S^1\) and \(\mathbb{G}_m\) separately): objects are \(\mathbb{P}^1\)-spectra (sequences \(E_n\) with bonding maps \(E_n\to \Omega_{\mathbb{P}^1}E_{n+1}\), or symmetric/orthogonal motivic spectra). The homotopy category \(\mathrm{SH}(S)\) is triangulated; as an \(\infty\)-category it is stable presentable symmetric monoidal.
Sphere spectrum \(\mathbb{1}_S\) (unit for smash \(\wedge\)). Homotopy sheaves \(\underline{\pi}_{p,q}(E)\): Nisnevich sheaves on \(\mathbf{Sm}/S\) associated to \(U\mapsto [\Sigma^{p,q}U_+,E]\).
Six functors (pointer). For morphisms of bases, \(\mathrm{SH}(-)\) supports a six-functor formalism (\(f_*,f^*,f_!,f^!,\wedge,\underline{\mathrm{Hom}}\)) under standard hypotheses (Ayoub, Cisinski–Déglise, …) — parallel to motives and to étale sheaves.
Connectivity (Morel)¶
Over a perfect field \(k\), Morel’s \(\mathbb{A}^1\)-connectivity theorem controls vanishing of \(\underline{\pi}_{p,q}\). Key computations for the sphere:
| Group | Identification (sketch) |
|---|---|
| \(\underline{\pi}_{n,n}(\mathbb{1})\) | related to Milnor–Witt \(K\)-theory sheaves \(\underline{\mathbf{K}}^{\mathrm{MW}}_n\) |
| \(\pi_{0,0}(\mathbb{1})(k)\) | GW\((k)\) (Grothendieck–Witt) for infinite perfect \(k\) (Morel) |
| \(\pi_{1,1}\), low stems | computed in ranges; motivic Adams spectral sequence |
Theorem (Morel). For infinite perfect \(k\), the endomorphism ring of \(\mathbb{1}\) in \(\mathrm{SH}(k)\) in bidegree \((0,0)\) is the Grothendieck–Witt ring \(\mathrm{GW}(k)\). This is the motivic upgrade of \(\pi_0\mathbb{S}\cong\mathbb{Z}\) in classical stable homotopy.
Algebraic \(K\)-theory pointer¶
There is a motivic spectrum \(\mathrm{KGL}\in\mathrm{SH}(S)\) such that for \(X\in\mathbf{Sm}/S\) regular (under standard hyp.), $$ \mathrm{Map}{\mathrm{SH}}(\Sigma^\infty)\;\simeq\;K(X) $$ (the algebraic }^1}X_+,\mathrm{KGL\(K\)-theory space/spectrum) — representability theorems of Voevodsky, Morel, Cisinski, … after Thomason–Trobaugh descent. Bott periodicity: \(\mathrm{KGL}\simeq\mathrm{KGL}\wedge\mathbb{P}^1\) (periodic). See K-theory.
Motivic cohomology pointer¶
Motivic Eilenberg–MacLane spectra \(\mathbf{H}\mathbb{Z}\) (and \(\mathbf{H}\mathbb{Z}/\ell\)) represent motivic cohomology $$ H^{p,q}(X,\mathbb{Z})\;\cong\;[\Sigma^\infty X_+,\Sigma^{p,q}\mathbf{H}\mathbb{Z}]_{\mathrm{SH}}. $$ Voevodsky’s isomorphism with higher Chow groups; Beilinson–Lichtenbaum / norm residue theorem (Voevodsky–Rost) identifies mod-\(\ell\) motivic cohomology with étale cohomology in the appropriate range. Slice filtration (Voevodsky) decomposes spectra into slices related to motivic cohomology — the motivic analogue of the Postnikov / chromatic picture.
| Object | Represents |
|---|---|
| \(\mathrm{KGL}\) | algebraic \(K\)-theory |
| \(\mathrm{MGL}\) | algebraic cobordism (Voevodsky; Levine–Morel closely related) |
| \(\mathbf{H}\mathbb{Z}\) | motivic cohomology |
| \(\mathbb{1}\) | stable \(\mathbb{A}^1\)-stems |
Realizations¶
| Realization | Target | Effect on \(\mathbb{P}^1\) |
|---|---|---|
| Betti (\(k=\mathbb{C}\)) | \(\mathrm{Sp}\) | \(S^2\) |
| Étale / \(\ell\)-adic | \(\ell\)-complete spectra / sheaves | depends on setup |
| Real (\(k=\mathbb{R}\)) | \(\mathrm{C}_2\)-spectra | genuine equivariant flavour |
See it¶
Purity / Thom spaces. For a vector bundle \(V\to X\) of rank \(r\), the Thom space \(\mathrm{Th}(V)=V/(V\setminus 0)\) is \(\mathbb{A}^1\)-equivalent to a \(\mathbb{P}^1\)-suspension of \(X_+\) when \(V\) is trivializable; in general \(\Sigma^{2r,r}\) twisted by the determinant in \(\mathrm{SH}\). Homotopy purity (Morel–Voevodsky): for smooth closed \(Z\subset X\) of codimension \(c\), $$ X/(X\setminus Z)\;\simeq_{\mathbb{A}^1}\;\mathrm{Th}(N_{Z/X}). $$ This is the engine for Gysin sequences, Euler classes in \(\mathrm{MW}\)-theory, and oriented cohomology theories on \(\mathbf{Sm}\).
Check¶
Why is \(\mathbb{G}_m\) not equivalent to \(S^1\) in \(\mathcal{H}_\bullet(k)\)?
Different bidegrees: \(\mathbb{G}_m=S^{1,1}\) vs \(S^{1,0}\). After Betti realization both become \(S^1\), but weight separates them motivically; e.g.\ \(\mathrm{Map}(S^{1,0},S^{1,1})\) encodes units / symbols not visible in pure topological degree.
Sketch how an elementary Nisnevich square gives a distinguished triangle in \(\mathrm{SH}\).
Unstably the square is a homotopy pushout; suspending to \(\mathrm{SH}\) yields a cofiber sequence \(V_+\wedge U_+\to\cdots\) and thus a Gysin / Mayer–Vietoris triangle for any \(E\in\mathrm{SH}\). Descent for \(E\)-cohomology follows.
Pitfalls¶
- Nisnevich vs étale cohomology: representing étale cohomology needs different localizations; motivic \(\mathbf{H}\mathbb{Z}/\ell\) matches étale only after BL range / hypercompleteness issues.
- Base change: \(\mathrm{SH}(S)\) for general \(S\) needs care (resolution of singularities, compactifications) for six functors in full strength.
- Effective covers / connectivity: not every object is a suspension of a smooth scheme; cellularity and effective homotopy \(t\)-structure (Morel, Levine, …) organize this.
- Char \(p\): some classical MV arguments used resolution of singularities; modern approaches use alterations / \(\infty\)-categorical presentations.
Refs¶
- Morel–Voevodsky, \(\mathbb{A}^1\)-homotopy theory of schemes
- Morel, \(\mathbb{A}^1\)-Algebraic Topology over a Field
- Voevodsky, Motivic cohomology with \(\mathbb{Z}/l\)-coefficients; related ICM notes
- Ayoub, Les six opérations de Grothendieck; Cisinski–Déglise monographs
Next¶
Motivic homotopy index · Six Functors & Base Change · Slice Filtration · Motivic Adams & Stable Stems · Algebraic Cobordism · K-theory · Algebraic \(K\)-theory · Algebraic geometry · Model categories · Higher categories · Algebraic topology