Let \(Y \xrightarrow{f} X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}}\), and define the graph of \(f\) as the following pullback:
Here \(\delta\) is the diagonal immersion of \(X_{/ {S}}\).
Note: Notes
Last updated: 2021-07-27
References:
Abstract:
Building on initial conjectures due to Beilinson, Voevodsky has initiated a rich variety of “motivic categories,” the universal one being Morel-Voevodsky’s homotopy category. This world, that is now called “motivic homotopy theory,” has produced a wide range of results, settling older conjectures as well as opening new tracks to follow.
This lecture series will aim at giving a survey of this world, from the pure motivic origin, through the functoriality developments and then to some of the exciting open questions.
Recall the Euler product expansion for the zeta function. General \(L{\hbox{-}}\)functions were studied around the 20s, followed by the Weil conjectures in the 40s, and then étale \(\ell{\hbox{-}}\)adic shaves by Grothendieck et al in the 60s. Letters from Grothendieck to Serre describe the notion of weights in relation to the Weil conjectures, and served as an impetus in the early 70s for pure motives.
A second line of study considered number fields and class number formulas, along with special values of \(L{\hbox{-}}\)functions, going back to Dirichlet. Lichtenbaum related special values to \(K{\hbox{-}}\)theory in the 70s, and this along with the theory of perverse sheaves in the early 80s led to the Beilinson conjecture and motivic complexes in the 90s.
As an aside, there is also a notion of \(p{\hbox{-}}\)adic \(L{\hbox{-}}\)functions and corresponding \(p{\hbox{-}}\)adic motives.
An outline for today:
Sheaves with transfers, which are modeled on étale homotopy sheaves
Homotopy sheaves over perfect fields
Motivic complexes
There are three main notions for étale sheaves:
We’ll fix \(S\) a regular Noetherian scheme.
For \(X,Y\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}}\), a finite correspondence \(\alpha\) from \(X\) to \(Y\) is a formal sum \begin{align*} \alpha = \sum_{i=1}^m m_i [Z_i] && \text{with } Z_i \subseteq X \underset{\scriptscriptstyle {S} }{\times} Y \text{ closed, integral} \end{align*} with \(Z_i\to X\) finite and dominant over a connected component of \(X\), i.e. an algebraic cycle in the product. These form an abelian group denoted \(c(X, Y) \in {\mathsf{Ab}}{\mathsf{Grp}}\), and can be composed without imposing any equivalence relation on algebraic cycles.
We can thus define a closed symmetric monoidal (additive) category enriched over abelian groups, the category of finite correspondences over \(S\): \begin{align*} \mathsf{C} &\mathrel{\vcenter{:}}={\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} \\ \mathrm{Ob}(\mathsf{C}) &\mathrel{\vcenter{:}}={\operatorname{Ob}}({\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} ) \\ \mathsf{C}(X, Y) &\mathrel{\vcenter{:}}= c(X, Y) .\end{align*} where the monoidal structure is the cartesian product over \(S\) on objects and on \(c(X, Y)\) is induced by the exterior product of algebraic cycles.
Writing \(XYZ \mathrel{\vcenter{:}}= X { \underset{\scriptscriptstyle {S} }{\times} } Y { \underset{\scriptscriptstyle {S} }{\times} } Z\), we have smooth projection maps \begin{align*} p: XYZ &\to XY \\ r: XYZ &\to XZ \\ q: XYZ &\to YZ .\end{align*} Given cycles \(\alpha\in c(X, Y), \beta\in c(Y, Z)\), these pull back to \(XYZ\) and intersect properly, with their intersection product given by Serre’s Tor formula.
Let \(Y \xrightarrow{f} X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}}\), and define the graph of \(f\) as the following pullback:
Here \(\delta\) is the diagonal immersion of \(X_{/ {S}}\).
Note that \(\Gamma_f \subseteq YX\) is a closed subscheme, and there is an associated algebraic cycle \begin{align*} [\Gamma_f]_{XY} \in c(Y, X) .\end{align*}
Letting \({\varepsilon}: YX\to XY\) be the permutation of factors, \({\varepsilon}_* [\Gamma_f] \in c(X, Y)\) is a finite correspondence denoted \(f^t\), the transpose of \(f\).
