# Preliminary Exam Topics Sarah Mayes

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1 Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition of morphism, kernel, image, and exact sequence of sheaves Definition of quotient sheaves A morphism is an isomorphism of sheaves if and only if it is an isomorphism on stalks [Hartshorne, II.1.1] Definition of the direct image sheaf Definition of inverse image sheaf 2. Schemes Prime spectrum of a ring, Spec A Definition of (locally) ringed space and morphism of (locally) ringed spaces Correspondence between homomorphisms of rings and morphisms of affine schemes [Hartshorne, II.2.3] Definition of schemes Gluing schemes [Hartshorne, II.3.5] Definition of Proj of a ring Definition of irreducible, reduced, and integral schemes A scheme is integral if and only if it is reduced and irreducible [Hartshorne, II.3.1] Definition of finite morphisms and morphisms of finite type Definition of open and closed immersions; definition of open and closed subschemes Reduced induced closed subscheme structure [Hartshorne, II.3.2.6] Fibred product of schemes [Hartshorne II.3.3] Definition of fibres of a morphism of schemes Definition of base extensions 1

2 3. Separated and proper morphisms Definition of separated morphisms A morphism is separated if and only if the image of the diagonal is closed [Hartshorne, II.4.2] Statement of the Valuative Criterion of Separatedness [Hartshorne, II.4.3] Definition of universally closed and proper morphisms Statement of the Valuative Criterion of Properness [Hartshorne, II.4.7] Definition of a projective morphism The intersection of affine subsets of a separated scheme is affine [Hartshorne, exercise 4.3] 4. Sheaves of modules Definition of sheaves of O X -modules, morphism of sheaves of modules, free and locally free sheaves of modules Definition of the inverse image sheaf Adjointness of the inverse and direct image functors Definition of the sheaf M associated to an A-module M on SpecA and basic properties of M [Hartshorne, II.5.2] Definition of quasi-coherent and coherent sheaves of modules Equivalence between the categories of A-modules and O X -modules for X = Spec A [Hartshorne, II.5.5] Coherence (quasi-coherence) of the kernel and image of morphism of coherent (quasi-coherent) sheaves. Definition of the ideal sheaf of a closed subscheme Correspondence between quasi-coherent sheaves of ideals and closed subschemes [Hartshorne, II.5.9] Definition of the sheaf associated to a graded S-module on Proj S For X= Proj S and F a sheaf of O X modules, the definition of sheaves O X (n) and twisted sheaves F(n); basic properties [Hartshorne, II.5.12] Definition of the graded S-module associated with a sheaf of O X - modules where X = Proj S. 2

3 Γ (O X ) = S when S = A[x 0,..., x r ] and X = Proj S [Hartshorne, II.5.13] Statement of correspondence between Γ (F) and F for quasicoherent sheaves on Proj S; structure of projective scheems over Spec A [Hartshorne, II.5.15] Definition of the twisting sheaf O(1) on P r Y Definition of very ample invertible sheaves Conditions for Γ(X, F) to be a finite-dimensional k-vector space [Hartshorne, II.5.19] 5. Divisors, invertible sheaves, and projective morphisms Definition of prime, Weil, and principal divisors Definition of the divisor class group Cl X of a noetherian integral separated scheme X regular in codimension 1. A noetherian domain A is a UFD if and only if X = Spec A is normal and Cl X = 0 Definition of the divisor class group of P n k and degree of a divisor [Hartshorne, II.6.2] Correspondence between Cl X and Cl U when Z = X U is a proper closed subset of X [Hartshorne, II.6.5] Definition of a Cartier divisor Correspondence between Cartier and Weil divisors [Hartshorne, II.6.11] Definition of the Picard group Definition of L(D), the invertible sheaf corresponding to a Cartier divisor; basic properties [Hartshorne, II.6.13] Correspondence between CaCl X and Pic X for arbitrary and integral schemes [Hartshorne, II.6.14, II.6.15] Every invertible sheaf on P n k is isomorphic to O(l) for some l [Hartshorne, II.6.17] Definition of the closed subscheme associated with an effective Cartier divisor Correspondence between global sections of invertible sheaves and morphisms to projective space [Hartshorne, II.7.1] Equivalent conditions for an invertible sheaf on a Noetherian scheme to be ample [Hartshorne, II.7.5] 3

4 Relation between ample and very ample invertible sheaves 6. Differential forms and vector bundles on varieties Definition of the module of differential forms Ω[X] The module of differential forms is locally free in neighbourhoods of simple points [Shafarevich, III.4.1, Theorem 1] Generators of Ω[X] when X is an affine variety [Shafarevich, III.4.2, Prop. 1] Algebraic description of Ω[X] for a smooth affine variety [Shafarevich, III.4.2, Prop. 2] Definition of r-dimensional regular differential forms Dimension of Ω r [X] [Shafarevich, III.4.3, Theorem 2] Definition of the module of r-dimensional rational differential forms Ω r (X) and its dimension over k(x) Definition of the canonical class, canonical divisor Definition of a vector bundle and sections of a vector bundle Correspondence between vector bundles and locally free sheaves [Shafarevich, VI.1.1, Theorem 2] Definition of the sheaf of p-differential forms, the cotangent bundle, and the tangent bundle 7. Blowing up Definition of the blow-up of a variety along a smooth subvariety [Shafarevich, VI.2.2] The blow-up of a variety is a smooth and irreducible quasiprojective variety [Shafarevich, VI.2.2] Definition and structure of the exceptional subvariety Inverse image of smooth irreducible subvarieties under the blowup map [Shafarevich, VI.2.2d] Definition of Proj of a sheaf of graded algebras over a scheme Definition of the blow-up of a Noetherian scheme with respect to a coherent sheaf of ideals Definition of the inverse image ideal sheaf Statement of the universal property of blowing up [Hartshorne, II.7.14] 4

