Schur functors and motives (Q1772112)

Let \(\mathcal A\) be \(\mathbb Q\)-linear tensor category i.e. \(\mathcal A\) is pseudo-abelian, \(\mathbb Q\)-linear and such that \(\otimes\) is \(\mathbb Q\)-bilinear. If \(\lambda\) is a partition of a natural number \(n\) then there is an idempotent (Young symmetrizer) \(c_{\lambda}\in {\mathbb Q}[\Sigma_n].\) Let \(X\in Ob(\mathcal A)\). The symmetric group \(\Sigma_n\) acts on \(X^{\otimes n}\) in an obvious way and this yields the Schur functor \(S_{\lambda}\) of \(\lambda ,\) where \(S_{\lambda}(X)= c_{\lambda}(X^{\otimes n}).\) An object of \(\mathcal A\) is Schur-finite if there exists \(n\) and a partition \(\lambda\) of \(n\) such that \(S_{\lambda}(X)=0.\) An object of \(\mathcal A\) is called even (resp. odd) if \({\bigwedge}^{n}X\) (resp. \(\text{Sym}^{n}X=0,\)) for some \(n,\) and Kimura-finite if \(X=X_{+}\oplus X_{-}\) where \(X_{+}\) is even and \(X_{-}\) is odd. Note that Kimura-finite object is a direct sum of two Schur-finite objects. The author studies the category of classical motives and the Voevodsky's category \({DM}^{\text{eff},-}_{\text{Nis}} (k,{\mathbb Q})\) [\textit{V.Voevodsky, A.Suslin} and \textit{E. M. Friedlander}, Cycles, transfers, and motivic homology theories. Ann. Math. Stud. 143 (2000; Zbl 1021.14006)]. The author shows that the motive of any curve is Kimura-finite. This result has also been proven by \textit{V.Guletski} [Finite dimensional objects in distinguished triangles, preprint, \texttt{http://www.math.uiuc.edu/K-theory/0637})]. The author also gives an example of a motive that is non-Kimura-finite but Schur-finite.