Abgeleitet-zahme Algebren und nichtkommutative nodale projektive Kurven
Zusammenfassung der Projektergebnisse
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.
Projektbezogene Publikationen (Auswahl)
- 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
