The Six-Functor Formalism in Motivic Homotopy¶
Job: develop the Ayoub / Cisinski–Déglise six-functor formalism on \(\mathrm{SH}(-)\) — the pullback / pushforward / exceptional pushforward / exceptional pullback / smash / internal Hom, the localization distinguished triangles, the purity (Thom) equivalence, the cdh descent, and the projection / Verdier duality formulas — with the Gysin sequence for a smooth blow-up as the worked example.
Core¶
The six functors¶
For a morphism \(f:X\to Y\) of schemes (under the standard hypotheses: \(S\) finite-dimensional Noetherian, \(f\) Nagata-compactifiable — factoring as a composition of a proper map and an open immersion), the assignment \(S\mapsto\mathrm{SH}(S)\) extends to a functor on schemes equipped (Ayoub; Cisinski–Déglise) with six operations:
| Functor | Direction | Adjunction |
|---|---|---|
| \(f^*:\mathrm{SH}(Y)\to\mathrm{SH}(X)\) | pullback (symmetric monoidal) | \(f^*\dashv f_*\) |
| \(f_*:\mathrm{SH}(X)\to\mathrm{SH}(Y)\) | pushforward (right adjoint to \(f^*\)) | \(f^*\dashv f_*\) |
| \(f_!:\mathrm{SH}(X)\to\mathrm{SH}(Y)\) | exceptional / proper pushforward (for \(f\) compactifiable) | \(f_!\dashv f^!\) |
| \(f^!:\mathrm{SH}(Y)\to\mathrm{SH}(X)\) | exceptional pullback (right adjoint to \(f_!\), the "dualizing" functor) | \(f_!\dashv f^!\) |
| \((-)\wedge(-)\) | symmetric monoidal product | \((-\wedge-)\dashv\underline{\mathrm{Hom}}\) |
| \(\underline{\mathrm{Hom}}(-,-)\) | internal Hom | \((-\wedge-)\dashv\underline{\mathrm{Hom}}\) |
For a smooth morphism \(f\) of relative dimension \(d\), \(f^!\simeq f^*\wedge\Sigma^{2d,d}\mathbb{1}\) (up to a twist by the relative dimension, the smooth pullback is self-dual up to the Thom twist). For a proper morphism, \(f_!=f_*\) (the exceptional and ordinary pushforward coincide).
Localization and the purity theorem¶
For a closed immersion \(i:Z\hookrightarrow X\) with open complement \(j:U=X\setminus Z\hookrightarrow X\), the localization distinguished triangles are $$ j_!j^* E\to E\to i_i^ E\xrightarrow{+1} j_!j^E[1], $$ $$ i_i^! E\to E\to j_j^E\xrightarrow{+1} i_*i^!E[1], $$ for every \(E\in\mathrm{SH}(X)\). The first expresses \(E\) in terms of its restriction to \(U\) and its localization at \(Z\); the second is the dual (the "co-localization"). These are the motivic analogs of the open / closed distinguished triangles in the Grothendieck–Verdier formalism.
Theorem (purity, Morel–Voevodsky; Ayoub). For a closed immersion \(i:Z\hookrightarrow X\) of codimension \(c\) with \(X, Z\) smooth over \(S\), there is a natural equivalence $$ i^!\simeq i^\wedge\Sigma^{2c,c}\mathbb{1}, $$ the *purity / Thom equivalence** (the "homotopy purity" — \(i^!\) is a twist of \(i^*\) by the Thom space of the normal bundle \(N_{Z/X}\)). Equivalently \(X/(X\setminus Z)\simeq_{\mathbb{A}^1}\mathrm{Th}(N_{Z/X})\) — the cofiber of the open complement is the Thom space of the normal bundle.
This is the foundational Gysin / localization tool: the Gysin map \(i_*:i^*E\to E\wedge\Sigma^{-2c,-c}\mathbb{1}\) (the pushforward with the Thom twist) is the algebraic analog of the Thom–Gysin map in topology.
Topologies and descent¶
| Topology | Generators | Descent in \(\mathrm{SH}\) |
|---|---|---|
| Zariski | open covers | yes (sheaf condition) |
| Nisnevich | elementary distinguished squares | yes (the defining topology) |
| étale | finite étale covers (char \(\ne\ell\) for \(\ell\)-adic) | \(\mathrm{SH}(S)[1/\ell]\) satisfies étale descent (the \(\ell\)-completion / the étale localization) |
| cdh | Nisnevich covers + abstract blow-up squares | yes (cdh descent: \(\mathrm{SH}\) satisfies the cdh sheaf condition, Voevodsky / Ayoub) |
The cdh topology is generated by Nisnevich covers and abstract blow-up squares (a cartesian square with \(X'\to X\) a proper birational morphism and \(Z\to X\) a closed subscheme such that \(X'\setminus Z'\to X\setminus Z\) is an isomorphism). The cdh descent theorem (Voevodsky, Ayoub) says \(\mathrm{SH}\) satisfies cdh descent — every motivic spectrum is a cdh-sheaf — giving the excision / blow-up formulas: for an abstract blow-up square, the square of \(f_*, f^*\) is homotopy-cartesian (the cdh descent distinguished triangle). This is the engine behind the blow-up formula in motivic homology and the cdh-local models of \(K\)-theory (the Haesemeyer resolution).
