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

Inspired by recent results of Ringel–Schmidmeier, Kussin–Lenzing–Meltzer, Xiao-Wu Chen, and Pu Zhang, given a field K  , m≥1m≥1, and a finite poset I≡(I,⪯)I≡(I,⪯) with a unique maximal element ⁎, we study the category Mon(I,Fm)Mon(I,Fm) of mono-representations of I over the Frobenius K  -algebra Fm:=K[t]/(tm)Fm:=K[t]/(tm) of K  -dimension m<∞m<∞, viewed as K  -vector spaces U⁎U⁎, with an m-nilpotent K  -linear operator t:U⁎→U⁎t:U⁎→U⁎, together with t-invariant subspaces Ui⊆Uj⊆U⁎Ui⊆Uj⊆U⁎, for all i⪯j⪯⁎i⪯j⪯⁎ in I. The problem of when the Krull–Schmidt K  -category Mon(I,Fm)Mon(I,Fm) 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)(I,m) (resp. all pairs), with m≥1m≥1, such that category Mon(I,Fm)Mon(I,Fm) is of wild (resp. tame) representation type. We reduce the problem to a Birkhoff type problem for the category fspr(I•,Fm)⊆Mon(I•,Fm)fspr(I•,Fm)⊆Mon(I•,Fm) of subprojective representations over FmFm of a larger poset I•⊃II•⊃I. The tame–wild dichotomy for the category Mon(I,Fm)Mon(I,Fm) is also proved.Surprisingly, in case when I=Ia,bI=Ia,b is the union of two incomparable chains I′I′ and I″I″ of length |I′|=a−1≥1|I′|=a−1≥1 and |I″|=b−1≥1|I″|=b−1≥1, with I′∩I″={⁎}I′∩I″={⁎}, the problem is equivalent with the wildness (resp. tameness) of the category coh-X(p)coh-X(p) of coherent sheaves over the weighted projective line X(p)X(p), for the weight triple p=(a,b,m)p=(a,b,m), with a,b,m≥2a,b,m≥2, studied by Kussin, Lenzing and Meltzer [19] in relation with the hypersurface singularity f=x1a+x2b+x3m.