##
**Resolutions in factorization categories.**
*(English)*
Zbl 1353.13016

Matrix factorizations were introduced by Eisenbud in 1980 as a tool for studying free resolutions of modules over local hypersurface rings. Since then, matrix factorizations have proven useful in such diverse fields as algebraic geometry, mathematical physics, and knot theory.

In the article under review, the authors introduce a new framework for the study of matrix factorizations, which they call a factorization category. We give the definition here. Fix an abelian category \(\mathcal{A}\), an autoequivalence \(\Phi\) of \(\mathcal{A}\), and a natural transformation \(w: \text{id}_{\mathcal{A}} \to \Phi\) such that \(w_{\Phi(A)} = \Phi(w_A)\) for all objects \(A \in \mathcal{A}\). A factorization of the triple \((\mathcal{A}, \Phi, w)\) is a pair \(E^{-1}, E^0\) of objects in \(\mathcal{A}\) along with a pair of morphisms \[ \phi_E^{-1}: \Phi^{-1}(E^0) \to E^{-1} \]

\[ \phi_E^{0}: E^{-1} \to E^{0} \] such that \(\phi_E^0 \phi_E^{-1} = \Phi^{-1}(w_{E^0})\) and \(\Phi(\phi_E^{-1}) \phi_E^0 = w_{E^{-1}}\). \(E^0\) and \(E^{-1}\) are called the components of the factorization. Morphisms of factorizations are defined component-wise in the evident way, so that factorizations form an abelian category. Applying formalism introduced by L. Positselski in [Mem. Am. Math. Soc. 996, 133 p. (2011; Zbl 1275.18002)], the authors introduce the notions of absolute derived, co-derived, and contra-derived categories of factorizations. The goal of the article is to study derived functors between such categories.

In Section 2, basics on factorization categories and their various derived categories are discussed. In Section 3, the authors discuss specific methods for “resolving” factorizations; that is, replacing them, up to various notions of derived isomorphism, with factorizations whose components are injective/projective. Section 4 contains a discussion of derived functors between the various derived categories of factorizations, and, in Section 5, the authors apply their results to some concrete examples arising in algebraic geometry.

In the article under review, the authors introduce a new framework for the study of matrix factorizations, which they call a factorization category. We give the definition here. Fix an abelian category \(\mathcal{A}\), an autoequivalence \(\Phi\) of \(\mathcal{A}\), and a natural transformation \(w: \text{id}_{\mathcal{A}} \to \Phi\) such that \(w_{\Phi(A)} = \Phi(w_A)\) for all objects \(A \in \mathcal{A}\). A factorization of the triple \((\mathcal{A}, \Phi, w)\) is a pair \(E^{-1}, E^0\) of objects in \(\mathcal{A}\) along with a pair of morphisms \[ \phi_E^{-1}: \Phi^{-1}(E^0) \to E^{-1} \]

\[ \phi_E^{0}: E^{-1} \to E^{0} \] such that \(\phi_E^0 \phi_E^{-1} = \Phi^{-1}(w_{E^0})\) and \(\Phi(\phi_E^{-1}) \phi_E^0 = w_{E^{-1}}\). \(E^0\) and \(E^{-1}\) are called the components of the factorization. Morphisms of factorizations are defined component-wise in the evident way, so that factorizations form an abelian category. Applying formalism introduced by L. Positselski in [Mem. Am. Math. Soc. 996, 133 p. (2011; Zbl 1275.18002)], the authors introduce the notions of absolute derived, co-derived, and contra-derived categories of factorizations. The goal of the article is to study derived functors between such categories.

In Section 2, basics on factorization categories and their various derived categories are discussed. In Section 3, the authors discuss specific methods for “resolving” factorizations; that is, replacing them, up to various notions of derived isomorphism, with factorizations whose components are injective/projective. Section 4 contains a discussion of derived functors between the various derived categories of factorizations, and, in Section 5, the authors apply their results to some concrete examples arising in algebraic geometry.

Reviewer: Michael Brown (Bonn)

### MSC:

13D09 | Derived categories and commutative rings |

18E30 | Derived categories, triangulated categories (MSC2010) |

### Citations:

Zbl 1275.18002### References:

[1] | Abouzaid, M.; Auroux, D.; Efimov, A.; Katzarkov, L.; Orlov, D., Homological mirror symmetry for punctured spheres, J. Amer. Math. Soc., 26, 4, 1051-1083 (2013) · Zbl 1276.53089 |

[2] | Ballard, M.; Deliu, D.; Favero, D.; Isik, U.; Katzarkov, L., Homological projective duality via variation of geometric invariant theory quotients, J. Eur. Math. Soc., to appear · Zbl 1400.14048 |

