Derived-tame algebras and non-commutative nodal projective curves
Final Report Abstract
In a joint work with Yuriy Drozd, I have settled the theory of non-commutative nodal curves and clarified the criterion of their tameness. This work led to a discovery of a new class of derived-tame algebras called quasi-gentle. In a more general context of non-commutative noetherian schemes, Drozd and myself established a general Morita theorem and gave a new proof of a conjecture of Caldararu on Morita equivalences of Azumaya algebras on noetherian schemes. Using an insight from the homological mirror symmetry, Lekili and Polishchuk discovered, that two tame non-commutative nodal curves can be Morita non-equivalent, but have equivalent derived categories of coherent sheaves. Elaborating ideas of Lekili and Polishchuk, Drozd and myself provided a version of the homological mirror symmetry for general tame non-commutative nodal curves of gentle type. In a joint work with Plamondon and Schroll, Sebastian Opper introduced a combinatorial model of a gentle algebra. Using the developed technique, he discovered (independently to Amiot, Plamondon and Schroll) a full derived invariant of a gentle algebra. Finally, Sebastian Opper gave an answer to an old question of Polishchuk on spherical objects on cycles of projective lines, posed more than fifteen years ago.
Publications
- On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems
I. Burban, Yu. Drozd
- A geometric model for the derived category of gentle algebras
S. Opper, P.-G. Plamondon, S. Schroll
- Non-commutative nodal curves and derived-tame algebras
I. Burban, Yu. Drozd
- Morita theory for non-commutative noetherian schemes
I. Burban, Yu. Drozd
- On auto-equivalences and complete derived invariants of gentle algebras
S. Opper