Base-change, projection, and Verdier duality¶
- Proper base change. For a cartesian square $$ \begin{matrix}X'\ar[r]^{g'}\ar[d]_{f'}&Y'\ar[d]^f\X\ar[r]_g&Y\end{matrix} $$ with \(f\) proper (or \(f\) compactifiable and the square Tor-independent), \(g^*f_*\simeq f'_*g'^*\) (and \(g^*f_!\simeq f'_!g'^*\) for the exceptional pushforward, the smooth / proper base change).
- Smooth base change. For \(g\) smooth, \(g^*f_*\simeq f'_*g'^*\) and \(g^*f_!\simeq f'_!g'^*\) (the smooth base change, a consequence of the smooth projection formula + the purity).
- Projection formula. For \(f:X\to Y\) compactifiable and \(E\in\mathrm{SH}(X)\), \(F\in\mathrm{SH}(Y)\), $$ f_!(E\wedge f^*F)\simeq f_!E\wedge F. $$
- Künneth formula. \(f_*(E\wedge F)\) under Künneth / tensor hypotheses (e.g. \(f\) proper and one factor \(\mathbf{H}\mathbb{Z}\)-local, the Künneth spectral sequence).
- Verdier duality. For \(p_X:X\to S\) the structure morphism, the dualizing motive is \(D_X=p_X^!\mathbb{1}_S\in\mathrm{SH}(X)\), and the Verdier dual of \(E\in\mathrm{SH}(X)\) is $$ D_X(E)=\underline{\mathrm{Hom}}(E,D_X)=\underline{\mathrm{Hom}}(E,p_X^!\mathbb{1}). $$ The duality \(D_X(D_X(E))\simeq E\) (for \(X\) smooth, the double-duality), and \(p_{X,!}(E\wedge D_X(F))\simeq\underline{\mathrm{Hom}}(p_X^*-, p_{X,!}E\wedge F)\) — the six-functor formalism recovers the Grothendieck–Verdier duality in the motivic setting, with \(p_X^!\mathbb{1}\) the Borel–Moore homology dualizing object.
See it¶
Gysin sequence for a smooth blow-up. Let \(h:\tilde X=\mathrm{Bl}_Z X\to X\) be the blow-up along a smooth center \(Z\subset X\) of codimension \(c\) (both \(X, Z\) smooth), with exceptional divisor \(E=\mathbb{P}(N_{Z/X})\to Z\) (the projectivized normal bundle, a \(\mathbb{P}^{c-1}\)-bundle over \(Z\)). The abstract blow-up square $$ \begin{matrix}E\ar[r]^{i_E}\ar[d]{q}&\tilde X\ar[d]^h\Z\ar[r] $$ is a cdh cover (the square is cartesian, }&X\end{matrix\(h\) proper birational, and \(\tilde X\setminus E\xrightarrow{\sim}X\setminus Z\)). The cdh descent gives the distinguished triangle $$ h_!h^M\to M\to i_i^M\xrightarrow{+1}h_!h^M[1] $$ for \(M\in\mathrm{SH}(X)\), and the properness of \(h\) (\(h_!=h_*\)) plus the purity \(i^*\simeq i^!\wedge\Sigma^{2c,c}\mathbb{1}\) identifies \(i_*i^*M\simeq i_*i^!M\wedge\Sigma^{2c,c}\mathbb{1}\). The exceptional divisor \(E\) is the projectivized normal bundle, and the Thom space \(\mathrm{Th}(N_{Z/X})\) fits into the Thom isomorphism \(\mathrm{Th}(N_{Z/X})\simeq\Sigma^{2c,c}\mathbb{P}(N_{Z/X}\oplus\mathcal{O})/\mathbb{P}(N_{Z/X})\), giving the blow-up formula in motivic homology: $$ [\mathrm{Bl}_Z X]=[X]+[\mathbb{P}^{c-1}]\cdot[Z]\quad\text{in }K_0(\mathrm{Var}/S)\text{ / in }\mathrm{SH}, $$ the relation \(h_*[\tilde X]=[X]+(\mathbb{L}+\cdots+\mathbb{L}^{c-1})\cdot[Z]\) (with \(\mathbb{L}=[\mathbb{A}^1]\) the Lefschetz motive) — the motivic blow-up formula. This is the formula underlying the relation of \(K\)-theory under blow-up (the deformation to the normal cone), the cdh-local descent of \(K\)-theory (Haesemeyer), and the computation of the Grothendieck ring of varieties under blow-up.
