Syzygies, regularity, and their interplay with additive combinatorics
In this thesis, we study some interactions between commutative algebra and additive combinatorics. Based on recent works by Eliahou and Mazumdar, Elias, and Colarte-Gómez, Elias and Miró-Roig, we associate with each finite set A ⊂ ℕᵈ a projective toric variety X⊂ ℙₖⁿ, where k is an infinite field an...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universidad de Valladolid |
| Repositorio: | UVaDOC. Repositorio Documental de la Universidad de Valladolid |
| OAI Identifier: | oai:uvadoc.uva.es:10324/81884 |
| Acceso en línea: | https://doi.org/10.35376/10324/81884 https://uvadoc.uva.es/handle/10324/81884 |
| Access Level: | acceso abierto |
| Palabra clave: | Algebra Graded free resolutions Resoluciones libres graduadas Additive combinatorics Combinatoria aditiva Castelnuovo-Mumford regularity Regularidad Conjuntos suma Sumsets 12 Matemáticas |
| Sumario: | In this thesis, we study some interactions between commutative algebra and additive combinatorics. Based on recent works by Eliahou and Mazumdar, Elias, and Colarte-Gómez, Elias and Miró-Roig, we associate with each finite set A ⊂ ℕᵈ a projective toric variety X⊂ ℙₖⁿ, where k is an infinite field and n = |A|-1. We focus on the study of the sumsets of A and the Castelnuovo-Mumford regularity of [k], the coordinate ring of X. In particular, we look at the cases when X is a curve, a smooth variety, and a surface with a single singular point. Moreover, when X is a curve C, we study the relation between the Betti numbers of k [C] and its affine charts. Finally, we provide an explicit method to compute the minimal graded free resolution of R/I as A-module, where I ⊂ R = k[x₁,…,xₙ] is a weighted homogeneous ideal and A, the polynomial ring in the last d variables, is a Noether normalization of R/I. |
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