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| 2. |
- Sædén Ståhl, Gustav
(författare)
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An intrinsic definition of the Rees algebra of a module
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Ingår i: Proceedings of the Edinburgh Mathematical Society. - 0013-0915 .- 1464-3839.
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Tidskriftsartikel (refereegranskat)abstract
- This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.
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| 3. |
- Sædén Ståhl, Gustav
(författare)
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AN INTRINSIC DEFINITION OF THE REES ALGEBRA OF A MODULE
- 2018
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Ingår i: Proceedings of the Edinburgh Mathematical Society. - : CAMBRIDGE UNIV PRESS. - 0013-0915 .- 1464-3839. ; 61:1, s. 13-30
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Tidskriftsartikel (refereegranskat)abstract
- This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.
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| 4. |
- Sædén Ståhl, Gustav, 1987-, et al.
(författare)
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Explicit projective embeddings of standard opens of the Hilbert scheme of points
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We describe explicitly how certain standard opens of the Hilbert scheme of points are embedded into Grassmannians. The standard opens of the Hilbert scheme that we consider are given as the intersection of a corresponding basic open affine of the Grassmannian and a closed stratum determined by a Fitting ideal.
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| 5. |
- Sædén Ståhl, Gustav, 1987-
(författare)
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Good Hilbert functors
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We introduce the good Hilbert functor and prove that it is algebraic. This functor generalizes various versions of the Hilbert moduli problem, such as the multigraded Hilbert scheme and the invariant Hilbert scheme. Moreover, we generalize a result concerning formal GAGA for good moduli spaces.
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| 6. |
- Sædén Ståhl, Gustav
(författare)
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Gotzmann's persistence theorem for finite modules
- 2017
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Ingår i: Journal of Algebra. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0021-8693 .- 1090-266X. ; 477, s. 278-293
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Tidskriftsartikel (refereegranskat)abstract
- We prove a generalization of Gotzmann's persistence theorem in the case of modules with constant Hilbert polynomial. As a consequence, we show that the defining equations that give the embedding of a Quot scheme of points into a Grassmannian are given by a single Fitting ideal.
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| 7. |
- Sædén Ståhl, Gustav
(författare)
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Gotzmann's persistence theorem for finite modules
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We prove a generalization of Gotzmann's persistence theorem in the case of modules with constant Hilbert polynomial. As a consequence, we show that the defining equations that give the embedding of a Quot scheme of points into a Grassmannian are given by a single Fitting ideal.
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| 8. |
- Sædén Ståhl, Gustav, 1987-
(författare)
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Hilbert schemes and Rees algebras
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The topic of this thesis is algebraic geometry, which is the mathematical subject that connects polynomial equations with geometric objects. Modern algebraic geometry has extended this framework by replacing polynomials with elements from a general commutative ring, and studies the geometry of abstract algebra. The thesis consists of six papers relating to some different topics of this field.The first three papers concern the Rees algebra. Given an ideal of a commutative ring, the corresponding Rees algebra is the coordinate ring of a blow-up in the subscheme defined by the ideal. We study a generalization of this concept where we replace the ideal with a module. In Paper A we give an intrinsic definition of the Rees algebra of a module in terms of divided powers. In Paper B we show that features of the Rees algebra can be explained by the theory of coherent functors. In Paper C we consider the geometry of the Rees algebra of a module, and characterize it by a universal property.The other three papers concern various moduli spaces. In Paper D we prove a partial generalization of Gotzmann’s persistence theorem to modules, and give explicit equations for the embedding of a Quot scheme inside a Grassmannian. In Paper E we expand on a result of Paper D, concerning the structure of certain Fitting ideals, to describe projective embeddings of open affine subschemes of a Hilbert scheme. Finally, in Paper F we introduce the good Hilbert functor parametrizing closed substacks with proper good moduli spaces of an algebraic stack, and we show that this functor is algebraic under certain conditions on the stack.
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| 9. |
- Sædén Ståhl, Gustav
(författare)
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Rees algebras of modules and coherent functors
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We show that several properties of the theory of Rees algebras of modules become more transparent using the category of coherent functors rather than working directly with modules. In particular, we show that the Rees algebra is induced by a canonical map of coherent functors.
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| 10. |
- Sædén Ståhl, Gustav
(författare)
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Rees algebras of modules and Quot schemes of points
- 2014
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of three articles. The first two concern a generalization of Rees algebras of ideals to modules. Paper A shows that the definition of the Rees algebra due to Eisenbud, Huneke and Ulrich has an equivalent, intrinsic, definition in terms of divided powers. In Paper B, we use coherent functors to describe properties of the Rees algebra. In particular, we show that the Rees algebra is induced by a canonical map of coherent functors.In Paper C, we prove a generalization of Gotzmann's persistence theorem to finite modules. As a consequence, we show that the embedding of the Quot scheme of points into a Grassmannian is given by a single Fitting ideal.
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