Check¶
Show that for a smooth closed immersion \(i:Z\hookrightarrow X\) of codimension \(c\), the purity equivalence \(i^!\simeq i^*\wedge\Sigma^{2c,c}\mathbb{1}\) implies the Gysin map \(i_*:i^*E\to E\wedge\Sigma^{-2c,-c}\mathbb{1}\).
The unit of the adjunction \(i_!\dashv i^!\) gives \(\mathrm{id}\to i^!i_!\), and the purity \(i^!\simeq i^*\wedge\Sigma^{2c,c}\mathbb{1}\) identifies \(i^!i_!E\) with \(i^*i_!E\wedge\Sigma^{2c,c}\mathbb{1}\). For \(i\) a closed immersion \(i_!=i_*\), so the counit \(i_!i^!\to\mathrm{id}\) gives \(i_*i^*E\wedge\Sigma^{2c,c}\mathbb{1}\to E\), equivalently the Gysin pushforward \(i_*:i^*E\to E\wedge\Sigma^{-2c,-c}\mathbb{1}\) (untwisting by \(\Sigma^{-2c,-c}\)). This is the algebraic Thom–Gysin map, the pushforward in the oriented cohomology theory represented by \(E\).
Why does the cdh descent hold in \(\mathrm{SH}\) but the cdh-local objects differ from the Nisnevich-local?
The cdh topology is finer than Nisnevich (it adds abstract blow-up squares), so cdh-descent is a stronger condition: a motivic spectrum satisfies cdh descent iff it satisfies Nisnevich descent AND the abstract-blow-up excision (the square of \(f_*, f^*\) is homotopy-cartesian for abstract blow-ups). The theorem (Voevodsky / Ayoub) is that every \(E\in\mathrm{SH}\) (with \(\mathbb{A}^1\)-localization built in) satisfies this stronger descent. The cdh-local objects (the cdh-sheaves that are not \(\mathbb{A}^1\)-local) differ: e.g. the cdh-local \(K\)-theory (the Haesemeyer cdh-sheafication) differs from the Nisnevich-local \(K\)-theory on singular varieties — the cdh-local version is insensitive to rational singularities, the Nisnevich-local is not. So the descent is for \(\mathrm{SH}\) (the \(\mathbb{A}^1\)-localized category), but the local objects at each topology differ.
Pitfalls¶
- The six-functor formalism requires the base to be finite-dimensional / Nagata-compactifiable (the compactifiability is needed for \(f_!\) — without a compactification \(f=\bar f\circ j\) with \(\bar f\) proper and \(j\) open, \(f_!\) is not defined). For arbitrary bases (e.g. infinite-dimensional), the formalism is partial.
- The cdh descent holds in \(\mathrm{SH}\) (the \(\mathbb{A}^1\)-localized category), but the cdh-local objects differ from the Nisnevich-local — the cdh-sheafication of a Nisnevich-sheaf changes it (e.g. for singular varieties). Do not conflate "cdh descent for \(\mathrm{SH}\)" with "every cdh-sheaf is motivic".
- The localization triangle requires the open-closed decomposition; for non-cartesian squares the base-change can fail without the Tor-independence / properness hypotheses.
- The purity equivalence is for \(X, Z\) smooth (the normal bundle is a vector bundle; the Thom space is well-defined); for singular \(Z\) or non-smooth \(X\), the purity is more subtle (the deformation to the normal cone, the cdh arguments).
- The Verdier dual \(p_X^!\mathbb{1}\) is the Borel–Moore dualizing object; for \(X\) smooth of relative dimension \(d\) over \(S\), \(p_X^!\mathbb{1}\simeq\Sigma^{2d,d}\mathbb{1}_X\) (the smooth purity), but for singular \(X\) the dualizing object is more complex (the Borel–Moore homology).
- The projection formula is \(f_!(E\wedge f^*F)\simeq f_!E\wedge F\) (with \(f_!\) the exceptional pushforward); the version with \(f_*\) (the ordinary pushforward) requires \(f\) proper (\(f_!=f_*\)) — do not apply the projection formula with \(f_*\) for non-proper \(f\).
Refs¶
- Ayoub, Les six opérations de Grothendieck et la formalisme des cycles évanescents (Astérisque 2007)
- Cisinski–Déglise, Triangulated Categories of Mixed Motives (Springer 2019); Déglise–Cisinski, " étale and cdh descent"
- Morel–Voevodsky, \(\mathbb{A}^1\)-homotopy theory of schemes (Publ. Math. IHÉS 1999); Voevodsky, "homotopy of schemes" (Sel. Math. 1998)
- Voevodsky, "The cdh-topology and the slice filtration" (doc. math. 2010); Haesemeyer, "Descent properties of smooth schemes" (2004)
Next¶
MH index · Foundations · Slice Filtration · Motivic Adams & Stable Stems · Algebraic Cobordism · K-theory · Algebraic geometry