Noetherian property of infinite EI categories.

*(English)*Zbl 1327.18003This article is a contribution to the following general and hard problem: Let \(C\) be a small category and \(k\) a noetherian ring, when is the category of functors from \(C\) to \(k\)-modules locally noetherian?

The paper restricts to the case where \(k\) is a field of characteristic \(0\) and \(C\) an \(EI\)-category (that is, all endomorphisms in \(C\) are isomorphisms). It shows (Theorem 3.7) that the answer to the previous question is positive if moreover the following assumptions are satisfied:

- the objects of \(C\) are the non-negative integers, \(C(i,j)\) is always a finite set and is empty if and only if \(i>j\);

- (transitivity condition) for all integer \(i>0\), the group \(C(i,i)\) acts transitively on the set \(C(i-1,i)\);

- (bijectivity condition) let us choose, for each \(i>0\), a morphism \(\alpha_i\in C(i-1,i)\) and denote by \(H_{i,j}\), for \(i<j\), the stabilizer of \(\alpha_j\alpha_{j-1}\dots\alpha_{i+1}\in C(i,j)\) under the action of the group \(C(j,j)\). Then for all non-negative integer \(i\), the canonical function \(H_{i,j}\backslash C(i,j)\to H_{i,j+1}\backslash C(i,j+1)\) is a bijection for \(j\) big enough with respect to \(i\) (the transitivity condition implies that this condition does not depend on the choice of the arrows \(\alpha_i\)).

The article gives several examples of applications, as the category \(C=FI\) of finite sets with injections, where the noetherianity result was already known by Church, Ellenberg and Farb [T. Church et al., Duke Math. J. 164, No. 9, 1833–1910 (2015; Zbl 1339.55004)] – and extended to the case where \(k\) is an arbitrary noetherian ring in the paper by the same three authors and Nagpal [T. Church et al., Geom. Topol. 18, No. 5, 2951–2984 (2014; Zbl 1344.20016)] –, the category of finite dimensional vector spaces over a finite field with linear injections (or linear injections with a complement). These cases were also proven, even in the case where \(k\) is an arbitrary noetherian ring, by independent methods relying on Gröbner bases, in the work of S. V. Sam and A. Snowden [“Gröbner methods for representations of combinatorial categories”, arXiv:1409.1670] and A. Putman and S. V. Sam [“Representation stability and finite linear groups”, arXiv:1408.3694].

The proof of the main theorem is rather direct, using usual idempotents of algebras of finite groups over a field of characteristic zero. Without this crucial assumption, it is very natural to ask whether the conclusion still holds (the important aforementioned particular cases suggest that it could be true).

Even if the approach is not very conceptual and the main examples were surpassed by other works, especially by Putman-Sam-Snowden, the assumptions of the main statement (transitivity and bijectivity) are interesting, and should be for example compared with Bass conditions in linear algebra; they are reminiscent of the set-up for stable homology given in the beginning of the paper by the reviewer and C. Vespa [Ann. Sci. Éc. Norm. Supér. (4) 43, No. 3, 395–459 (2010; Zbl 1221.20036)] and the preprint by N. Wahl and O. Randal-Williams [“Homological stability for automorphism groups”, arXiv:1409.3541], which are themselves reminiscent of classical constructions in algebraic \(K\)-theory.

The paper restricts to the case where \(k\) is a field of characteristic \(0\) and \(C\) an \(EI\)-category (that is, all endomorphisms in \(C\) are isomorphisms). It shows (Theorem 3.7) that the answer to the previous question is positive if moreover the following assumptions are satisfied:

- the objects of \(C\) are the non-negative integers, \(C(i,j)\) is always a finite set and is empty if and only if \(i>j\);

- (transitivity condition) for all integer \(i>0\), the group \(C(i,i)\) acts transitively on the set \(C(i-1,i)\);

- (bijectivity condition) let us choose, for each \(i>0\), a morphism \(\alpha_i\in C(i-1,i)\) and denote by \(H_{i,j}\), for \(i<j\), the stabilizer of \(\alpha_j\alpha_{j-1}\dots\alpha_{i+1}\in C(i,j)\) under the action of the group \(C(j,j)\). Then for all non-negative integer \(i\), the canonical function \(H_{i,j}\backslash C(i,j)\to H_{i,j+1}\backslash C(i,j+1)\) is a bijection for \(j\) big enough with respect to \(i\) (the transitivity condition implies that this condition does not depend on the choice of the arrows \(\alpha_i\)).

The article gives several examples of applications, as the category \(C=FI\) of finite sets with injections, where the noetherianity result was already known by Church, Ellenberg and Farb [T. Church et al., Duke Math. J. 164, No. 9, 1833–1910 (2015; Zbl 1339.55004)] – and extended to the case where \(k\) is an arbitrary noetherian ring in the paper by the same three authors and Nagpal [T. Church et al., Geom. Topol. 18, No. 5, 2951–2984 (2014; Zbl 1344.20016)] –, the category of finite dimensional vector spaces over a finite field with linear injections (or linear injections with a complement). These cases were also proven, even in the case where \(k\) is an arbitrary noetherian ring, by independent methods relying on Gröbner bases, in the work of S. V. Sam and A. Snowden [“Gröbner methods for representations of combinatorial categories”, arXiv:1409.1670] and A. Putman and S. V. Sam [“Representation stability and finite linear groups”, arXiv:1408.3694].

The proof of the main theorem is rather direct, using usual idempotents of algebras of finite groups over a field of characteristic zero. Without this crucial assumption, it is very natural to ask whether the conclusion still holds (the important aforementioned particular cases suggest that it could be true).

Even if the approach is not very conceptual and the main examples were surpassed by other works, especially by Putman-Sam-Snowden, the assumptions of the main statement (transitivity and bijectivity) are interesting, and should be for example compared with Bass conditions in linear algebra; they are reminiscent of the set-up for stable homology given in the beginning of the paper by the reviewer and C. Vespa [Ann. Sci. Éc. Norm. Supér. (4) 43, No. 3, 395–459 (2010; Zbl 1221.20036)] and the preprint by N. Wahl and O. Randal-Williams [“Homological stability for automorphism groups”, arXiv:1409.3541], which are themselves reminiscent of classical constructions in algebraic \(K\)-theory.

Reviewer: Aurelien Djament (Nantes)