Topological and scheme-theoretic properties of the $D$-graded Proj construction
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912971803852800 |
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| author | Goebler, Felix |
| author_facet | Goebler, Felix |
| contents | We generalize the topological description of the $\mathbb{N}$-graded Proj construction to the multigraded Proj construction for factorially graded rings that are graded by finitely generated abelian groups $D$. However, there is one big structural difference: While the classical description is given by the space of homogeneous prime ideals not containing the irrelevant ideal, we characterize the multigraded Proj setting using $D$-prime ideals, i.e.\ ideals that have the prime property, but only for homogeneous factorizations. In particular, we establish a multigraded version of the Nullstellensatz. Additionally, we present algebraic conditions for separability in terms of factorially graded rings, and observe that Proj$^D(S)$ is not separated in many cases. Finally, building on Mayeux-Riche's definition of Serre twists, we give a criterion for their freeness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12812 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological and scheme-theoretic properties of the $D$-graded Proj construction Goebler, Felix Algebraic Geometry Commutative Algebra We generalize the topological description of the $\mathbb{N}$-graded Proj construction to the multigraded Proj construction for factorially graded rings that are graded by finitely generated abelian groups $D$. However, there is one big structural difference: While the classical description is given by the space of homogeneous prime ideals not containing the irrelevant ideal, we characterize the multigraded Proj setting using $D$-prime ideals, i.e.\ ideals that have the prime property, but only for homogeneous factorizations. In particular, we establish a multigraded version of the Nullstellensatz. Additionally, we present algebraic conditions for separability in terms of factorially graded rings, and observe that Proj$^D(S)$ is not separated in many cases. Finally, building on Mayeux-Riche's definition of Serre twists, we give a criterion for their freeness. |
| title | Topological and scheme-theoretic properties of the $D$-graded Proj construction |
| topic | Algebraic Geometry Commutative Algebra |
| url | https://arxiv.org/abs/2602.12812 |