We determine the singularity category of an arbitrary finite‐dimensional gentle algebra normalΛ. It is a finite product of n‐cluster categories of type A1. Equivalently, it may be described as the stable module category of a self‐injective gentle algebra. If normalΛ is a Jacobian algebra arising from a triangulation MJX-tex-caligraphicscriptT of an unpunctured marked Riemann surface, then the number of factors equals the number of inner triangles of MJX-tex-caligraphicscriptT.

Abstract. Let R be an isolated Gorenstein singularity with a non-commutative resolution A = End R (R ⊕ M ). In this paper, we show that the relative singularity category ∆ R (A) of A has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category D sg (R) of Buchweitz and Orlov as a certain canonical quotient category. If R has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that D sg (R) determines ∆ R (Aus(R)), where Aus(R) is the corresponding Auslander algebra. The proofs of these results use dg algebras, A ∞ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.

We study objects in triangulated categories which have a two-dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical. This general result is then applied to algebraic geometry.Comment: 21 pages. Identical to published version. There is a separate article with examples from representation theory, see arXiv:1502.0683

We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many IwanagaGorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander-Solberg and Kong type results.

Abstract. We study composition series of derived module categories in the sense of Angeleri Hügel, König & Liu for quasi-hereditary algebras. More precisely, we show that having a composition series with all factors being derived categories of vector spaces does not characterise derived categories of quasi-hereditay algebras. This gives a negative answer to a question of Liu & Yang and the proof also confirms part of a conjecture of Bobiński & Malicki. In another direction, we show that derived categories of quasi-hereditary algebras can have composition series with lots of different lengths and composition factors. In other words, there is no Jordan-Hölder property for composition series of derived categories of quasi-hereditary algebras.