Several of the operations from the six functor formalism appear here:
We now enlarge \({\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}}\) to a larger abelian category. This uses the fact that the Yoneda embedding will be a fully faithful functor \begin{align*} X\mapsto c({-}, X) \mathrel{\vcenter{:}}={\mathbb{Z}}^{\mathrm{tr}}_{/ {S}} (X) \end{align*} landing in a cocomplete abelian category extending the 6 functors.
A presheaf with transfers \({\mathcal{F}}\) over \(S\) is an additive functor \begin{align*} {\mathcal{F}}: {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} ^{\operatorname{op}}\to {\mathsf{Ab}}{\mathsf{Grp}} .\end{align*} We then define a category of presheaves with transfers over \(S\): \begin{align*} \mathsf{C} &\mathrel{\vcenter{:}}={\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }_{/ {S}} \\ {\operatorname{Ob}}(\mathsf{C}) &\mathrel{\vcenter{:}}=\text{Presheaves with transfers } {\mathcal{F}}\\ \mathsf{C}({\mathcal{F}}, {\mathcal{G}}) &\mathrel{\vcenter{:}}=\text{Natural transformations } \eta: {\mathcal{F}}\to{\mathcal{G}} .\end{align*}
Let \(f\in {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} (Y, X)\). Note that by contravariance of presheaves \({\mathcal{F}}\) we always get maps \begin{align*} {\mathcal{F}}(f) \in {\mathsf{Ab}}{\mathsf{Grp}}({\mathcal{F}}(X), {\mathcal{F}}(Y)) .\end{align*} The data of a transfer is the additional following operation on \({\mathcal{F}}\), yielding a “wrong way” map: \begin{align*} f_* \mathrel{\vcenter{:}}={\mathcal{F}}(f^t) \in {\mathsf{Ab}}{\mathsf{Grp}}( {\mathcal{F}}(Y), {\mathcal{F}}(X)) .\end{align*}
A Nisnevich cover of \(X\in {\mathsf{Sch}}\) is a family of étale morphisms \(\left\{{ W_i \xrightarrow{p_i} X }\right\}_{i\in I}\) where for \(x\in X\), \(p_i(w) = x\) for some \(w\in W_i\) inducing a trivial residual extension \(\kappa(w) / \kappa(x)\).
For \({\mathcal{F}}: {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} ^{\operatorname{op}}\to {\mathsf{Ab}}{\mathsf{Grp}}\) any abelian presheaf, \({\mathcal{F}}\) is a sheaf for the Nisnevich topology iff \({\mathcal{F}}(\Delta)\) is a cartesian square for every distinguished square \(\Delta\) of the following form:
Here \(j\) is an open immersion, has reduced closed complement \(Z\), \(p\) is étale, and \(p^{-1}(Z) \xrightarrow{\sim} Z\).
There is a canonical embedding \begin{align*} \gamma: {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} &\to {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} \\ X &\mapsto X \\ (Y\to X) &\mapsto [\Gamma_f]_{XY} \in c(Y, X) .\end{align*} A sheaf with transfers is a presheaf \({\mathcal{F}}\in {\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }_{/ {S}}\) such that \begin{align*} {\mathcal{F}}\circ \gamma: \in {\mathsf{Sh}}\qty{ {\mathsf{Sch}}^{{\mathrm{Nis}}}_{/ {S}} , {\mathsf{Ab}}{\mathsf{Grp}}} ,\end{align*} i.e. the composition \({\mathcal{F}}\circ \gamma\) is a sheaf on the Nisnevich site of schemes (a Nisnevich sheaf). These form a category denoted \({\mathrm{tr}}{\mathsf{Sh}}_{/ {S}}\), and there is an adjunction \begin{align*} \adjunction {\mathop{\mathrm{Forget}}} {a^{\mathrm{tr}}} {{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}} } {{\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }_{/ {S}} } \end{align*} where \({ \left.{{a^{\mathrm{tr}}({\mathcal{F}})}} \right|_{{{\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} }} } = \left( F\circ \gamma \right)^{\scriptscriptstyle \mathrm{sh}}\).
The smooth site on \({\mathsf{Sch}}_{/ {S}}\) is big in the following sense: to give a Nisnevich sheaf in this site is equivalent to an assignment \begin{align*} {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} &\to {\mathsf{Sh}}({\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} , {\mathsf{Ab}}{\mathsf{Grp}}) \\ X &\mapsto {\mathcal{F}}_X \\ (Y\xrightarrow{f} X) &\mapsto (f^*({\mathcal{F}}_Y) \xrightarrow{\tau_f} {\mathcal{F}}_X) .\end{align*} Noting that \(\tau_f\) is not generally an isomorphism, it somehow measures a defect of base change. In particular, \({\mathsf{Sh}}({\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} )\) is a much bigger category than \({\mathsf{Sh}}({\mathsf{Sch}}^{\mathrm{Nis}}_{/ {S}} )\).
