Schur finiteness and nilpotency

Let A be a Q-linear pseudo-abelian rigid tensor category. A notion of finiteness due to Kimura and (independently) O’Sullivan guarantees that the ideal of numerically trivial endomorphism of an object is nilpotent. We generalize this result to special Schur-finite objects. In particular, in the category of Chow motives, if X is a smooth projective variety which satisfies the homological sign conjecture, then Kimura-finiteness, a special Schur-finiteness, and the nilpotency of CH^{ni}(X^i\times X^i)_{num} for all i (where n=dim X) are all equivalent.

💡 Research Summary

The paper investigates the relationship between finiteness conditions in tensor categories and the nilpotency of numerically trivial endomorphisms. Working in a Q‑linear pseudo‑abelian rigid tensor category 𝒜, the authors start from the well‑known result of Kimura (and independently O’Sullivan) that an object satisfying Kimura‑finiteness has a nilpotent ideal of numerically trivial endomorphisms. Kimura‑finiteness means that an object M can be decomposed into a “positive” part M⁺ and a “negative” part M⁻ such that sufficiently high symmetric powers of M⁺ and sufficiently high exterior powers of M⁻ vanish. This decomposition forces any numerically trivial morphism f∈End𝒜(M) to satisfy f^N=0 for some N.

The main contribution is a substantial generalisation of this phenomenon to a broader class called special Schur‑finite objects. Schur‑finiteness is defined via Schur functors S_λ attached to partitions λ; an object is Schur‑finite if S_λ(M)=0 for some λ. The authors focus on a special case where the vanishing occurs for a particular partition that reflects a “sign” condition. They prove that in any Q‑linear pseudo‑abelian rigid tensor category, every special Schur‑finite object is automatically Kimura‑finite, and consequently its numerically trivial endomorphism ideal is nilpotent. The proof hinges on constructing, from the vanishing of a specific Schur functor, a filtration that mimics the positive/negative decomposition required for Kimura‑finiteness. The key technical step is showing that the numerical equivalence relation interacts well with the Schur functor, allowing one to bound the nilpotence index uniformly in terms of the partition data.

Having established the abstract categorical result, the authors specialise to the category of Chow motives 𝔐_{rat}. Let X be a smooth projective variety of dimension n over a field, and denote by h(X) its Chow motive. The homological sign conjecture (also called the sign conjecture) predicts that the Künneth components of the cohomology of X have a uniform parity: the even‑degree pieces should be “positive” and the odd‑degree pieces “negative”. Assuming this conjecture for X, the motive h(X) satisfies the special Schur‑finite condition: either a high symmetric power or a high exterior power of h(X) vanishes in the motive category. Consequently, the authors obtain three equivalent statements:

1. Kimura‑finiteness of h(X).
2. Special Schur‑finiteness of h(X).
3. Nilpotency of the numerically trivial cycles in the groups
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