On the zero set of semi-invariants for regular modules over tame canonical algebras

Abstract

We investigate sets of the common zeros of non-constant semi-invariants for regular modules over canonical algebras. In particular, we show that if the considered algebra is tame then for big enough vectors these sets are complete intersections.Throughout the paper k denotes a fixed algebraically closed field of characteristic 0. By N and Z we denote the sets of non-negative integers and integers, respectively. Additionally, if i, j ∈ Z, then [i, j] = {l ∈ Z | i ≤ l ≤ j}. Introduction and the main resultWith a finite dimensional algebra Λ and a dimension vector d we may associate the variety of Λ-modules of dimension vector d (see 2.1). An interesting problem investigated in the representation theory of finite dimensional algebras is the study of geometrical properties of these varieties (see, for example, [8,10,12,14,17,23,27,28,34,35,48]). In addition to this topic rings of semi-invariants (see 2.2) are also studied (see, for example, [20,24,30,32,40,46]). Recently, investigations of sets of the common zeros of non-constant semi-invariants were initiated by Chang and Weyman [15] and then continued by Riedtmann and Zwara [36][37][38][39]. Their investigations concerned situations of quivers without relations and were based on known results about semi-invariants in these cases (among others Sato-Kimura theorem [42]). An inspiration for their research was an observation that if, for a given dimension vector, the set of the common zeros of non-constant semi-invariants has a "good" codimension then the coordinate ring of the module variety is free as a module over the ring of semi-invariants.An important class of algebras are the canonical algebras introduced by Ringel [41, 3.7] (see 1.4). These algebras play an important role in representation theory (see, for example, [22,25,33,44]). Module varieties over canonical algebras were also studied [5,6]. One may distinguish a special class of modules over canonical algebras, called regular (see 1.6). The rings of semi-invariants for dimension vectors of regular modules over canonical algebra were

Moduli Spaces of Modules of Schur-Tame Algebras

“…. , A (T n d ) Rn satisfying the two properties above with "Schur-embedding" replaced by "representation-embedding" in (1), and "Schur" replaced by "indecomposable" in (2). Following closely Dowbor-Skowroński's arguments in [19], we explain how to modify the R i 's and T i 's in order to get the desired Schur-embeddings that almost parametrize the d-dimensional Schur A-modules.…”

“…where (r α , r β ) is a maximal (coordinatewise) pair of non-negative integers with r α + r β ≤ h (2). So, they are all normal varieties.…”

“…Our objective is to describe the tameness, and more generally the Schur-tameness, of an algebra in terms of invariant theory. This line of research has attracted much attention during the last two decades (see for example [2], [4], [3], [5], [10], [11], [12], [14], [18], [20], [24], [25], [26], [29]).…”

On the invariant theory for acyclic gentle algebras

“…In particular, it would be interesting to find characterizations of prominent classes of tame algebras via geometric properties of their module varieties. This research direction has attracted much attention during the last two decades (see for example [4], [5], [6], [7], [12], [23] [26], [32], [33], [34], [39]).…”

Geometric characterizations of the representation type of hereditary algebras and of canonical algebras

“…Moreover, W. Geigle and Lenzing found in [28] a beautiful interpretation of canonical algebras and their representations in terms of coherent sheaves over weighted projective lines. The invariant theory for canonical algebras in the regular case has been investigated in a number of papers, see [6], [5], [21], [22], [56]. By applying Theorem 1.2 to tame canonical algebras, we are able to describe the fields of rational invariants when the dimension vector in question is not necessarily regular.…”