As before, the last two examples don’t form sheaves with transfers:
We have \(f^*, p_\sharp, \otimes\) on \({\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}}\), and these can be extended to sheaves:
\(f_*(F) \mathrel{\vcenter{:}}= F \circ f^*\), which yields a base change/direct image adjunction: \begin{align*} \adjunction{f^*}{f_*}{{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}} }{{\mathrm{tr}}{\mathsf{Sh}}_{/ {T}} } .\end{align*}
\(p:T\to S\) yields a forget base/base change adjunction: \begin{align*} \adjunction{p_\sharp}{p^*}{{\mathrm{tr}}{\mathsf{Sh}}_{/ {T}} }{{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}} } ,\end{align*} where for open immersions, \(p_\sharp\) is \(p_!\), the exceptional direct image
\({}_{h}\otimes\) on \({\mathrm{tr}}{\mathsf{Sh}}_{/ {S}}\) yields a closed symmetric monoidal structure \begin{align*} \adjunction{{-}\otimes^{\mathrm{tr}}{\mathcal{F}}}{\underline{\mathop{\mathrm{Hom}}}^{\mathrm{tr}}({\mathcal{F}}, {-})}{{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}} }{{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}} } ,\end{align*} where \(\underline{\mathop{\mathrm{Hom}}}^{\mathrm{tr}}\) is an internal hom.
Let \({\mathcal{F}}\in {\mathrm{tr}}{\mathsf{Sh}}(S)\) and let \(p:{\mathbb{A}}^1_{/ {X}} \to X\) be the canonical projection. We say \({\mathcal{F}}\) is \({\mathbb{A}}^1{\hbox{-}}\)invariant or a homotopy sheaf if for any \(X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}}\), there is an induced isomorphism \begin{align*} p^*: F(X) \xrightarrow{\sim} F({\mathbb{A}}^1_{/X}) .\end{align*} These form a category denote \({\mathsf{HI}}^{\mathrm{tr}}(S)\).
Let \(\alpha, \beta \in {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} (X, Y)\) be two morphisms. We then say \(\alpha\) is homotopic to \(\beta\) and write \(\alpha\sim\beta\) iff there exists a \(H\) satisfying the following: \begin{align*} H &\in {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} ({\mathbb{A}}^1\times X, Y) \\ \alpha &= H \circ s_0 \\ \beta &= H \circ s_1 ,\end{align*} where \(s_0, s_1\) are the zero and unit sections of \({\mathbb{A}}^{1}_{/X}\in \mathsf{Ring}{\mathsf{Sch}}_{/X}\). This yields an equivalence relation, and we set \begin{align*} \pi_S(X, Y) \mathrel{\vcenter{:}}={\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} (X, Y)/\sim .\end{align*}
The sheaves \({\mathbb{G}}_m({-})\) and \(\mathop{\mathrm{Hom}}({-}, A)\) are \({\mathbb{A}}^1{\hbox{-}}\)invariant.
Let \(S\in {\mathsf{Aff}}{\mathsf{Sch}}\) be regular and \(C\in{\mathsf{Aff}}{\mathsf{Sch}}_{/ {S}}\) an affine curve admitting a good compactification \(\tilde C\):
Then for any \(X\in {\mathsf{sm}}{\mathsf{Aff}}{\mathsf{Sch}}_{/ {S}}\), there is a canonical isomorphism of groups: \begin{align*} \pi_S(X, C) &\xrightarrow{\sim} {\operatorname{Pic}}(X { \underset{\scriptscriptstyle {S} }{\times} } \tilde C { \underset{\scriptscriptstyle {S} }{\times} } C_\infty) \\ \alpha &\mapsto [{\mathcal{O}}(\alpha)] ,\end{align*} where \({\mathcal{O}}(\alpha)\) is the line bundle associated to \(\alpha\), viewed as a Cartier divisor in \(X { \underset{\scriptscriptstyle {S} }{\times} } \tilde C\).
