Resolutions in factorization categories

Ballard, Matthew; Deliu, Dragos; Favero, David; Isik, M. Umut; Katzarkov, Ludmil

doi:10.1016/j.aim.2016.02.008

Cited by 28 publications

(61 citation statements)

References 26 publications

(22 reference statements)

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“…Proof. By [BDFIK,Lemma 2.26], there are two finite dimensional k-vactor spaces V and V ′ , and a triangle of the following form in Dcoh(k, 0) = DMF(k, 0)…”

Section: 2mentioning

confidence: 99%

Relative singular locus and Balmer spectrum of matrix factorizations

Hirano

2018

Trans. Amer. Math. Soc.

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For a separated Noetherian scheme X with an ample family of line bundles and a non-zero-divisor W ∈ Γ(X, L) of a line bundle L on X, we classify certain thick subcategories of the derived matrix factorization category DMF(X, L, W ) of the Landau-Ginzburg model (X, L, W ). Furthermore, by using the classification result and the theory of Balmer's tensor triangular geometry, we show that the spectrum of the tensor triangulated category (DMF(X, L, W ), ⊗ 1 2 ) is homeomorphic to the relative singular locus Sing(X 0 /X), introduced in this paper, of the zero scheme X 0 ⊂ X of W .1.2. Relative singular locus. To state our main results, we introduce a new notion of relative singular locus. Let i : T ֒→ S be a closed immersion of Noetherian schemes. We define the relative singular locus, denoted by SingT is worse than the mildness of the singularity of p in S. In fact, for a quasi-projective variety X over C and a regular function f ∈ Γ(X, O X ) which is non-zero-divisor, we have the following equality of subsets of the associated complex analytic space (X an , O an X ); Sing loc (f −1 (0)/X) ∩ X an = Crit(f an ) ∩ Zero(f an ),

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“…Proof. By [BDFIK,Lemma 2.26], there are two finite dimensional k-vactor spaces V and V ′ , and a triangle of the following form in Dcoh(k, 0) = DMF(k, 0)…”

Section: 2mentioning

confidence: 99%

Relative singular locus and Balmer spectrum of matrix factorizations

Hirano

2018

Trans. Amer. Math. Soc.

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“…Instead, we can appeal to a closely related category defined by Positselski [Po1] called the absolute derived category D abs [Fact(B, Z, g)] (see also [BFK2,BFK3]). As the details are a bit technical, we just mention that this category is a Verdier localization of a category defined the same way as DGrB(g) except that the pairs consist of any two finitely generated graded B-modules as opposed to finitely generated projective graded B-modules.…”

Section: There Is An Isomorphism Of Griffiths Groupsmentioning

confidence: 99%

On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type

Favero

Iliev

Katzarkov

2014

Pure and Applied Mathematics Quarterly

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A deep result of Voisin asserts that the Griffiths group of a general non-rigid Calabi-Yau (CY) 3-fold is infinitely generated. This theorem builds on an earlier method of hers which was implemented by Albano and Collino to prove the same result for a general cubic sevenfold. In fact, Voisin's method can be utilized precisely because the variation of Hodge structure on a cubic 7-fold behaves just like the variation of Hodge structure of a Calabi-Yau 3-fold. We explain this relationship concretely using Kontsevitch's noncommutative geometry. Namely, we show that for a cubic 7-fold, there is a noncommutative CY 3-fold which has an isomorphic Griffiths group. This serves as partial confirmation of seminal work of Candelas, Derrick, and Parkes describing a cubic 7-fold as a mirror to a rigid CY 3-fold.Similarly, one can consider other examples of Fano manifolds with with the same type of variation of Hodge structure as a Calabi-Yau threefold (FCYs). Among the complete intersections in weighted projective spaces, there are only three classes of smooth FCY manifolds; the cubic 7-fold X 3 , the fivefold quartic double solid X 4 , and the fivefold intersection of a quadric and a cubic X 2.3 . We settle the two remaining cases, following Voisin's method to demonstrate that the Griffiths group for a general complete intersection FCY manifolds, X 4 and X 2.3 , is also infinitely generated.Received December 11, 2012. David Favero, Atanas Iliev and Ludmil KatzarkovIn the case of X 4 , we also show that there is a noncommutative CY 3-fold with an isomorphic Griffiths group. Finally, for X 2.3 there is a noncommutative CY 3-fold, B, such that the Griffiths group of X 2.3 surjects on to the Griffiths group of B. We finish by discussing some examples of noncommutative covers which relate our noncommutative CYs back toq honest algebraic varieties such as products of elliptic curves and K3-surfaces.

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“…Another advantage of derived categories of the second kind is that they are defined for curved differential graded (CDG) structures as well as for conventional differential graded structures [25]. Hence the important role that such derived category constructions play, in particular, in the theory of matrix factorizations [22,5,1].…”

mentioning

confidence: 99%

“…Furthermore, there is a natural equivalence between the coderived category of left CDG-comodules and the contraderived category of left CDG-contramodules over C [25, Section 5]: (1) D co (C-comod) ≃ D ctr (C-contra).…”

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Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras

Positselski

2017

Algebr Represent Theor

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We present the definition of a dedualizing complex of bicomodules over a pair of cocoherent coassociative coalgebras C and D. Given such a complex B • , we construct an equivalence between the (bounded or unbounded) conventional, as well as absolute, derived categories of the abelian categories of left comodules over C and left contramodules over D. Furthermore, we spell out the definition of a dedualizing complex of bisemimodules over a pair of semialgebras, and construct the related equivalence between the conventional or absolute derived categories of the abelian categories of semimodules and semicontramodules. Artinian, co-Noetherian, and cocoherent coalgebras are discussed as a preliminary material.

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Resolutions in factorization categories

Cited by 28 publications

References 26 publications

Relative singular locus and Balmer spectrum of matrix factorizations

Relative singular locus and Balmer spectrum of matrix factorizations

On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type

Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras

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