Geometry of regular modules over canonical algebras

Bobiński, Grzegorz

doi:10.1090/s0002-9947-07-04174-8

Cited by 8 publications

(10 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, if U is the set of points of mod d Λ (k) corresponding to the τ -periodic modules and U = ∅, then U = P(d). Geometry of this set and related problems (including study of semiinvariants) are objects of interest, especially in the case of Ringel's canonical algebras [20] -see [2,3,6,7,12,13,21] for some results. The following theorem is the main result of the paper.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

The closure of the set of periodic modules over a concealed canonical algebra is regular in codimension one

Bobiński

Zwara

2017

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Introduction and The Main Resultsmentioning

confidence: 99%

The closure of the set of periodic modules over a concealed canonical algebra is regular in codimension one

Bobiński

Zwara

2017

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…In particular, the geometry of module varieties with separating families of tubes, or more generally coils, has attracted much attention (see [4][5][6][7]26,27] for some results in this direction).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

On the additive categories of generalized standard almost cyclic coherent Auslander–Reiten components

Malicki

Skowroński

2007

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…Recall that if mod Λ (d) is irreducible, then it is a complete intersection of dimension a(d) (see, for example, [6]). Hence we get the following corollary, which determines our strategy of the proof:…”

Section: 5mentioning

confidence: 99%

“…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).…”

mentioning

confidence: 99%

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

Bobiński

2008

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

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

show abstract

Geometry of regular modules over canonical algebras

Abstract: Abstract. We classify canonical algebras such that for every dimension vector of a regular module the corresponding module variety is normal (respectively, a complete intersection). We also prove that for the dimension vectors of regular modules normality is equivalent to irreducibility.

Cited by 8 publications

References 21 publications

The closure of the set of periodic modules over a concealed canonical algebra is regular in codimension one

The closure of the set of periodic modules over a concealed canonical algebra is regular in codimension one

On the additive categories of generalized standard almost cyclic coherent Auslander–Reiten components

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

Contact Info

Product

Resources

About