5 Morphisms of Noetherian schemes and blowing up [Hartshorne, II.7.15] 8. Cohomology of sheaves Definition of flabby sheaves Properties of flabby sheaves relating to exact sequences [Ueno, 6.2, 6.5] The canonical flabby resolution of a sheaf of additive groups Statement of commutative diagram induced by an exact sequence of sheaves of additive groups and their canonical flabby resolutions Definition of cohomology groups of a sheaf of additive groups Statement of independence of cohomology groups on choice of flabby resolutions [Ueno, 6.8] Existence of long exact sequence of cohomology groups induced from exact sequence of sheaves of additive groups [Ueno, 6.9] For a quasicoherent O X -module F over an affine scheme X, H n (X, F) = 0 [Ueno, 6.10] Injective resolution of a module Statement that Ĩ is flabby over Spec R for an injective R-module I Definition of Čech cohomology groups Existence of long exact sequence of Čech cohomology groups induced from exact sequence of sheaves of additive groups (for separated scheme) [Ueno, 6.14] Statement of Leray s theorem [Ueno, 6.15] For a quasicoherent sheaf on a separated scheme X, Ȟn (X, F) = H n (X, F) [Ueno, 6.16] Cohomology of the invertible sheaf O X (m) of projective space P n R over a noetherian ring R [Ueno, 6.19] Statements on finiteness of cohomology of projective space [Ueno, 6.21, 6.22] Cohomological criterion for ampleness of an invertible sheaf over a proper scheme on Spec A [Ueno, 6.26] 9. General commutative algebra Definition of a (discrete) valuation ring 5

6 Valuation rings lie between subrings and fields [Matsumura, 10.2] A valuation ring is integrally closed [Matsumura, 10.3] The integral closure of a ring as the intersection of valuation rings [Matsumura, 10.4] Equivalent conditions for a valuation ring to be a DVR; uniformising elements [Matsumura, 11.1] Nakayama s lemma [Matsumura, 2.2] Definition of the Krull dimension of a ring Hilbert s Nullstellensatz [Matsumura, 5.4] Definition of systems of parameters, regular systems of parameters Equivalent conditions for elements in the maximal ideal of a local ring to be part of a regular system of parameters [Matsumura, 14.2] Definition of a regular local ring A regular local ring is an integral domain [Matsumura, 14.3] Definition of Cohen-Macaulay local rings in terms of systems of parameters Equivalent conditions for a ring to be Cohen-Macaulay [Hochster, p. 12] Prime avoidance [Hochster, p. 8] The integral closure of an ideal and the Rees ring [Hochster, p. 17] Definition of a symbolic power of a prime ideal [Hochster, p. 19] Definition of the absolute integral closure of a domain [Hochster, p. 26] Base change and Froebenius functors; basic properties [Hochster, pp ] 10. Tight closure Definition of tight closure of ideals (for Noetherian rings of prime characteristic) Nine properties of tight closure (given in Hochster, pp ) Definition of (weak) F-regularity Three more properties of tight closure (given in Hochster, p. 58) Briançon-Skoda theorem Definition of test elements and test ideals [Hochster, pp ] Definition of F-finite rings [Hochster, p. 76] Result on existence of test elements [Hochster, p. 77] 6

7 References [1] Hartshorne, R., 2006: Algebraic Geometry. Springer. [2] Hochster, M., 2007: Foundations of Tight Closure Theory. [Available online at hochster/711f07/fndtc.pdf.] [3] Matsumura,H., 1986: Commutative ring theory. Cambridge University Press. [4] Shafarevich, I.R., 1977: Basic algebraic geometry. Springer-Verlag. [5] Ueno, K., 2001: Algebraic Geometry 2: Sheaves and cohomology. American Mathematical Society. Primary references for each topic: 1. Sheaves 2. Schemes Hartshorne, Section II.1 Hartshorne, Sections II.2, II.3 3. Separated and proper morphisms Hartshorne, Section II.4 4. Sheaves of modules Hartshorne, Section II.5 5. Divisors, invertible sheaves, and projective morphisms Hartshorne, Sections II.6, II.7 6. Differential forms and vector bundles on varieties Shafarevich, Sections III.4 and VI.1 7. Blowing up Shafarevich, Section VI.2 Hartshorne, Section II.7 7

8 8. Cohomology of sheaves Ueno, Chapter 6 9. General commutative algebra Matsumura Hochster 10. Tight Closure Hochster 8

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