Fix \(k \in \mathsf{Perf}\mathsf{Field}\), then a function field \(E\) over \(k\) is a separable finitely generated field extension \(E_{/ {k}}\). One can define the fiber of a homotopy sheaf \(F\) at \(E_{/ {k}}\) as a filtered colimit over smooth finitely generated sub \(k{\hbox{-}}\)algebras \(A\): \begin{align*} F(E_{/ {k}} ) \mathrel{\vcenter{:}}=\colim_{A_{/ {k}} \leq E_{/ {k}} } F(\operatorname{Spec}A) .\end{align*} This yields a fiber functor: it is exact and commutes with coproducts.
We define the category \({\mathsf{HI}}^{\mathrm{tr}}(S) \leq {\mathrm{tr}}{\mathsf{Sh}}(S)\) to be the category of all homotopy sheaves, which is (Grothendieck) abelian and bicomplete. The forgetful functor is exact, so there is an adjunction \begin{align*} \adjunction {h_0} {\mathop{\mathrm{Forget}}} {{\mathrm{tr}}{\mathsf{Sh}}_{/ {k}} } {{\mathsf{HI}}^{\mathrm{tr}}_{/ {k}} } .\end{align*}
If \(F\) is a homotopy sheaf and \(Z \xrightarrow{i} X\) is a codimension 1 closed immersion in \({\mathsf{sm}}{\mathsf{Sch}}_{/ {k}}\) with \(j: X\setminus Z\to X\) an open immersion, then there is a SES of Nisnevich sheaves over \(X_{\mathrm{Nis}}\): \begin{align*} 0\to {\mathcal{F}}_X \to j_* {\mathcal{F}}_{X\setminus Z} \to i_* {\mathcal{F}}_{-1, Z} \to 0 .\end{align*}
If \(k\in \mathsf{Perf}\mathsf{Field}\) and \({\mathcal{F}}\in {\mathsf{HI}}^{\mathrm{tr}}_{/ {k}}\), then for all \(m\) and all \(X \in {\mathsf{sm}}{\mathsf{Sch}}_{/ {k}}\), there is an isomorphism \begin{align*} p^*: H^n_{\mathrm{Nis}}(X; {\mathcal{F}}) \xrightarrow{\sim} H_{\mathrm{Nis}}^n({\mathbb{A}}^1_{/X}; {\mathcal{F}}) ,\end{align*} so the presheaf \(H^n_{\mathrm{Nis}}({-}, {\mathcal{F}})\) is homotopy invariant.
For \(Z\hookrightarrow X\) smooth closed of codimension \(m\), then \begin{align*} H_Z^n(X; {\mathcal{F}}) \xrightarrow{\sim} H_{\mathrm{Nis}}^{n-m}(Z; {\mathcal{F}}_{-m}) .\end{align*} Here the LHS is Nisnevich cohomology with support.
For \(X\) smooth, \({\mathcal{F}}_x\) is Cohen-Macaulay and there is a Cousin complex \(C^*(X; {\mathcal{F}})\), also called the Gersten complex of \({\mathcal{F}}\), and one can compute Nisnevich cohomology as \begin{align*} H^n_{\mathrm{Nis}}(X; {\mathcal{F}}) \xrightarrow{\sim} H^n(C^*(X; {\mathcal{F}})) .\end{align*}
Write \({\mathbb{S}}^{n} \mathrel{\vcenter{:}}={\mathbb{G}}_m{ {}^{ \scriptstyle {}_{h} {\otimes_{}^{n}} } }\), then for a function field \(E_{/ {k}}\), there is an isomorphism of sheaves \begin{align*} {\mathbb{S}}^n(E) \xrightarrow{\sim} \underline{ {\mathsf{K}}^{\scriptstyle\mathrm{M}} _n}(E) ,\end{align*} so this identifies with the \(n\)th unramified \({\mathsf{K}}{\hbox{-}}\)theory of \(E\). Using the Gersten resolution of \({\mathbb{S}}^{n}\), one obtains an isomorphism of groups \begin{align*} H_{\mathrm{Nis}}^n(X; {\mathbb{S}}^n) \xrightarrow{\sim} {\operatorname{CH}}^n(X) ,\end{align*} the Chow group of codimension \(n\) cycles modulo rational equivalence.
Recall the Beilinson conjectures (84/86), and Bloch’s higher Chow groups (86). Today we’ll discuss the \({\mathbb{A}}^1{\hbox{-}}\)homotopy category \({\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}\) and the category of motives \({\mathsf{DM}}_{/ {S}}\). We’ll be working \({\mathsf{Top}}_*\), the infinity category of pointed spaces, and \(\mathbf{D} {{\mathsf{Ab}}}\), the (infinity) derived category of abelian groups.
