Change of rings and singularity categories

Oppermann, Steffen; Psaroudakis, Chrysostomos; Stai, Torkil

doi:10.1016/j.aim.2019.04.029

Cited by 14 publications

(12 citation statements)

References 38 publications

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“…This approach is simple but gives some interesting results. For instance, Theorem 3.6 is a “bimodule version” of a result of Oppermann–Psaroudakis–Stai [19, Proposition 3.7.1], which recovers some known results from the literature on singular equivalences (see Example 3.8 and Proposition 3.9) and gives some examples of singular equivalences of Morita type with level (see Remark 4.5 and Example 4.6).…”

Section: Introductionsupporting

confidence: 64%

On singular equivalences of Morita type with level and Gorenstein algebras

Dalezios

2021

Bull. London Math. Soc.

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Rickard proved that for certain self‐injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Broué. In this paper we study singular equivalences of finite‐dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex of bimodules induces a singular equivalence of Morita type with level, in the sense of Wang. This recovers Rickard's theorem in the self‐injective case.

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Section: Introductionsupporting

confidence: 64%

On singular equivalences of Morita type with level and Gorenstein algebras

Dalezios

2021

Bull. London Math. Soc.

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“…A closely related triangulated category to the above is the singularity category, which is the stable category of the Frobenius category consisting of finitely presented Gorenstein projective objects. The study of how the singularity category moves along ring homomorphisms has been studied in [OPS19], albeit from a different viewpoint.…”

Section: Transfer Of Gorenstein Flat-cotorsion Modulesmentioning

confidence: 99%

Purity and ascent for Gorenstein Flat Cotorsion modules

Bird

2021

Preprint

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The extension of scalars functor along a finite ring homomorphism is a classic example of a functor which preserves purity and pure injectivity. We consider how this functor behaves when restricted to the class of Gorenstein flat modules over a right coherent rings, and give particular attention to the Frobenius category of Gorenstein flat and cotorsion modules by showing there is an induced triangulated functor on the stable categories. This enables a comparison between pure injectivity for Gorenstein flat modules and pure injectivity in the triangulated categories as well as an investigation in how purity for Gorenstein flat modules is transferred along the homomorphism. Throughout motivating examples from commutative algebra are considered, including over hypersurfaces where a pure injective analogue of Knörrer periodicity for Gorenstein flat modules is developed.

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“…We will now recall the more recent notion of 0-cocompactness as introduced in [37], together with its dual. A bit of terminology is involved:…”

Section: (Co)compactness and 0-(co)compactnessmentioning

confidence: 99%

“…In [37], coveting a more potent dual of (t 1 ), we introduced the weaker notion of 0-cocompactness ad hoc, and showed that if X is a set of 0-cocompact objects, then ⊥ X, ( ⊥ X) ⊥ (t 2 ) is a (stable) t-structure in T. The point was that the definition was forgiving enough to apply to the homotopy categories studied in that paper.…”

Section: Introductionmentioning

confidence: 99%

Partial Serre duality and cocompact objects

Oppermann¹,

Psaroudakis²,

Stai³

2021

Preprint

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A successful theme in the development of triangulated categories has been the study of compact objects. A weak dual notion called 0-cocompact objects was introduced in [37], motivated by the fact that sets of such objects cogenerate co-t-structures, dual to the t-structures generated by sets of compact objects. In the present paper, we show that the notion of 0-cocompact objects also appears naturally in the presence of certain dualities.We introduce "partial Serre duality", which is shown to link compact to 0-cocompact objects. We show that partial Serre duality gives rise to an Auslander-Reiten theory, which in turn implies a weaker notion of duality which we call "non-degenerate composition", and throughout this entire hierarchy of dualities the objects involved are 0-(co)compact.Furthermore, we produce explicit partial Serre functors for multiple flavors of homotopy categories, thus illustrating that this type of duality, as well as the resulting 0-cocompact objects, are abundant in prevalent triangulated categories.

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Change of rings and singularity categories

Cited by 14 publications

References 38 publications

On singular equivalences of Morita type with level and Gorenstein algebras

On singular equivalences of Morita type with level and Gorenstein algebras

Purity and ascent for Gorenstein Flat Cotorsion modules

Partial Serre duality and cocompact objects

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