Tilting and cotilting modules over concealed canonical algebras

Abstract

Abstract. We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be interpreted in terms of Gabriel localizations of the corresponding category of quasi-coherent sheaves over a noncommutative curve of genus zero. In the tubular case, we have to distinguish between tilting modules of rational and irrational slope. For rational slope the situation is analogous to the domestic case. In contrast, for any irrational slope, there is just one tilting module of that slope up to equivalence. We also provide a dual description of infinite dimensional cotilting modules and a classification result for the indecomposable pure-injective modules.

Large Tilting Sheaves over Weighted Noncommutative Regular Projective Curves

“…, p t , that is, p = 1 if X is elliptic, and p > 1 if X is tubular. Further, the slope of a non-zero object E ∈ H is defined by µ 7). By semistability we have the following result, similar to Atiyah's classification [12].…”

“…Altogether, the pairs (B, V ) correspond to Serre subcategories of coh X, and tilting sheaves are closely related with Gabriel localization, like in the case of tilting modules over commutative noetherian rings, cf. also [7,Sec. 5].…”

“…Since noncommutative curves of genus zero are derived-equivalent to canonical algebras in the sense of Ringel and Crawley-Boevey [57], our results are closely related to the classification of large tilting modules over canonical algebras. The module case is treated more directly in [7], where we also address the dual concept of cotilting modules and the classification of pure-injective modules.…”

Strictly atomic modules in definable categories

“…If r is irrational, then the category D r of modules of slope r contains no finite-dimensional nonzero module. A good deal of information has been obtained about these categories in [4,13,16] and [20] but there is currently no description of the indecomposable pure-injectives in D r . Here, with that aim in mind, we shed a little more light on the structure of D r .…”

Decidability of theories of modules over tubular algebras

“…Acknowledgements I would like to thank Mike Prest for many helpful conversations. I would also like to thank Lidia Angeleri-Hügel for her showing me a draft of [AHK15] and for a very useful conversation about some of the material which is now in section 6. Although I do not directly use the description of the pure-injective indecomposable modules over tubular algebras given in [AHK15], the existence of this result was useful.…”