Consider infinity functors \begin{align*} F: {\mathsf{sm}}{\mathsf{Sch}}^{\operatorname{op}}_{/ {S}} \to {\mathsf{Top}}_* \end{align*} and define \begin{align*} F(X, Z) \mathrel{\vcenter{:}}={\operatorname{hofib}}( F(X) \to F(X\setminus Z)) .\end{align*} Then the (pointed) \({\mathbb{A}}^1\) homotopy category of \(S\), denoted \({\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}}\), consists of such functors \(F\) that satisfy
Excision: for all \((Y, T) \to (X, Z)\) excisive, there is a weak equivalence \begin{align*} F(Y, T) \underset{\scriptscriptstyle W}{\rightarrow}F(X, Z) .\end{align*}
Homotopy invariance: The canonical projection \({\mathbb{A}}^1_{/ {X}}\) induces a weak equivalence \begin{align*} F(X) \underset{\scriptscriptstyle W}{\rightarrow}F({\mathbb{A}}^1_{/ {X}} ) .\end{align*}
This category admits a monoidal structure, which we’ll denote by the smash product \(X\wedge Y\).
The excision axiom can be replaced by the following condition: for distinguished squares \(\Delta\), the image \(F(\Delta)\) is homotopy cartesian:
We can similarly ask for (infinity additive) functors \(K:{\mathsf{sm}}{\mathsf{Sch}}^{\operatorname{op}}_{/ {S}} \to \mathbf{D} {{\mathsf{Ab}}}\) satisfying these properties.
We can use infinity categorical localization theory to build this category. For a scheme \(S\), view a pointed space over \(S\) as a presheaf valued in pointed simplicial sets, viewed as an infinity category. We can then construct \begin{align*} {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} &= \underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} , \mathsf{sSet}_*) \left[ { \scriptstyle W^{-1}} \right] \mathrel{\vcenter{:}}=\mathsf{C} \left[ { \scriptstyle W^{-1}} \right] \\ \\ W &\mathrel{\vcenter{:}}=\left\{{ {\mathbb{Z}}_S^*({\mathbb{A}}^1_{/ {X}} ) \to {\mathbb{Z}}_S^*(X) {~\mathrel{\Big|}~}X\in {\operatorname{Ob}}(\mathsf{C}) }\right\} .\end{align*}
One can similarly do this for \({\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} , \mathbf{D} { {\mathsf{Ab}}} ) = \mathbf{D} { {\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }_{/ {S}} }\). Effective motives \({\mathsf{DM}}^{\mathsf{eff}}_{/ {S}}\) can be constructed by replacing presheaves with \(\mathbf{D} {{\mathsf{Sh}}^{\mathrm{tr}}_{/ {S}} }\) and localizing at \({\mathbb{Z}}_S^{\mathrm{tr}}({\mathbb{A}}^1_{/ {X}} ) \to {\mathbb{Z}}_S^{{\mathrm{tr}}}(X)\).
\({\mathbb{A}}^1{\hbox{-}}\)local spaces are those \(S{\hbox{-}}\)spaces for which the realization induces a weak equivalence on mapping spaces: \begin{align*} {\left\lvert {{\mathbb{A}}^1_{/ {X}} } \right\rvert} _+ \to X_+ \leadsto \mathop{\mathrm{Hom}}(X_+, Y ) \underset{\scriptscriptstyle W}{\rightarrow} \mathop{\mathrm{Hom}}(\qty{ {\mathbb{A}}^1_{/ {X}} } _+, Y) \quad \forall Y\in {\operatorname{Ob}}(\mathsf{C}) .\end{align*}
Fix \(k\in\mathsf{Perf}\mathsf{Field}\) and consider complexes of sheaves \(K \in\mathsf{Ch}\qty{ {\mathsf{Sh}}^{\mathrm{tr}}_{/ {S}} }\). We can define cohomology sheaves \(H^*(K)\) by taking kernels mod images in \(\underset{ \mathsf{pre} } {\mathsf{Sh} }^{\mathrm{tr}}_{/ {S}}\) and Nisnevich-sheafifying to get a sheaf \begin{align*} \underline{H}^i(K) \mathrel{\vcenter{:}}= \left( H^i(K) \right)^{\scriptscriptstyle \mathrm{sh}} \in {\mathsf{Sh}}^{\mathrm{tr}}_{/ {S}} .\end{align*} This gives a way to take cohomology of complexes of sheaves with transfers.
