Mixed Weil cohomologies (Q411727)

Let \(k\) be a perfect field and let \(K\) be a field of characteristic 0. A Weil cohomology is a cohomology defined on smooth and projective varieties over \(k\) with values in \(K\)-vector spaces satisfing the usual properties: Künneth formula, Poincaré duality, cycle map etc. According to Grothendieck's conjecture about the existence of an abelian category of mixed motives \({\mathcal M}{\mathcal M}\) a mixed Weil cohomology should define an exact tensor functor from \({\mathcal M}{\mathcal M}\) to the category of (super) vector spaces over \(K\), such that its restriction to pure motives is a Weil cohomology. The aim of this paper is to provide a simple set of axioms for a cohomology theory to induce a symmetric monoidal realization functor \(R\) from Voevodsky's triangulated category \(DM_{gm}(k)_{{\mathbb{Q}}}\) of mixed motives, to the bounded derived category \(D^b(K)\) of finite-dimensional \(K\)-vector spaces.NEWLINENEWLINE Let \({\mathcal A}\) be the category of smooth affine \(k\)-schemes and let \(E\) be a presheaf of commutative differential graded \(K\)-algebras on \({\mathcal A}\). Given any smooth affine scheme \(X\), a closed subset \(Z\subset X\) such that \(U= X- Z\) is affine define, for any integer \(n\) NEWLINE\[CARRIAGE_RETURNNEWLINEH^n_Z(X, E)= H^{n-1}(\text{Cone}(E(X)- E(U))).CARRIAGE_RETURNNEWLINE\]NEWLINE A presheaf of differential graded \(K\) algebras \(E\) is said to be a \(K\)-linear mixed Weil theory on \({\mathcal A}\) if it satisfies the following axiomsNEWLINE{\parindent=7mmNEWLINE\begin{itemize}\item[(i)]\(\dim_KH^i(\text{Spec}k, E)= 1\) for \(i=0\) and \(0\) otherwise;NEWLINE\item[(ii)]\(\dim_KH^i({\mathbf A}^1_k, E)= 1\) for \(i=0\) and \(0\) otherwise;NEWLINE\item[(iii)]\(\dim_KH^i({\mathbf G}_m, E)= 1\), if \(i=0\), \(1\) and \(0\) otherwise;NEWLINE\item[(iv)]\(\bigoplus_{p+q=n}H^p(X,E)\otimes H^q(Y,E)\simeq H^n(X\times Y,E)\) (Künneth formula);NEWLINE\item[(v)]For any \(X\) in \({\mathcal A}\) the cohomology groups of the complex \(E(X)\) are isomorphic to the Nisnevich hypercohomology groups of \(X\) with coefficients in \(E_{Nis}\) (Nisnevich descent).NEWLINENEWLINE\end{itemize}}NEWLINEThe author shows that property (iv) is shown to be equivalent to the excision isomorphism \(H^*_T(Y, E)\simeq H^*_Z(X,E)\) for a commutative diagram NEWLINE\[CARRIAGE_RETURNNEWLINE\begin{tikzcd} T\rar["j"]\dar["g" '] & Y\dar["f"]\\ Z \rar["i" '] & X\end{tikzcd}CARRIAGE_RETURNNEWLINE\]NEWLINE where \(i\) and \(j\) are closed immersions, \(X,Y,X- Z, Y- T\) are smooth affine, \(f\) is étale. \(f^{-1}(x- z)= Y- T)\), \(g\) is an isomorphism.NEWLINENEWLINE With these definition and properties one gets cohomology groups \(H^n(X, E)\) for any smooth scheme \(X\), such thatNEWLINE{\parindent=7mmNEWLINE\begin{itemize}\item[(1)]the \(K\)-vector spaces \(\bigoplus_n H^n(X, E)\) are finite-dimensional;NEWLINE\item[(2)]there is a cycle class map from the motivic cohomology \(H^q(X,{\mathbb{Q}}(p)\to H^q(X,E) (p)\) which is compatible with cup product;NEWLINE\item[(3)]there are cohomology groups with compact support \(H^q(X,E)\) such that Poincaré duality holds.NEWLINENEWLINE\end{itemize}}NEWLINEThen there exists a symmetric monoidal triangulated functor \(R:DM_{gm}(k)_{{\mathbb{Q}}}\to D^b(K)\) such that, for any smooth \(k\)-scheme \(X\) one has the following canonical identifications NEWLINE\[CARRIAGE_RETURNNEWLINER(M_{gm}(X)^\vee)\simeq R(M_{gm}(X))^\vee\simeq{\mathbb{R}}\Gamma(X,E),CARRIAGE_RETURNNEWLINE\]NEWLINE where \(\vee\) denotes the dual. Moreover, for any object \(M\) in \(DM_{gm}(k)_{{\mathbb{Q}}}\) and any integer \(p\) one has \(R(M(p))= R(M)(p)\).