Finite dimensionality of Motives

在此期间有6个主要成果。1.考虑一个刚性张量范畴，设V是一个Schur有限对象。然后存在一个最小的杨图，它杀死V，比如λ。即μ杀死V，则μ包含λ。这样，我们可以定义V的Schur维数为λ.2。推广张量范畴中对象的有限维概念，我们定义了有限维态射的概念。一个对象V是有限维的当且仅当V的单位态射是有限维的。这样，我们得到了一个新的、方便的有限维判别准则.当X的动机是有限维的，那么它的动机zeta，X的对称乘积的Chow动机的形式和，成为有理数。在这项与Javier Elizondo教授的联合工作中，我们研究了如果我们用Chow品种代替对称产品会发生什么。令我们惊讶的是，即使X是射影平面，形式和也不是有理的。在与Jacob Murre教授的合作中，我们证明了Picard函子的唯一性，假设Chow动机Pic（X）是有限维的。Uwe Jannsen教授证明了如果循环映射是单射的，那么它是双射的。我们将这个结果推广到Chow动机。在与Nobuyoshi Takahashi教授和Kenichiro Kimura教授的联合工作中，我们研究了Schur有限对象V的动机zeta是否变得合理。答案取决于V的舒尔维数。如果舒尔维数是一个钩子，那么动机zeta是有理数。一般来说，动机zeta总是决定性的理性，但不一定是一致理性的。

There are 6 major results during this period.1. Consider a rigid tensor category, and let V be a Schur finite object. Then there exists a smallest Young diagram which kills V, say λ. namely μ kills V, then μ contains λ. In this way, we can define the Schur dimension of V to be λ.2. Generalizing the notion of finite dimensionality of an object in a tensor category, we defined the notion of finite dimensional morphism. An object V is finite dimensional if and only if the identity morphism of V is finite dimensional. In this way, we obtain a new and convenient criterion for finite dimensionality.3. When the motive of X is finite dimensional, then its motivic zeta, formal sum of the Chow motives of the symmetric products of X, becomes rational. In this joint work with Professor Javier Elizondo, we studied what happens if we replace the symmetric products by Chow varieties. To our surprise, the formal sum is not rational, even when X is the projective plane.4. In a joint work with Prefessor Jacob Murre, we proved the uniqueness of the Picard Functor, assuming that the Chow motive Pic(X) is finite dimensional.5. Professor Uwe Jannsen prove that if the cycle map is injective, then it is bijective. We generalized this result to Chow motives.6. In this joint work with Professor Nobuyoshi Takahashi and Professor Kenichiro Kimura, we studied, if the motivic zeta of Schur finite object V becomes rational. The answer depends on the Schur dimension of V. If the Schur dimension is a hook, then the motivic zeta is rational. In general, the motivic zeta is always determinantally ration], but not necessarily uniformly rational.