Algebraic Geometry
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Geometry of Multigraded Rings and Embeddings Into Toric Varieties
GEOMETRY OF MULTIGRADED RINGS AND EMBEDDINGS INTO TORIC VARIETIES ALEX KÜRONYA AND STEFANO URBINATI Dedicated to Bill Fulton on the occasion of his 80th birthday. ABSTRACT. We use homogeneous spectra of multigraded rings to construct toric embeddings of a large family of projective varieties which preserve some of the birational geometry of the underlying variety, generalizing the well-known construction associated to Mori Dream Spaces. CONTENTS Introduction 1 1. Divisorial rings and Cox rings 5 2. Relevant elements in multigraded rings 8 3. Homogeneous spectra of multigraded rings 11 4. Ample families 18 5. Gen divisorial rings and the Mori chamber decomposition in the local setting 23 6. Ample families and embeddings into toric varieties 26 References 31 INTRODUCTION arXiv:1912.04374v1 [math.AG] 9 Dec 2019 In this paper our principal aim is to construct embeddings of projective varieties into simplicial toric ones which partially preserve the birational geometry of the underlying spaces. This we plan to achieve by pursuing two interconnected trains of thought: on one hand we study geometric spaces that arise from multigraded rings, and on the other hand we analyze the relationship between local Cox rings and embeddings of projective varieties into toric ones. The latter contributes to our understanding of birational maps between varieties along the lines of [HK, KKL, LV]. It has been known for a long time how to build a projective variety or scheme starting from a ring graded by the natural numbers. This construction gave rise to the very satisfactory theory 2010 Mathematics Subject Classification. Primary: 14C20. Secondary: 14E25, 14M25, 14E30. -
Special Sheaves of Algebras
Special Sheaves of Algebras Daniel Murfet October 5, 2006 Contents 1 Introduction 1 2 Sheaves of Tensor Algebras 1 3 Sheaves of Symmetric Algebras 6 4 Sheaves of Exterior Algebras 9 5 Sheaves of Polynomial Algebras 17 6 Sheaves of Ideal Products 21 1 Introduction In this note “ring” means a not necessarily commutative ring. If A is a commutative ring then an A-algebra is a ring morphism A −→ B whose image is contained in the center of B. We allow noncommutative sheaves of rings, but if we say (X, OX ) is a ringed space then we mean OX is a sheaf of commutative rings. Throughout this note (X, OX ) is a ringed space. Associated to this ringed space are the following categories: Mod(X), GrMod(X), Alg(X), nAlg(X), GrAlg(X), GrnAlg(X) We show that the forgetful functors Alg(X) −→ Mod(X) and nAlg(X) −→ Mod(X) have left adjoints. If A is a nonzero commutative ring, the forgetful functors AAlg −→ AMod and AnAlg −→ AMod have left adjoints given by the symmetric algebra and tensor algebra con- structions respectively. 2 Sheaves of Tensor Algebras Let F be a sheaf of OX -modules, and for an open set U let P (U) be the OX (U)-algebra given by the tensor algebra T (F (U)). That is, ⊗2 P (U) = OX (U) ⊕ F (U) ⊕ F (U) ⊕ · · · For an inclusion V ⊆ U let ρ : OX (U) −→ OX (V ) and η : F (U) −→ F (V ) be the morphisms of abelian groups given by restriction. For n ≥ 2 we define a multilinear map F (U) × · · · × F (U) −→ F (V ) ⊗ · · · ⊗ F (V ) (m1, . -
SHEAVES of MODULES 01AC Contents 1. Introduction 1 2
SHEAVES OF MODULES 01AC Contents 1. Introduction 1 2. Pathology 2 3. The abelian category of sheaves of modules 2 4. Sections of sheaves of modules 4 5. Supports of modules and sections 6 6. Closed immersions and abelian sheaves 6 7. A canonical exact sequence 7 8. Modules locally generated by sections 8 9. Modules of finite type 9 10. Quasi-coherent modules 10 11. Modules of finite presentation 13 12. Coherent modules 15 13. Closed immersions of ringed spaces 18 14. Locally free sheaves 20 15. Bilinear maps 21 16. Tensor product 22 17. Flat modules 24 18. Duals 26 19. Constructible sheaves of sets 27 20. Flat morphisms of ringed spaces 29 21. Symmetric and exterior powers 29 22. Internal Hom 31 23. Koszul complexes 33 24. Invertible modules 33 25. Rank and determinant 36 26. Localizing sheaves of rings 38 27. Modules of differentials 39 28. Finite order differential operators 43 29. The de Rham complex 46 30. The naive cotangent complex 47 31. Other chapters 50 References 52 1. Introduction 01AD This is a chapter of the Stacks Project, version 77243390, compiled on Sep 28, 2021. 1 SHEAVES OF MODULES 2 In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Z- modules. Basic references are [Ser55], [DG67] and [AGV71]. We work out what happens for sheaves of modules on ringed topoi in another chap- ter (see Modules on Sites, Section 1), although there we will mostly just duplicate the discussion from this chapter. -
