Motivic Homotopy Theory¶
Job: build a homotopy theory of schemes in which the affine line \(\mathbb{A}^1\) plays the role of the unit interval, yielding stable motivic homotopy \(\mathrm{SH}(S)\), motivic spheres, and bridges to algebraic \(K\)-theory and motivic cohomology.
Core¶
Morel–Voevodsky \(\mathbb{A}^1\)-homotopy theory localizes spaces (simplicial sheaves) on a base scheme \(S\) so that \(X\to X\times\mathbb{A}^1\) becomes a weak equivalence. After stabilization with respect to \(\mathbb{P}^1\) (or equivalently \(S^1\) and \(\mathbb{G}_m\)), one obtains the stable motivic homotopy category \(\mathrm{SH}(S)\) — a triangulated / stable \(\infty\)-category hosting motivic spectra, algebraic cobordism, and the motivic sphere spectrum \(\mathbb{1}\).
| Layer | Content |
|---|---|
| Unstable | \(\mathbf{Spc}(S)\) / \(\mathcal{H}(S)\): \(\mathbb{A}^1\)-local simplicial Nisnevich sheaves |
| Stable | \(\mathrm{SH}(S)\): \(\mathbb{P}^1\)-spectra; bigraded homotopy sheaves |
| Spheres | \(S^{p,q}=S^{p-q}\wedge\mathbb{G}_m^{\wedge q}\); \(\mathbb{P}^1\simeq S^{2,1}\) |
| Connectivity | Morel’s \(\mathbb{A}^1\)-connectivity; \(\pi_{n,n}\mathbb{1}\) and Milnor–Witt \(K\)-theory |
| Interfaces | \(K\)-theory spectrum \(\mathrm{KGL}\); motivic cohomology \(\mathbf{H}\mathbb{Z}/\ell\); slice filtration |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | \(\mathbb{A}^1\)-localization; \(\mathrm{SH}(S)\); spheres; connectivity; \(K\)-theory / motivic cohomology pointers |
| Six Functors & Base Change | \(f^*,f_*,f_!,f^!,\wedge,\underline{\mathrm{Hom}}\); localization triangles; purity; cdh descent; Verdier duality |
| Slice Filtration | Voevodsky slices; slice spectral sequence; \(s_0\mathbb{1}=\mathbf{H}\mathbb{Z}\); \(\mathrm{MGL}\) slices; effective \(t\)-structures |
| Motivic Adams & Stable Stems | motivic Adams / Adams–Novikov; motivic Steenrod algebra; Milnor–Witt stems; \(\mathbb{R}\)-motivic / \(C_2\) / Kervaire |
| Algebraic Cobordism | \(\mathrm{MGL}\) universality; oriented spectra / FGL; Conner–Floyd; Hopkins–Morel slice; prismatic / \(\mathrm{TC}\) |
Dependency order¶
- Algebraic geometry (schemes, Nisnevich topology).
- Algebraic topology + model categories (simplicial sheaves, stabilization).
- Higher categories (optional modern packaging of \(\mathrm{SH}\)).
- K-theory as primary computational target; then arithmetic geometry applications.
Interfaces¶
| Area | Contact |
|---|---|
| K-theory | \(\mathrm{KGL}\) represents algebraic \(K\)-theory after \(\mathbb{A}^1\) |
| Algebraic geometry | base \(S\); smooth schemes as motivic spaces |
| Model categories | original MV constructions via model structures |
| Homological algebra | DM, motives as linear counterparts |
| Langlands | geometric Langlands uses related sheaf categories (distinct package) |
| Topology / AT | Betti realization \(\mathrm{SH}(\mathbb{C})\to\mathrm{Sp}\) |
Notation (paper-standard)¶
- \(\mathcal{H}(S)\), \(\mathcal{H}_\bullet(S)\); \(\mathrm{SH}(S)\); \(\mathbb{1}_S\) sphere spectrum
- \(S^{p,q}\), \(\pi_{p,q}\), \(\underline{\pi}_{p,q}\) (Nisnevich sheaves)
- \(\mathrm{KGL}\), \(\mathrm{MGL}\) (algebraic cobordism), \(\mathbf{H}A\) Eilenberg–MacLane
- \(f\)-, \(s\)-, \(c\)-motivic variants (field \(k\); real / complex realizations)
See it¶
Why \(\mathbb{A}^1\) and Nisnevich? Zariski-local \(\mathbb{A}^1\)-invariance fails to see interesting homotopy (too coarse); étale topology makes \(\mathbb{A}^1\)-localization interact poorly with \(\mathbb{A}^1\)-homotopy sheaves for \(K\)-theory. Nisnevich is the Goldilocks topology: algebraic \(K\)-theory satisfies Nisnevich descent and \(\mathbb{A}^1\)-invariance on regular schemes (after Bass, Thomason–Trobaugh, …), so it is representable in \(\mathrm{SH}\).
Check¶
What is the motivic sphere \(S^{2,1}\) equivalent to?
\(\mathbb{P}^1\) (pointed at \(\infty\)) is \(\mathbb{A}^1\)-equivalent to \(S^1\wedge\mathbb{G}_m\simeq S^{2,1}\). Stabilization in \(\mathrm{SH}\) inverts smashing with \(\mathbb{P}^1\).
How does Betti realization relate \(\mathrm{SH}(\mathbb{C})\) to classical spectra?
Complex realization sends smooth \(X/\mathbb{C}\) to \(X(\mathbb{C})^{\mathrm{an}}\) (singular complex), extending to \(\mathrm{SH}(\mathbb{C})\to\mathrm{Sp}\); \(\mathbb{P}^1\mapsto S^2\). Homotopy groups map to classical stable stems with weight information collapsed/filtered.
Pitfalls¶
- Bigading: \(\pi_{p,q}\) has topological degree \(p\) and weight \(q\); forgetting weight loses Morel’s theorems.
- Perfect fields / resolutions: many foundational computations assume perfect \(k\) or resolve singularities carefully.
- DM vs SH: Voevodsky motives \(\mathrm{DM}\) are linear; \(\mathrm{SH}\) is the nonlinear stable homotopy category — related by motives functor / EM spectra, not identical.
- Unstable vs stable: \(\mathbb{A}^1\)-fundamental group (Morel) is unstable; do not confuse with \(\pi_{1,0}\) or \(\pi_{1,1}\) of \(\mathbb{1}\).
Refs¶
- Morel–Voevodsky, \(\mathbb{A}^1\)-homotopy theory of schemes (IHES)
- Morel, \(\mathbb{A}^1\)-Algebraic Topology over a Field (LNM)
- Voevodsky, notes on motivic cohomology / DM
- Dundas–Levine–Østvær–Röndigs–Østvær surveys; Cisinski–Déglise, Triangulated Categories of Motives
Next¶
Foundations · Six Functors & Base Change · Slice Filtration · Motivic Adams & Stable Stems · Algebraic Cobordism · K-theory · Algebraic geometry · Model categories · Higher categories · Algebraic topology · Arithmetic geometry