Abstract
We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by dropping acyclicity, and by incorporating band modules. A special class of gentle algebras are Jacobian algebras arising from triangulations of unpunctured marked surfaces. For these we obtain a bijection between the set of generically tau -reduced decorated irreducible components and the set of laminations of the surface. As an application, we get that the set of bangle functions (defined by Musiker–Schiffler–Williams) in the upper cluster algebra associated with the surface coincides with the set of generic Caldero-Chapoton functions (defined by Geiß–Leclerc–Schröer).
Highlights
Introduction and main resultsWe study some geometric aspects of the representation theory of gentle algebras
Gentle algebras are special biserial, which implies that their module categories can be described combinatorially, see [56] and [11]
The generically τ -reduced decorated components parametrize the generic CalderoChapoton functions, which belong to the coefficient-free upper cluster algebra U(S,M) associated with (S, M)
Summary
We study some geometric aspects of the representation theory of gentle algebras. This class of finite-dimensional algebras was defined by Assem and Skowronski [5], who were classifying the iterated tilted algebras of path algebras of extended Dynkin type A. A special class of gentle algebras are Jacobian algebras arising from triangulations of unpunctured marked surfaces (S, M) For these we obtain a bijection between the set of generically τ -reduced decorated irreducible components and the set of laminations of the surface. The generically τ -reduced decorated components parametrize the generic CalderoChapoton functions, which belong to the coefficient-free upper cluster algebra U(S,M) associated with (S, M) In many cases, these generic Caldero-Chapoton functions are known to form a basis, called the generic basis, of U(S,M), see for example [30] and [49]. We use the bijection mentioned above to show that the generic basis coincides with Musiker–Schiffler–Williams’ bangle basis (see [45, Corollary 1.3]) of the coefficient-free cluster algebra A(S,M) associated with (S, M) It is known in most cases (for example, if |M| ≥ 2) that A(S,M) = U(S,M), see [42,43]. In the following subsections we describe our results in more detail
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