Differentials
Section 2.8 - Differentials Daniel Murfet October 5, 2006 In this section we will define the sheaf of relative differential forms of one scheme over another. In the case of a nonsingular variety over C, which is like a complex manifold, the sheaf of differential forms is essentially the same as the dual of the tangent bundle in differential geometry. However, in abstract algebraic geometry, we will define the sheaf of differentials first, by a purely algebraic method, and then define the tangent bundle as its dual. Hence we will begin this section with a review of the module of differentials of one ring over another. As applications of the sheaf of differentials, we will give a characterisation of nonsingular varieties among schemes of finite type over a field. We will also use the sheaf of differentials on a nonsingular variety to define its tangent sheaf, its canonical sheaf, and its geometric genus. This latter is an important numerical invariant of a variety. Contents 1 K¨ahler Differentials 1 2 Sheaves of Differentials 2 3 Nonsingular Varieties 13 4 Rational Maps 16 5 Applications 19 6 Some Local Algebra 22 1 K¨ahlerDifferentials See (MAT2,Section 2) for the definition and basic properties of K¨ahler differentials. Definition 1. Let B be a local ring with maximal ideal m.A field of representatives for B is a subfield L of A which is mapped onto A/m by the canonical mapping A −→ A/m. Since L is a field, the restriction gives an isomorphism of fields L =∼ A/m. Proposition 1. -
Relative Affine Schemes
Relative Affine Schemes Daniel Murfet October 5, 2006 The fundamental Spec(−) construction associates an affine scheme to any ring. In this note we study the relative version of this construction, which associates a scheme to any sheaf of algebras. The contents of this note are roughly EGA II §1.2, §1.3. Contents 1 Affine Morphisms 1 2 The Spec Construction 4 3 The Sheaf Associated to a Module 8 1 Affine Morphisms Definition 1. Let f : X −→ Y be a morphism of schemes. Then we say f is an affine morphism or that X is affine over Y , if there is a nonempty open cover {Vα}α∈Λ of Y by open affine subsets −1 Vα such that for every α, f Vα is also affine. If X is empty (in particular if Y is empty) then f is affine. Any morphism of affine schemes is affine. Any isomorphism is affine, and the affine property is stable under composition with isomorphisms on either end. Example 1. Any closed immersion X −→ Y is an affine morphism by our solution to (Ex 4.3). Remark 1. A scheme X affine over S is not necessarily affine (for example X = S) and if an affine scheme X is an S-scheme, it is not necessarily affine over S. However, if S is separated then an S-scheme X which is affine is affine over S. Lemma 1. An affine morphism is quasi-compact and separated. Any finite morphism is affine. Proof. Let f : X −→ Y be affine. Then f is separated since any morphism of affine schemes is separated, and the separatedness condition is local. -
Lie Algebroid Cohomology As a Derived Functor
LIE ALGEBROID COHOMOLOGY AS A DERIVED FUNCTOR Ugo Bruzzo Area di Matematica, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy; Department of Mathematics, University of Pennsylvania, 209 S 33rd st., Philadelphia, PA 19104-6315, USA; Istituto Nazionale di Fisica Nucleare, Sezione di Trieste Abstract. We show that the hypercohomology of the Chevalley-Eilenberg- de Rham complex of a Lie algebroid L over a scheme with coefficients in an L -module can be expressed as a derived functor. We use this fact to study a Hochschild-Serre type spectral sequence attached to an extension of Lie alge- broids. Contents 1. Introduction 2 2. Generalities on Lie algebroids 3 3. Cohomology as derived functor 7 4. A Hochshild-Serre spectral sequence 11 References 16 arXiv:1606.02487v4 [math.RA] 14 Apr 2017 Date: Revised 14 April 2017 2000 Mathematics Subject Classification: 14F40, 18G40, 32L10, 55N35, 55T05 Keywords: Lie algebroid cohomology, derived functors, spectral sequences Research partly supported by indam-gnsaga. u.b. is a member of the vbac group. 1 2 1. Introduction As it is well known, the cohomology groups of a Lie algebra g over a ring A with coeffi- cients in a g-module M can be computed directly from the Chevalley-Eilenberg complex, or as the derived functors of the invariant submodule functor, i.e., the functor which with ever g-module M associates the submodule of M M g = {m ∈ M | ρ(g)(m)=0}, where ρ: g → End A(M) is a representation of g (see e.g. [16]). In this paper we show an analogous result for the hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme X, with coefficients in a representation M of L (the notion of hypercohomology is recalled in Section 2). -
