Path Coalgebras of Quivers with Relations and a Tame-Wild Dichotomy Problem for Coalgebras

Tame comodule type, roiter bocses, and a geometry context for coalgebras

“…Assume that K is an algebraically closed field and C is a basic infinite-dimensional cocommutative K-coalgebra with a unique simple subcoalgebra S. If S is finitely copresented and C is not fc-wild then (i) C is a subcoalgebra of the path K-coalgebra K (L 2 , Ω) (see [21] (Example 6.18), [22], [24]), where L 2 is the two loop quiver…”

“…and Ω ⊆ KL 2 is the ideal of the path algebra KL 2 generated by the two zero-relations β 1 β 2 and β 2 β 1 , and (ii) K (L 2 , Ω) is a string coalgebra in the sense of [22] (Sec. 6), (iii) the coalgebras K (L 2 , Ω) and C are of tame comodule type, and K (L 2 , Ω) is of nonpolynomial growth.…”

“…A K-coalgebra C is defined to be of K-tame comodule type [25] (or K-tame, in short) if the category C-comod of finite-dimensional left C-comodules is of K-tame representation type ( [18], Sec. 14.4, [22]), i.e., for every vector v ∈ K 0 (C) ∼ = Z (I C ) , there exist C-K[t]-bicomodules L (1) , . .…”

“…Throughout this paper, we freely use the coalgebra representation theory notation and terminology introduced in [2,16,21,22,28]. The reader is referred to [1,8,10,18] for representation theory terminology and notation, and to [3,4,7,9,13] for a background on the representation theory of bocses.…”

“…Throughout this paper, we use the terminology and notation introduced in [21,22,28]. We fix a field K. Given a K-coalgebra C, we denote by C-Comod and C-comod the categories of left C-comodules and left C-comodules of finite K-dimension.…”