Journal•ISSN: 1937-0652
Algebra & Number Theory
Mathematical Sciences Publishers
About: Algebra & Number Theory is an academic journal published by Mathematical Sciences Publishers. The journal publishes majorly in the area(s): Mathematics & Conjecture. It has an ISSN identifier of 1937-0652. Over the lifetime, 859 publications have been published receiving 14217 citations. The journal is also known as: ANT & Algebra and number theory.
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TL;DR: In this article, the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface is investigated, and some applications of their results to graph theory, arithmetic geometry, and tropical geometry are provided.
Abstract: We investigate the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface. We also provide some applications of our results to graph theory, arithmetic geometry, and tropical geometry.
263 citations
TL;DR: In this paper, the authors associate an algebra A(Γ) to a triangulation Γ of a surface S with a set of boundary marking points, and prove that the algebra is cluster-tilted if and only if the surface S is a disc or an annulus.
Abstract: We associate an algebra A(Γ) to a triangulation Γ of a surface S with a set of boundary marking points. This algebra A(Γ) is gentle and Gorenstein of dimension one. We also prove that A(Γ) is cluster-tilted if and only if it is cluster-tilted of type A or A , or if and only if the surface S is a disc or an annulus. Moreover all cluster-tilted algebras of type A or A are obtained in this way.
212 citations
TL;DR: For the case of a singularity which is an isolated point of a C ∗ -action and admits a symmetric obstruction theory compatible with the C ∆ -action, the answer is (−1) d, where d is the dimension of the Zariski tangent space as discussed by the authors.
Abstract: Recall that in an earlier paper by one of the authors DonaldsonThomas type invariants were expressed as certain weighted Euler characteristics of the moduli space. The Euler characteristic is weighted by a certain canonical Z-valued constructible function on the moduli space. This constructible function associates to any point of the moduli space a certain invariant of the singularity of the space at the point. In the present paper, we evaluate this invariant for the case of a singularity which is an isolated point of a C ∗ -action and which admits a symmetric obstruction theory compatible with the C ∗ -action. The answer is (−1) d , where d is the dimension of the Zariski tangent space. We use this result to prove that for any threefold, proper or not, the weighted Euler characteristic of the Hilbert scheme of n points on the threefold is, up to sign, equal to the usual Euler characteristic. For the case of a projective Calabi-Yau threefold, we deduce that the DonaldsonThomas invariant of the Hilbert scheme of n points is, up to sign, equal to the Euler characteristic. This proves a conjecture of Maulik-NekrasovOkounkov-Pandharipande.
191 citations
TL;DR: In this paper, it was shown that the center Z(C) of a fusion category is equivalent to a G-equivariantization of the relative center ZD(C).
Abstract: Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center ZD(C). We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara–Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara–Yamagami categories. Finally, we prove a general result about the existence of zeroes in S-matrices of weakly integral modular categories.
138 citations
TL;DR: In this paper, the authors define the triangulated category of relative singularities of a closed subscheme in a scheme, and prove a version of the Thomason-Trobaugh-Neeman localization theorem for coherent matrix factorizations.
Abstract: We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its full subcategory. The latter category is identified with the kernel of the direct image functor corresponding to the closed embedding of the zero locus and acting between the conventional (absolute) triangulated categories of singularities. Similar results are obtained for matrix factorizations of infinite rank; and two different “large” versions of the triangulated category of relative singularities, corresponding to the approaches of Orlov and Krause, are identified in the case of a Cartier divisor. A version of the Thomason–Trobaugh–Neeman localization theorem is proven for coherent matrix factorizations and disproven for locally free matrix factorizations of finite rank. Contravariant (coherent) and covariant (quasicoherent) versions of the Serre–Grothendieck duality theorems for matrix factorizations are established, and pull-backs and push-forwards of matrix factorizations are discussed at length. A number of general results about derived categories of the second kind for curved differential graded modules (CDG-modules) over quasicoherent CDG-algebras are proven on the way. Hochschild (co)homology of matrix factorization categories are discussed in an appendix.
122 citations