\(K\) is \({\mathbb{A}}^1{\hbox{-}}\)local iff for all \(\underline{H^n}(K)\) is \({\mathbb{A}}^1{\hbox{-}}\)local in \({\mathsf{HI}}^{\mathrm{tr}}_{/ {k}}\) for all \(n\)
Define standard cosimplicial scheme as \begin{align*} \Delta^n\mathrel{\vcenter{:}}=\operatorname{Spec}\qty{k[x_0, \cdots, x_n] \over \left\langle{ \sum x_i }\right\rangle } \in {\mathsf{Sch}}_{/ {k}} \end{align*} and for \(K\in \mathsf{Ch}\qty{ {\mathrm{tr}}{\mathsf{Sh}}_{/ {k}} }\) a complex of sheaves with transfers, the Suslin singular complex is the complex of sheaves defined as \begin{align*} \underline{C_{*}^S}(K), && {\mathsf{\Gamma}\qty{X_{/ {S}} } } \mathrel{\vcenter{:}}={ \operatorname{Tot} }^{\Pi}K(\Delta^\bullet { \underset{\scriptscriptstyle {k} }{\times} } X) .\end{align*} for \(X\in{ {\mathsf{sm}}{\mathsf{Sch}}}_{/ {S}}\).
The Suslin singular complex \(\underline{C_*^S}(K)\) is \({\mathbb{A}}^1{\hbox{-}}\)local, and the functor \(\underline{C_*^S}({-})\) is an isomorphism in \({\mathsf{DM}}\)?
The homological motive of a smooth scheme \(X\in{ {\mathsf{sm}}{\mathsf{Sch}}}_{/ {k}}\) is \begin{align*} M(X) \mathrel{\vcenter{:}}= C_*(S){\mathbb{Z}}^{\mathrm{tr}}_{/ {k}} (X) .\end{align*}
The Tate twist is defined as \begin{align*} {\mathbb{Z}}(1) \mathrel{\vcenter{:}}=\operatorname{coker}\qty{ M\left\{{1}\right\} \to M({\mathbb{G}}_m)}[-1] .\end{align*}
\({\mathbb{Z}}(1) = {\mathbb{G}}_m[-1]\in [0, 1]\) is supported in homotopy degree 0 and 1 (and in fact just in degree 1), and generally \({\mathbb{Z}}(n) = {\mathbb{Z}}(1){ {}^{ \scriptstyle\otimes_{k}^{n} } } \in (-\infty, n]\) is supported in degree at most \(n\).
For all \(n>0\), \({\mathbb{Z}}(n) \in [1, n]\), so it is in fact only supported in positive degrees. Moreover, for \(E_{/ {k}} \in{\mathsf{fn}}\mathsf{Field}\), \begin{align*} H^{i> n }\qty{ C_E(\Delta_E^*, {\mathbb{G}}_m^n)_{\mathbb{Q}}} = 0 .\end{align*} By Bloch-Kato, the integral and rational cases are equivalent.
For \(X\in{ {\mathsf{sm}}{\mathsf{Sch}}}_{/ {k}}\), the motivic cohomology is given by \begin{align*} H_{ \mathrm{mot}} ^{n, i}(X) \mathrel{\vcenter{:}}= H^n_{\mathrm{Nis}}(X; {\mathbb{Z}}(i)) .\end{align*} The grading \(n\) is the degree, and \(i\) is the twist.
