The Algebras with Discrete Derived Category

By D. Vossieck. 2001

131 citations recorded by the source corpus.

Abstract We classify the finite-dimensional algebras over an algebraically closed field whose bounded derived category does not admit an infinite continuous family of pairwise non-isomorphic indecomposable complexes.

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

View Paper: https://www.semanticscholar.org/paper/27997b4cd6f4eac2cbf14543f854697882847281

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Citations in the corpus

Combinatorial derived invariants for gentle algebras (2006, methodology): “Recall that the derived classification of gentle algebras with at most one cycle is known; see [3,4,18] and [6].”
Indecomposables in Derived Categories of Gentle Algebras (2003, background): “We have found that there is a connection between the derived category of a gentle algebra and a matrix problem presented by V . M. Bondarenko (see [B]).”
Classification of discrete derived categories (2004, background): “Following [21] the derived category D(mod ¤) is said to be discrete provided for every vector n = (ni)i2Z of natural numbers there are only ̄nitely many isomorphism classes of indecomposable complexes in D(mod ¤) of homology dimension vector n.”
A geometric model for the derived category of gentle algebras (2018, unknown): “TEGORY OF GENTLE ALGEBRAS 3 In the context of a classiﬁcation of thick subcategories of discrete derived categories, a geometric model was given in [12]. Discrete derived categories were classiﬁed in [37], where it is shown that they correspond to bounded derived categories of a class of gentle alge”
Stable endomorphism algebras of modules over special biserial algebras (2002, background): “In this case, it turns out that A is either derived hereditary of Dynkin type or a gentle algebra, see [19].”
ON THE NOTION OF DERIVED TAMENESS (2002, background): “It is remarkable, that for the discrete case also (1) implies (3) without the hypothesis of finite global dimension, but this is based on a complete classification of the algebras of that type [23].”
Discrete derived categories I: homomorphisms, autoequivalences and t-structures (2013, methodology): “One precise definition of such smallness is given by Vossieck [35]; here we present a slight generalisation of his notion: a derived”
Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra (2014, background): “• Vossieck [23] introduced the family of derived-discrete algebras, for which we now understand various non-trivial homological properties [10, 11, 12, 13, 14].”
The almost split triangles for perfect complexes over gentle algebras (2009, background): “Further, if A is a derived discrete algebra, then either A is piecewise hereditary of Dynkin type or A is a one-cycle gentle algebra which does not satisfy the clock condition [28].”