Matrix factorizations over projective schemes

Burke, Jesse; Walker, Mark E.

doi:10.4310/hha.2012.v14.n2.a3

Cited by 9 publications

(16 citation statements)

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“…Different notions of relative singularity categories were introduced and studied by Positselski [78] and also by Burke & Walker [22]. We thank Greg Stevenson for bringing this unfortunate coincidence to our attention.…”

Section: Classical Vs Relative Singularity Categories For Gorensteinmentioning

confidence: 99%

Relative singularity categories I: Auslander resolutions

Kalck

Yang

2016

Advances in Mathematics

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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.

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Section: Classical Vs Relative Singularity Categories For Gorensteinmentioning

confidence: 99%

Relative singularity categories I: Auslander resolutions

Kalck

Yang

2016

Advances in Mathematics

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“…It is well known (see [Orl04], [BW12], [EfPo15], [BRTV]) that the dg-category of relative singularities Sing(B, f ) associated to a 1-dimensional affine flat Landau-Ginzburg model over a regular local ring is equivalent to the dg-category of matrix factorizations MF(B, f ) introduced by Eisenbud ([Eis80]). In this section we shall recall what matrix factorizations are.…”

Section: Matrix Factorizationsmentioning

confidence: 99%

“…Remark 3.4. There exists a second definition of matrix factorizations for non-affine LG-models (X, f ), see [BW12], [Efi18], [Orl12]. If X is a separated scheme with enough vector bundles, the two definitions agree.…”

Section: Matrix Factorizationsmentioning

confidence: 99%

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On the structure of dg categories of relative singularities

Pippi

2019

Preprint

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In this paper we show that every object in the dg-category of relative singularities Sing(B, f ) associated to a pair (B, f ), where B is a ring and f ∈ B n , is equivalent to a retract of a K(B, f )-dg module concentrated in n + 1 degrees. When n = 1, we show that Orlov's comparison theorem, which relates the dg-category of relative singularities and that of matrix factorizations of an LG-model, holds true without any regularity assumption on the potential. Contents 1 Preliminaries 2 2 The structure of Sing(B,f) 9 3 Orlov's theorem 12Acknowledgments This paper's results are part of my PhD project under the supervision of B. Toën and G. Vezzosi. I would like to thank them both for uncountable many conversations on the subject. I would also like to acknowledge D. Beraldo, V. Melani and M. Robalo for many useful remarks and suggestions.

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“…In Section 2 we recall the full definition of morphisms and give details on the following: The first equivalence of this theorem has been proven in various contexts by various authors in [24,28,29]. The version we use here is from our previous paper [13]. The second equivalence is essentially due to Orlov [26,Theorem 2.1].…”

Section: Introductionmentioning

confidence: 98%

Matrix factorizations in higher codimension

Burke

Walker

2015

Trans. Amer. Math. Soc.

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We observe that there is an equivalence between the singularity category of an affine complete intersection and the homotopy category of matrix factorizations over a related scheme. This relies in part on a theorem of Orlov. Using this equivalence, we give a geometric construction of the ring of cohomology operators, and a generalization of the theory of support varieties, which we call stable support sets. We settle a question of Avramov about which stable support sets can arise for a given complete intersection ring. We also use the equivalence to construct a projective resolution of a module over a complete intersection ring from a matrix factorization, generalizing the well-known result in the hypersurface case.

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Matrix factorizations over projective schemes

Cited by 9 publications

References 10 publications

Relative singularity categories I: Auslander resolutions

Relative singularity categories I: Auslander resolutions

On the structure of dg categories of relative singularities

Matrix factorizations in higher codimension

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