Motivic zeta functions in additive monoidal categories (Q2905333)

Let \({\mathcal M}_{\text{rat}}(k)\) be the category of pure Chow motives; with \(\mathbb Q\)-coefficients, over a field \(k\). A motive \(M\in{\mathcal M}_{\text{rat}}(k)\) is finite-dimensional if \(M= M^+\oplus M^-\), with \(\bigwedge^N M^+= 0\) and \(\text{Sym}^N M^-= 0\) for some \(N\gg 0\). Let \(K_0({\mathcal M}_{\text{rat}}(k))\) be the quotient of the free abelian group generated by \([M]\), for \(M\in{\mathcal M}_{\text{rat}}(k)\), modulo the relations \([M_1\oplus M_2]= [M_1]+ [M_2]\). Let \({\mathcal V}^\pm\) be the tensor category of super vector spaces, i.e., the category of \(\mathbb{Z}/2\mathbb{Z}\)-graded vector spaces. Let \({\mathcal P}_k\) be the category of smooth projective variety over \(k\); the functors \(h:{\mathcal P}_k\to{\mathcal M}_{\text{rat}}(k)\) and \(\Phi:{\mathcal M}_{\text{rat}}(k)\to sV^\pm\), with \(\Phi(M)= H^*(M)\) give rise to natural ring homomorphisms NEWLINE\[NEWLINEK_0({\mathcal P}_k)\to K_0({\mathcal M}_{\text{rat}}(k)\to K_0({\mathcal V}^\pm).NEWLINE\]NEWLINE If \(M\in{\mathcal M}_{\text{rat}}(k)\) is a finite-dimensional motive, then its motivic zeta function NEWLINE\[NEWLINE\sum_{0\leq n\leq\infty} [\text{Sym}^n M] t^n\in K_0({\mathcal M}_{\text{rat}}(k))NEWLINE\]NEWLINE is a rational function because NEWLINE\[NEWLINE\sum_{0\leq n\leq\infty} [\text{Sym}^n M]t^n= {\sum_{0\leq n\leq N}[\text{Sym}^n M^-] t^n\over \sum_{0\leq n\leq N}[\bigwedge^n M^+](-t)^n}.NEWLINE\]NEWLINE In order to study finite-dimensionality and the motivic zeta function in a triangulated category of mixed motives one has to consider the more general definition of Schur finiteness, which has the two-out-of-three property, i.e if \(M_1\to M_2\to M_3\to M_1[1]\) is a distinguished triangle with two of the \(M_i\)'s Schur finite, then the third is also Schur finite. An object \(X\), in a \(\mathbb{Q}\)-linear, pseudo-abelian, symmetric, monoidal category \({\mathcal C}\) is Schur finite if there exists a partition, or a Young diagram, \(\lambda\) such that \(S_\lambda X= 0\). Here \(S_\lambda: {\mathcal C}\to{\mathcal C}\) is the Schur functor associated to partitions \(\lambda\). In particular \(\text{Sym}^n X= S_{(n)}X\).NEWLINENEWLINE In this paper the authors consider several variants for the rationality of a formal power series \(f(t)= \sum_n a_n t^n\) in \(A[[t]]\), where \(A\) is a commutative ring with unit; uniform rationality, global rationality, determinantal rationality, pointwise rationality, and prove that the motivic zeta function of a Schur finite object is determinantally rational. Also if \(X\in{\mathcal C}\) is annihilated by a hook diagram (i.e. \(S_\lambda X= 0\), where the Young diagram \(\lambda\) is a hook) the motivic zeta function is uniformly rational. On the other hand uniform rationality does not hold in general for Schur finite objects.