Irreducible components of varieties of modules

By W. Crawley‐Boevey, J. Schröer. 2001

125 citations recorded by the source corpus.

We prove some basic results about irreducible components of varieties of modules for an arbitrary finitely generated associative algebra. Our work generalizes results of Kac and Schofield on representations of quivers, but our methods are quite different, being based on deformation theory.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/431fbcc456fee695c7773500d178a406394c2e36

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Citations in the corpus

Indecomposable parabolic bundles (2004, background): “Given dimension types (β, e) and (γ, f), we define ext((β, e),(γ, f)) to be the general dimension of Ext1(X, Y ) with X in KI(β, e) and Y in KI(γ, f). By the main theorems of [ 10 ],”
Generic bases for cluster algebras from the cluster category (2011, background): “kd i isomorphic to M. We will need the following information on the dimension of morphism and extension spaces, and on minimal projective presentations. 4 PIERRE-GUY PLAMONDON Lemma 2.1 (Lemma 4.2 of [10]). The functions rep d 1 (A) ×rep d 2 (A) −→ Z sending a pair (M1,M2) to the dimensions of the v”
General Presentations of Algebras (2009, background): “Unfortunately, the representation spaces of path algebras with relations may not be irreducible (or even reduced), so one has to work with general representations in a given irreducible component as in [5].”
Generic bases for cluster algebras and the Chamber Ansatz (2010, background): “able irreducible component. (Here we set d := dimΛ(Tk).) With these deﬁnitions, each irreducible component Z admits an essentially unique decomposition into indecomposable irreducible components, see [CBS]. GENERIC BASES FOR CLUSTER ALGEBRA AND THE CHAMBER ANSATZ 41 (2) We say, that an irreducible c”
Generic Variables in Acyclic Cluster Algebras (2010, background): “sume that d i and d′ i are of the same sign for every i ∈ Q 0, in particular, [d+d ′] + = [d] ++[d ′] +. As Ext1 CQ (d,d′) = 0, we have Ext1 kQ(d,d ′) = 0 and Ext1 kQ(d ′,d) = 0. It thus follows from [CBS02] that the morphism φ : ˆ rep(Q,[d] +)×rep(Q,[d′] +) −→ rep(Q,[d+d′] +) (U,V) → U ⊕V is domina”
Semicanonical bases and preprojective algebras (2004, background): “1 and [14] the square of any vector of S∗ belongs to S∗.”
Kac–Moody groups and cluster algebras (2010, background): “ j )(k,j)∈I(f)) ∼=Hom Λ(V i,X) as a B i-module. Thus, in fact Im(π2) = F(∆,f), where F(∆,f) is the subset of mod(B i,f) consisting of the ∆-ﬁltered B i-modules with dimension vector f. It is shown in [CBS] that F(∆,f) is irreducible and open in mod(B i,f). Finally, for π2(X,(ϕ (k) j ) (k,j )∈I f ) =”
Affine Mirković-Vilonen polytopes (2011, methodology): “Moreover, it follows from Crawley-Boevey and Schröer’s theory of canonical decomposition [18] that each Xk is the general point of an irreducible component Zk ⊆ Λ(nkδ).”
Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity (2003, background): “By the main theorems of [10], one has the following canonical decomposition.”