MR4456599 Pippi, Massimo On the structure of dg categories of relative singularities. High. Struct. 6 (2022), no. 1, 375–402. 14F08 (14A22 14B05 18G80) arXiv:1911.01332

It follows from a classical result due to Auslander and Buchsbaum that the bounded derived category of coherent sheaves $D^b(X)$ over a regular scheme $X$ coincides with its full triangulated subcategory $\mathrm{Perf}(X)$ of perfect complexes. The quotient $\mathrm{Sing}(X) = D^b(X)/\mathrm{Perf}(X)$, called the triangulated singularity category of $X$, is hence a measure of singularities of $X$. Similarly, dg-categories of relative singularities should measure the difference of singularities of two schemes along a morphism $X_0\to X$. Matrix factorization of an element $W$ in a commutative ring $B$ is a pair of maps $d_0: E_0\to E_1$ and $d_1:E_1\to E_0$ among finitely generated projective $B$-modules satisfying $d_0\circ d_1 = W\cdot\mathrm{id}_{E_1}$, $d_1\circ d_0 = W\cdot\mathrm{id}_{E_0}$. Matrix factorizations have been introduced by D. Eisenbud [Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260:35-64 (1980) MR570778] to study modules over the factor ring $B/(W)$ and the maximal Cohen-Macaulay modules in particular, and were later very useful in the study of hypersurface singularities.

Both constructions appear in a number of related contexts and in particular in the mathematical description of derived dg-categories of D-branes in topological Landau-Ginzburg models of type B. While for Calabi-Yau varieties the category of D-branes of type B involves complexes with finite and coherent cohomologies, in more general cases anomalies (in the sense of physics) lead to a twisting where differential $d$ of complexes is deformed and instead of $d^2$ vanishing, $d^2$ is a multiplication by a regular function, the (super)potential $W$, and $\mathbf{Z}$-grading collapses to $\mathbf{Z}_2$-grading. Such differentials are essentially matrix factorizations for the given potential. For potential with isolated singularities, the corresponding triangulated category is equivalent to a (finite) product of categories of singularities of fibers of the potential map. Similar relations between the categories of matrix factorizations and singularity categories were stated and proved in various levels of generality from the early days of subject, in particular in [Dmitri O. Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, Sbornik: Mathematics 197:12 (2006) 1827-1840] and [R. O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings. Unpublished manuscript, 155 pp., 1987].

The work under review is in the setting of dg-categories and relative geometry over an affine base scheme $S = \mathrm{Spec}(A)$ where $A$ is Noetherian and $B$ is an $A$-algebra. In particular, they consider the dg-category $\mathrm{dgCat}_S$ of $A$-linear dg-categories and its enrichment ${\mathbf{dgCat}}_S^{\mathrm{idm}}$ with a cofibrantly generated model category where weak equivalences are Morita equivalences (dg-functors inducing quasi-equivalences on the derived categories of perfect dg-modules); the latter category is also enriched with a symmetric monoidal structure, then denoted ${\mathbf{dgCat}}_S^{\mathrm{idm},\otimes}$. The dg-category of relative singularities $\mathrm{Sing}(B,W)$ is defined as the kernel of a dg-functor $i_*: \operatorname{Sing}(Spec(B)\times_{\mathbf{A}^1_S}^h S)\to\operatorname{Sing}(S)$ induced by the inclusion $i : Spec(B)\times_{\mathbf{A}^1_S}^h S\to S)$, where the fiber product $\times_{\mathbf{A}^1_S}^h$ is taken in the derived sense in order to remove some unnatural regularity assumptions.

A symmetric monoidal category $\mathrm{LG}_S(n)^{\oplus}$ of $n$-dimensional Landau-Ginzburg models over $S$ is defined in the paper, where the objects are $S$-schemes flat over $S$ equipped with a morphism $\mathbf{f}:X\to\mathbf{A}^n_S$ over $S$. The dg-category of relative singularities is defined for a more general sequence $\underline{f}\in B^n$ instead of a single potential $W\in B$. It has been shown in the paper how to describe the objects in the dg-category of relative singularities $\mathrm{Sing}(B,\underline{f})$ as homotopy retracts of dg-modules over the corresponding Koszul algebra $K(B,\underline{f})$ concentrated in $n+1$ degrees. Then this is further elaborated for $n=1$ and applied to compare, within ${\mathbf{dgCat}}_S^{\mathrm{idm},\otimes}$, the dg-category of relative singularities with the dg-category of matrix factorization corresponding to the same (geometric) Landau-Ginzburg model. The correspondence which to a 1-dimensional Landau-Ginzburg model $(B,W)$ over $S$ assigns a dg-category of relative singularities $\mathrm{Sing}(B,W)$ extends to a lax monoidal $\infty$-functor $\mathrm{Sing}(-,-):\mathrm{LG}_S(1)^{\oplus}\to{\mathbf{dgCat}}_S^{\mathrm{idm},\otimes}$, likewise for assigning the corresponding dg-category $\mathrm{MF}(B,W)$ of matrix factorizations. In [A. Blanc, M. Robalo, B. Toën, G. Vezzosi, Motivic realizations of singularity categories and vanishing cycles, Journal de lÉcole polytechnique Mathématiques 5 (2018), pp. 651-747, MR3877165] a lax monoidal $\infty$-natural transformation $\mathrm{Sing}(-,-)\to\mathrm{MF}(-,-)$ has been constructed. A consequence of the technical results proven in the paper under review is that this transformation is a lax monoidal $\infty$-natural equivalence, amounting to a refined monoidal version of the comparison theorem of Orlov.

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