Let \({\mathbb{Z}}(m)\in \mathbf{D} {{\mathsf{Sh}}^{\mathrm{tr}}_{/ {k}} }\), then for \(X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {k}}\) we have \begin{align*} H^{n, i}_M(X) = \mathop{\mathrm{Hom}}_{{\mathsf{DM}}^\mathsf{eff}}(M(X), {\mathbb{Z}}(i)[n] ) .\end{align*} Taking the sheaf defined in top diagonal bidegree, this can be identified with unramified Milnor \({\mathsf{K}}{\hbox{-}}\)theory: \begin{align*} \operatorname{\mathcal{H}}^n({-}; {\mathbb{Z}}(n)) = \operatorname{\mathcal{K^M}}_n({-}) .\end{align*}
Let \(X: { {\mathsf{sm}}{\mathsf{Sch}}}^{\operatorname{op}}_{/ {S}} \to {\mathsf{Top}}_*\), which is a “space” in an infinity categorical sense, and consider \(f:T\to S\) a morphisms of schemes. We can form \(f^*: { {\mathsf{sm}}{\mathsf{Sch}}}_{/ {S}} \to { {\mathsf{sm}}{\mathsf{Sch}}}_{/ {T}}\) which induces an adjunction \begin{align*} \adjunction {f^*} {f_*} {{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} } {{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {T}} } .\end{align*} For \(p:T\to S\) smooth, we can define \(p_\sharp\) and \(p^*\) similarly, yielding \begin{align*} \adjunction {p_\sharp} {p^*} {{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} } {{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {T}} } .\end{align*} There is also a stable lift of the tensor product to a smash product \({-}\wedge{-}\), yielding \begin{align*} \adjunction {{-}\wedge{-}} {\underline{\mathop{\mathrm{Hom}}}({-}, {-})} {{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} } {{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} } .\end{align*}
There are also formulas for things like \(f^*(K\wedge X_+)\), as well as (smooth) base change and projection.
Let \(i: Z\hookrightarrow S\) be closed and \(U\mathrel{\vcenter{:}}= S\setminus Z\) with \(j: U \underset{\scriptscriptstyle O}{\hookrightarrow}S\) an open immersion. Then for all \(X\in {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}}\), there is a homotopy cofiber sequence \begin{align*} j_\sharp j^*(X) \to X\to i_* i^* X ,\end{align*} where the maps are given by units/counits of the corresponding adjunctions.
This can be restated as a geometric version of \({\mathbb{A}}^1{\hbox{-}}\)homotopy equivalence: that there is a weak equivalence \begin{align*} {X \over X \setminus(X{ \underset{\scriptscriptstyle {S} }{\times} } Z)} \underset{\scriptscriptstyle W}{\rightarrow}i_*((X_Z)_+) .\end{align*} We don’t have the 6 functor formalism unstably.
One can take spheres in \({\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}}\) to be the pointed space \begin{align*} ({\mathbb{P}}^1, \infty ) \simeq S^1 \wedge({\mathbb{G}}_m, 1) .\end{align*} This yields a definition of loop spaces: \begin{align*} {\Omega}_{{\mathbb{P}}^1}({-}) \mathrel{\vcenter{:}}={\mathbf{R}}\underline{\mathop{\mathrm{Hom}}}({\mathbb{P}}^1, {-}) ,\end{align*} where one needs to derive this construction.
The stable homotopy category \({\mathsf{SH}}_{/ {S}}\) is defined as the limit \begin{align*} \cdots \xrightarrow{{\Omega}_{{\mathbb{P}}^1}} {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} \xrightarrow{{\Omega}_{{\mathbb{P}}^1}} {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} \xrightarrow{{\Omega}_{{\mathbb{P}}^1}} {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} ,\end{align*} which is a construction that works for presentable monoidal infinity categories.
This makes \({\mathbb{P}}^1\) a monoidally invertible object, and it turns out to invert \({\mathbb{G}}_m\) and the classical sphere \(S^1\). This is because if we define \({\mathbb{S}}^{n, m} \mathrel{\vcenter{:}}= S^n \wedge{\mathbb{G}}_m^m\), we have \begin{align*} {\mathbb{P}}^1 &\simeq{\mathbb{S}}^{1, 1} && \mathrel{\vcenter{:}}= S^1 \wedge{\mathbb{G}}_m \\ {\mathbb{A}}^n\setminus\left\{{0}\right\}&\simeq{\mathbb{S}}^{n-1, n} && \mathrel{\vcenter{:}}= S^n \wedge{\mathbb{G}}_m{ {}^{ \scriptstyle\otimes_{k}^{n} } } .\end{align*}
A concrete model is given by sequences of objects, called \({\mathbb{P}}^1{\hbox{-}}\)spectra. These are sequences of pointed spaces \(X\mathrel{\vcenter{:}}=(X_m)\) with \({\mathbb{A}}^1{\hbox{-}}\)homotopy equivalences \begin{align*} X_m \underset{\scriptscriptstyle W}{\rightarrow}{\Omega}_{{\mathbb{P}}^1}(X_{m+1}) .\end{align*} This is somehow related to \({\mathbb{P}}^1\wedge X_m \xrightarrow{\sigma_m} X_{m+1}\). \({\mathsf{SH}}_{/ {S}}\) satisfies the following universal property: it is the universal presentable monoidal infinity category receiving a functor \begin{align*} {\Sigma}^\infty : {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} \to {\mathsf{SH}}_{/ {S}} \end{align*} such that \({\mathbb{P}}^1\wedge({-})\) is invertible. It turns out that the category \({\mathsf{SH}}_{/ {S}}\) admits a diagram relating it to all of the categories that have appeared thus far.