[3] | Ballard, M.; Favero, D.; Katzarkov, L., A category of kernels for equivariant factorizations and its implications for Hodge theory, Publ. Math. Inst. Hautes Études Sci., 120, 1, 1-111 (2014) · Zbl 1401.14086 |

[4] | Ballard, M.; Favero, D.; Katzarkov, L., Variation of geometric invariant theory quotients and derived categories, J. Reine Angew. Math., to appear · Zbl 1432.14015 |

[5] | Ballard, M.; Favero, D.; Katzarkov, L., A category of kernels for equivariant factorizations and its implications for Hodge theory, II: further implications, J. Math. Pures Appl., 102, 4, 702-757 (2014) · Zbl 1326.14036 |

[6] | Baranovsky, V.; Pecharich, J., On equivalences of derived and singular categories, Cent. Eur. J. Math., 8, 1, 1-14 (2010) · Zbl 1191.14004 |

[7] | Becker, H., Models for singularity categories · Zbl 1348.16009 |

[8] | Bondal, A.; Larsen, M.; Lunts, V., Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. IMRN, 29, 1461-1495 (2004) · Zbl 1079.18008 |

[9] | Bondal, A.; Van den Bergh, M., Generators and representability of functors in commutative and non-commutative geometry, Mosc. Math. J., 3, 1, 1-36 (2003) · Zbl 1135.18302 |

[11] | Efimov, A., Homological mirror symmetry for curves of higher genus, Adv. Math., 230, 2, 493-530 (2012) · Zbl 1242.14039 |

[12] | Efimov, A., Homotopy finiteness of some DG categories from algebraic geometry · Zbl 1478.14010 |

[13] | Efimov, A.; Positselski, L., Coherent analogues of matrix factorizations and relative singularity categories · Zbl 1333.14018 |

[14] | Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc., 260, 1, 35-64 (1980) · Zbl 0444.13006 |

[15] | Gelfand, S.; Manin, Yu., Methods of Homological Algebra, Springer Monographs in Mathematics (2003), Springer-Verlag: Springer-Verlag Berlin · Zbl 1006.18001 |

[16] | Isik, M. U., Equivalence of the derived category of a variety with a singularity category, Int. Math. Res. Not. IMRN, 12, 2787-2808 (2013) · Zbl 1312.14052 |

[17] | Khovanov, M.; Rozansky, L., Matrix factorizations and link homology, Fund. Math., 199, 1, 1-91 (2008) · Zbl 1145.57009 |

[18] | Khovanov, M.; Rozansky, L., Matrix factorizations and link homology. II, Geom. Topol., 12, 3, 1387-1425 (2008) · Zbl 1146.57018 |

[19] | Lin, K.; Pomerleano, D., Global matrix factorizations · Zbl 1285.14019 |

[20] | Orlov, D., Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova, 246, 240-262 (2004) · Zbl 1101.81093 |

[21] | Orlov, D., Derived categories of coherent sheaves and triangulated categories of singularities, (Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin. Vol. II. Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin. Vol. II, Progr. Math., vol. 270 (2009), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA), 503-531 · Zbl 1200.18007 |

[22] | Orlov, D., Matrix factorizations for nonaffine LG-models, Math. Ann., 353, 1, 95-108 (2012) · Zbl 1243.81178 |

[23] | Polishchuk, A.; Positselski, L., Hochschild (co)homology of the second kind I, Trans. Amer. Math. Soc., 364, 10, 5311-5368 (2012) · Zbl 1285.16005 |

[24] | Positselski, L., Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence · Zbl 1275.18002 |

[25] | Preygel, A., Thom-Sebastiani and duality for matrix factorizations |

[26] | Segal, E., Equivalences between GIT quotients of Landau-Ginzburg B-models, Comm. Math. Phys., 304, 2, 411-432 (2011) · Zbl 1216.81122 |

[27] | Seidel, P., Homological mirror symmetry for the genus two curve, J. Algebraic Geom., 20, 4, 727-769 (2011) · Zbl 1226.14028 |

[28] | Sheridan, N., Homological mirror symmetry for Calabi-Yau hypersurfaces in projective space · Zbl 1344.53073 |

[29] | Shipman, I., A geometric approach to Orlov’s theorem, Compos. Math., 148, 5, 1365-1389 (2012) · Zbl 1253.14019 |

[30] | Toën, B., The homotopy theory of dg-categories and derived Morita theory, Invent. Math., 167, 3, 615-667 (2007) · Zbl 1118.18010 |

[31] | Verdier, J. L., Categories derivées, SGA 4 1/2, Lecture Notes in Math., vol. 569 (1977), Springer-Verlag |

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