4 Sheaves of Modules, Vector Bundles, and (Quasi-)Coherent Sheaves
4 Sheaves of modules, vector bundles, and (quasi-)coherent sheaves “If you believe a ring can be understood geometrically as functions its spec- trum, then modules help you by providing more functions with which to measure and characterize its spectrum.” – Andrew Critch, from MathOver- flow.net So far we discussed general properties of sheaves, in particular, of rings. Similar as in the module theory in abstract algebra, the notion of sheaves of modules allows us to increase our understanding of a given ringed space (or a scheme), and to provide further techniques to play with functions, or function-like objects. There are particularly important notions, namely, quasi-coherent and coherent sheaves. They are analogous notions of the usual modules (respectively, finitely generated modules) over a given ring. They also generalize the notion of vector bundles. Definition 38. Let (X, ) be a ringed space. A sheaf of -modules, or simply an OX OX -module, is a sheaf on X such that OX F (i) the group (U) is an (U)-module for each open set U X; F OX ✓ (ii) the restriction map (U) (V ) is compatible with the module structure via the F !F ring homomorphism (U) (V ). OX !OX A morphism of -modules is a morphism of sheaves such that the map (U) F!G OX F ! (U) is an (U)-module homomorphism for every open U X. G OX ✓ Example 39. Let (X, ) be a ringed space, , be -modules, and let ' : OX F G OX F!G be a morphism. Then ker ', im ', coker ' are again -modules. If is an - OX F 0 ✓F OX submodule, then the quotient sheaf / is an -module. -
On Sheaf Theory
Lectures on Sheaf Theory by C.H. Dowker Tata Institute of Fundamental Research Bombay 1957 Lectures on Sheaf Theory by C.H. Dowker Notes by S.V. Adavi and N. Ramabhadran Tata Institute of Fundamental Research Bombay 1956 Contents 1 Lecture 1 1 2 Lecture 2 5 3 Lecture 3 9 4 Lecture 4 15 5 Lecture 5 21 6 Lecture 6 27 7 Lecture 7 31 8 Lecture 8 35 9 Lecture 9 41 10 Lecture 10 47 11 Lecture 11 55 12 Lecture 12 59 13 Lecture 13 65 14 Lecture 14 73 iii iv Contents 15 Lecture 15 81 16 Lecture 16 87 17 Lecture 17 93 18 Lecture 18 101 19 Lecture 19 107 20 Lecture 20 113 21 Lecture 21 123 22 Lecture 22 129 23 Lecture 23 135 24 Lecture 24 139 25 Lecture 25 143 26 Lecture 26 147 27 Lecture 27 155 28 Lecture 28 161 29 Lecture 29 167 30 Lecture 30 171 31 Lecture 31 177 32 Lecture 32 183 33 Lecture 33 189 Lecture 1 Sheaves. 1 onto Definition. A sheaf S = (S, τ, X) of abelian groups is a map π : S −−−→ X, where S and X are topological spaces, such that 1. π is a local homeomorphism, 2. for each x ∈ X, π−1(x) is an abelian group, 3. addition is continuous. That π is a local homeomorphism means that for each point p ∈ S , there is an open set G with p ∈ G such that π|G maps G homeomorphi- cally onto some open set π(G). -
The Algebraic Theory of D-Modules
THE ALGEBRAIC THEORY OF D-MODULES DMITRY GOUREVITCH Abstract. These are draft notes of the course I gave in fall semester 2015 at the Weizmann Institute of Science, under the guidance of Joseph Bernstein. Notes by Alexander Shamov. Typos are probable, mistakes possible, comments welcome. Contents 1. D-modules on affine spaces 1 1.1. Dimension 2 1.2. Proof of Bernstein’s inequality 4 2. Holonomic modules, and an application. 5 2.1. Another proof of Bernstein inequality 5 3. Associated varieties and singular support 7 3.1. Digression on several definitions of dimension of algebraic varieties 8 3.2. The geometric filtration 9 3.3. Involutivity of the associated variety 9 3.4. Irreducible non-holonomic Dn-modules 10 4. Operations on D-modules 10 5. Homological properties 13 6. D-modules on smooth affine varieties 18 7. D-modules on general separated smooth varieties 20 8. Kashiwara’s lemma and its corollaries 21 8.1. Corollaries 22 9. D-modules on the projective space 24 9.1. Twisted differential operators on the projective space 25 10. The Bernstein-Kashiwara theorem on distributional solutions of holonomic modules 26 10.1. Proof of Theorem 10.6 29 11. Derived categories 30 References 34 1. D-modules on affine spaces Notation 1.1. Dn := hx1, . , xn, ∂1, . , ∂ni is the algebra of polynomial differential operators. Definition 1.2. Let M be a finitely generated Dn-module. Then a solution of M is a ∞ n −∞ n homomorphism from M to some Dn-module of functions (say, C (R ) or C (R )). -
