Viewing Quasi-Coherent Sheaves of Ideals as Ideals of a Ring
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929648178298880 |
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| author | Arodirik, Ayçin Iplikçi |
| author_facet | Arodirik, Ayçin Iplikçi |
| contents | This paper presents a technique for viewing quasi-coherent sheaves of ideals of a given blowup as regular ideals of a ring. In the paper, we first describe (Zariski) models as integral schemes that are separated and of finite type over an integral domain $D$. We then construct a ring $D^*$ for a given projective model (e.g. blowup of $D$ over a finitely generated ideal) by intersecting Nagata function rings. The spectrum of $D^*$ contains the projective model, but similar to the Proj-construction, it includes additional prime ideals. We characterize the relevant ideals and construct a faithfully flat morphism of schemes from the spectrum of $D^*$ to the model. Finally, using Abhyankar's definition of ideals on models, we identify the relevant ideals of $D^*$ with the quasi-coherent sheaves of ideals of the corresponding projective model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18694 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Viewing Quasi-Coherent Sheaves of Ideals as Ideals of a Ring Arodirik, Ayçin Iplikçi Commutative Algebra Algebraic Geometry This paper presents a technique for viewing quasi-coherent sheaves of ideals of a given blowup as regular ideals of a ring. In the paper, we first describe (Zariski) models as integral schemes that are separated and of finite type over an integral domain $D$. We then construct a ring $D^*$ for a given projective model (e.g. blowup of $D$ over a finitely generated ideal) by intersecting Nagata function rings. The spectrum of $D^*$ contains the projective model, but similar to the Proj-construction, it includes additional prime ideals. We characterize the relevant ideals and construct a faithfully flat morphism of schemes from the spectrum of $D^*$ to the model. Finally, using Abhyankar's definition of ideals on models, we identify the relevant ideals of $D^*$ with the quasi-coherent sheaves of ideals of the corresponding projective model. |
| title | Viewing Quasi-Coherent Sheaves of Ideals as Ideals of a Ring |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2412.18694 |