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Summary
This paper presents a technique for viewing quasi-coherent sheaves of ideals of a given blowup as regular ideals of a ring. It constructs a ring D∗ by intersecting Nagata function rings and characterizes the relevant ideals.
Highlights
- Introduces a technique for viewing quasi-coherent sheaves of ideals as regular ideals of a ring.
- Constructs a ring D∗ by intersecting Nagata function rings.
- Characterizes the relevant ideals of D∗.
- Establishes a bijective correspondence between relevant ideals of D∗ and quasi-coherent sheaves of ideals on a projective model.
- Shows that the scheme corresponding to a projective model is a separated and finite-type scheme over the base ring.
- Proves that the domination map from the spectrum of D∗ to the projective model is a faithfully flat morphism of schemes.
- Demonstrates that the quasi-coherent sheaves of ideals on a blow-up scheme correspond to the relevant ideals of the corresponding D∗.
Key Insights
- The paper provides a new perspective on quasi-coherent sheaves of ideals by representing them as regular ideals of a ring, which can simplify their study and manipulation.
- The construction of the ring D∗ is a key step in this process, and its properties are crucial in establishing the correspondence between quasi-coherent sheaves and regular ideals.
- The characterization of relevant ideals of D∗ is essential in understanding which ideals can be represented as quasi-coherent sheaves, and the results have implications for algebraic geometry and commutative algebra.
- The establishment of a bijective correspondence between relevant ideals and quasi-coherent sheaves provides a powerful tool for studying these objects and has potential applications in various areas of mathematics.
- The paper's results on the scheme corresponding to a projective model and the domination map from the spectrum of D∗ provide new insights into the geometry and algebra of these objects.
- The demonstration of the correspondence between quasi-coherent sheaves on a blow-up scheme and relevant ideals of D∗ highlights the significance of this research for algebraic geometry and its applications.
- The paper's findings have the potential to influence future research in algebraic geometry, commutative algebra, and related areas, and may lead to new breakthroughs and applications in these fields.
Mindmap
Citation
Arodirik, A. I. (2024). Viewing Quasi-Coherent Sheaves of Ideals as Ideals of a Ring (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.18694