LECTURE 3 MATH 256B Today We Will Discuss Some More
LECTURE 3 MATH 256B LECTURES BY: PROFESSOR MARK HAIMAN NOTES BY: JACKSON VAN DYKE Today we will discuss some more examples and features of the Proj construction. 1. Examples of Proj Example 1. As we saw last time, Proj k [x0; ··· ; xn] is classical projective space n Pk . This is covered by affines Xxi , i.e. the same ad hoc affines we found to show it was a scheme to begin with. Explicitly these now look like: ∼ n Xxi = Spec k [x0=xi; ··· ; xn=xi] = Ak where we omit xi=xi. Note that Xxi \ Xxj = Xxixj . Note that classically we only saw this for k = k, but this construction works more generally. Example 2. For any k-algebra A, we can consider Y = Spec A, and Proj A [x0; ··· ; xn]. This will be covered by n-dimensional affines over A in exactly the same way: n Xxi = AA = Spec A [v1; ··· ; vn] : But we can also think of this as A [x] = A ⊗k k [v] so n n AA = Spec A × Ak and therefore n Proj A [x0; ··· ; xn] = Y × Pk : In general, the Proj of something with nontrivial degree 0 part will be the product of Spec of this thing with projective space. 2. Functoriality Recall that since we have the quotient map R ! R=I we get that Spec R=I ,! Spec R is a closed subscheme. In light of this, we might wonder whether or not Proj (R=I) ,! Proj (R) is a closed subscheme. In fact it's not even quite clear if there is an induced map at all. -
SERRE DUALITY and APPLICATIONS Contents 1
SERRE DUALITY AND APPLICATIONS JUN HOU FUNG Abstract. We carefully develop the theory of Serre duality and dualizing sheaves. We differ from the approach in [12] in that the use of spectral se- quences and the Yoneda pairing are emphasized to put the proofs in a more systematic framework. As applications of the theory, we discuss the Riemann- Roch theorem for curves and Bott's theorem in representation theory (follow- ing [8]) using the algebraic-geometric machinery presented. Contents 1. Prerequisites 1 1.1. A crash course in sheaves and schemes 2 2. Serre duality theory 5 2.1. The cohomology of projective space 6 2.2. Twisted sheaves 9 2.3. The Yoneda pairing 10 2.4. Proof of theorem 2.1 12 2.5. The Grothendieck spectral sequence 13 2.6. Towards Grothendieck duality: dualizing sheaves 16 3. The Riemann-Roch theorem for curves 22 4. Bott's theorem 24 4.1. Statement and proof 24 4.2. Some facts from algebraic geometry 29 4.3. Proof of theorem 4.5 33 Acknowledgments 34 References 35 1. Prerequisites Studying algebraic geometry from the modern perspective now requires a some- what substantial background in commutative and homological algebra, and it would be impractical to go through the many definitions and theorems in a short paper. In any case, I can do no better than the usual treatments of the various subjects, for which references are provided in the bibliography. To a first approximation, this paper can be read in conjunction with chapter III of Hartshorne [12]. Also in the bibliography are a couple of texts on algebraic groups and Lie theory which may be beneficial to understanding the latter parts of this paper. -
The Relative Proj Construction
The Relative Proj Construction Daniel Murfet October 5, 2006 Earlier we defined the Proj of a graded ring. In these notes we introduce a relative version of this construction, which is the Proj of a sheaf of graded algebras S over a scheme X. This construction is useful in particular because it allows us to construct the projective space bundle associated to a locally free sheaf E , and it allows us to give a definition of blowing up with respect to an arbitrary sheaf of ideals. Contents 1 Relative Proj 1 2 The Sheaf Associated to a Graded Module 5 2.1 Quasi-Structures ..................................... 14 3 The Graded Module Associated to a Sheaf 15 3.1 Ring Structure ...................................... 21 4 Functorial Properties 21 5 Ideal Sheaves and Closed Subchemes 25 6 The Duple Embedding 27 7 Twisting With Invertible Sheaves 31 8 Projective Space Bundles 36 1 Relative Proj See our Sheaves of Algebras notes (SOA) for the definition of sheaves of algebras, sheaves of graded algebras and their basic properties. In particular note that a sheaf of algebras (resp. graded algebras) is not necessarily commutative. Although in SOA we deal with noncommutative algebras over a ring, here “A-algebra” will refer to a commutative algebra over a commutative ring A. Example 1. Let X be a scheme and F a sheaf of modules on X. In our Special Sheaves of Algebras (SSA) notes we defined the following structures: • The relative tensor algebra T(F ), which is a sheaf of graded OX -algebras with the property 0 that T (F ) = OX .