For \(f:T\to S\) a morphism of schemes, separated of finite type, there is a triangulated adjunction \begin{align*} \adjunction{f_!}{f^!}{{\mathsf{SH}}_{/ {S}} } {{\mathsf{SH}}_{/ {T}} } \end{align*} such that
Moreover \(\mathop{\mathrm{Th}}(T_f)\) is tensor-invertible in \({\mathsf{SH}}_{/ {S}}\) with inverse \(\mathop{\mathrm{Th}}(-T_f)\).
There is a base change formula, and \(p^* f_! \cong g_! q^*\) for cartesian squares:
There is a projection formula \begin{align*} f_!(C \otimes f^*(F)) \cong f_!(C) \otimes F .\end{align*} Moreover, \({\mathsf{SH}}({-})\) satisfies a generalized Beilinson formalism.
\({\mathsf{SH}}_{/ {S}}\) is triangulated, and there are several ways to construct a triangulated rationalization \({\mathsf{SH}}_{/ {S}} \otimes{\mathbb{Q}}\). This decomposes as \begin{align*} {\mathsf{SH}}_{/ {S}} \otimes{\mathbb{Q}}\xrightarrow{\sim} \qty{{\mathsf{SH}}_{/ {S}} }_{{\mathbb{Q}}^+} \times \qty{{\mathsf{SH}}_{/ {S}} }_{{\mathbb{Q}}^-} .\end{align*}
For \(S\) regular, the plus part is equivalent to \(\qty{ {\mathsf{DM}}_{/ {S}} }_{\mathbb{Q}}\). Writing \(S^0 \mathrel{\vcenter{:}}= S\otimes_{\mathbb{Z}}{\mathbb{Q}}\), the minus part is equivalent to the Witt sheaf \(\operatorname{\mathcal{W}}^{{\mathbb{Q}}}_{S^0}\), which is connected to quadratic forms. Reindexing and setting \(\tilde {\mathbb{S}}^{n, i} \mathrel{\vcenter{:}}= S^{n-i} \wedge{\mathbb{G}}_m{ {}^{ \scriptstyle\otimes_{k}^{i} } }\), one can define cohomotopy groups \begin{align*} \qty{\pi^{n, i}_{/ {S}} }_{\mathbb{Q}} &\mathrel{\vcenter{:}}= [ {\mathbb{S}}, S^{n-i} \wedge{\mathbb{G}}_m^i ] _{\qty{{\mathsf{SH}}_{/ {S}} } _{\mathbb{Q}}} \\ & =[ {\mathbb{S}}, \tilde {\mathbb{S}}^{n, i}] _{\qty{{\mathsf{SH}}_{/ {S}} } _{\mathbb{Q}}} \\ &= [\one, \one(s)[i] ] \\ & \xrightarrow{\sim} {\mathsf{gr}\,}_\gamma^i \qty{ (K_{2i-n}) _{/ {S}} }_{\mathbb{Q}}\oplus H^{n-i}_{\mathrm{Nis}}(S^0; \operatorname{\mathcal{W}} ) ,\end{align*} where \({\mathsf{gr}\,}\) is a grading.
For \(E\in\mathsf{Field}\), this yields \begin{align*} \pi^{n, i}(E)_{\mathbb{Q}}= H_{ \mathrm{mot}} (E)_{\mathbb{Q}}\oplus W(E)_{\mathbb{Q}} .\end{align*}
There is a Grothendieck-Verdier duality: for \(f:X\to S\) smooth finite type with \(S\) regular, then \(f^!(\one_S) \simeq\mathop{\mathrm{Th}}(Lf)\). If \({\mathbb{E}}\) is a compact (constructible) object of \({\mathsf{SH}}_{/ {S}}\) the \({\mathbb{E}} {}^{ \vee }= \underline{\mathop{\mathrm{Hom}}}({\mathbb{E}}, D_*)\) and there is an isomorphism \({\mathbb{E}}\to ({\mathbb{E}} {}^{ \vee }) {}^{ \vee }\).