Tame-wild dichotomy of Birkhoff type problems for nilpotent linear operators

JOURNAL OF ALGEBRA

卷:424

Article

Simson, Daniel

关键词: Poset;    Bocs;    Mono-representations;    Tame-wild dichotomy;    Bimodule matrix problem;    Birkhoff type problem;    Frobenius triangulated category;    Weighted projective line;    Cohen-Macaulay module;   

DOI  :  10.1016/j.jalgebra.2014.11.008

来源: Elsevier

【 摘 要 】

Inspired by recent results of Ringel-Schmidmeier, Kussin- Lenzing-Meltzer, Xiao-Wu Chen, and Pu Zhang, given a field K, m >= 1, and a finite poset I equivalent to (I, less than or similar to) with a unique maximal element *, we study the category Mon(I, F-m) of mono-representations of I over the Frobenius K-algebra F-m := K[t]/(t(m)) of K-dimension m < infinity, viewed as K-vector spaces U-*, with an m-nilpotent K-linear operator t: U-* -> U-*, together with t-invariant subspaces U-i subset of U-j subset of U-*, for all i less than or similar to j less than or similar to * in I. The problem of when the Krull-Schmidt K-category Mon(I, F-m) is of wild (resp. tame) representation type is called a wild (resp. tame) Birkhoff type problem for m-nilpotent operators. In case when K is algebraically closed, we give a complete solution of the problem by describing all minimal pairs (I, m) (resp. all pairs), with m >= 1, such that category lvton(I, F-m) is of wild (resp. tame) representation type. We reduce the problem to a Birkhoff type problem for the category fspr(I-center dot, F-m) subset of Mon(I-center dot, F-m) of subprojective representations over F-m, of a larger poset I-center dot superset of I. The tame-wild dichotomy for the category Mon(I, F-m) is also proved. Surprisingly, in case when I = I-a,I-b is the union of two incomparable chains I' and I '' of length If vertical bar I'vertical bar = a-1 >= 1 and vertical bar I ''vertical bar = b-1 >= 1, with I' boolean AND I '' = {*}, the problem is equivalent with the wildness (resp. tameness) of the category coh-X(p) of coherent sheaves over the weighted projective line X(p), for the weight triple p = (a, b, m), with a, b,m >= 2, studied by Kussin, Lenzing and Meltzer [19] in relation with the hypersurface singularity f =x(1)(a) + x(2)(b) + x(3)(m). (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free