Global matrix factorizations

By Kevin Lin, D. Pomerleano. 2011

69 citations recorded by the source corpus.

We study matrix factorization and curved module categories for Landau-Ginzburg models (X,W) with X a smooth variety, extending parts of the work of Dyckerhoff. Following Positselski, we equip these categories with model category structures. Using results of Rouquier and Orlov, we identify compact generators. Via To\"en's derived Morita theory, we identify Hochschild cohomology with derived endomorphisms of the diagonal curved module; we compute the latter and get the expected result. Finally, we show that our categories are smooth, proper when the singular locus of W is proper, and Calabi-Yau when the total space X is Calabi-Yau.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/528aedfb2c711b7260d096fc7f9e95108463d72f

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, background): “equivalent to the category of singularities of the associated hypersurface in the case when the ambient scheme is regular [Orl12]. K. Lin and D. Pomerleano also tackled non-aﬃne matrix factorizations [LP11]. Contemporaneously, A. Preygel, using genuinely new ideas rooted in derived algebraic geometr”
A geometric approach to Orlov’s theorem (2010, methodology): “We use a slightly modified version of the notation in [LP11] and many ideas from [Pos09].”
Homological mirror symmetry for punctured spheres (2011, background): “anifolds, and by recent work [2, 19], is known in some generality to be homologically smooth in the sense of Kontsevich [29] (homological smoothness also holds for categories of matrix factorizations [31, 35, 30]). In this paper, we give the ﬁrst non-trivial veriﬁcation that these categories are ind”
Variation of geometric invariant theory quotients and derived categories (2012, background): “sequence of factorizations, 0 → M i li∗l ∗ i F→ M i1<i2 li1,i2∗l ∗ i1,i2F→···→l1,...,p∗l1,...,pF→0. Denote this factorization by Cˇ(U,F). We have the following result, see also Proposition 2.11 of [LP11]. Lemma 2.4.16. Let Ebe a locally-free factorization and let Fbe a quasi-coherent factorizatio”
Coherent analogues of matrix factorizations and relative singularity categories (2011, methodology): “One generalizes the above definition, replacing free modules with projective modules [Kapustin and Li 2003; Orlov 2004], with locally free sheaves [Orlov 2012], and finally with coherent sheaves [Lin and Pomerleano 2013].”
Hochschild (co)homology of the second kind I (2010, methodology): “ of ﬁnitely generated CDG-modules, it follows that the coherent sheaf Σ(M)⊠k Σ(N) on X0×k X0, viewed as an object of Dabs(B⊗k Bop-mod cdg fg ), is isomorphic to M⊗k N. 62 Applying the same lemma from [8] again, we conclude that Σ(M ⊗k N) ≃Σ(M) ⊠k Σ(N) in Db Sing(Y0). The assertion of Lemma B is prov”
Matrix factorizations for nonaffine LG–models (2011, methodology): “When this paper was finished I was informed that a category of matrix factorizations in the nonaffine case is also discussed in a forthcoming paper of Kevin Lin and Daniel Pomerleano [ 6 ].”
Matrix factorizations and Cohomological Field Theories (2011, methodology): “ns is obtained if we allow E0 and E1 to be arbitrary quasicoherent sheaves. This category is featured prominently in more recent treatments of matrix factorizations in the non-aﬃne case (see [57] and [44]). We will mostly use it in the case when W = 0. Note that in this case we have a natural class ”