Math 256B Notes
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EXERCISES on LIMITS & COLIMITS Exercise 1. Prove That Pullbacks Of
EXERCISES ON LIMITS & COLIMITS PETER J. HAINE Exercise 1. Prove that pullbacks of epimorphisms in Set are epimorphisms and pushouts of monomorphisms in Set are monomorphisms. Note that these statements cannot be deduced from each other using duality. Now conclude that the same statements hold in Top. Exercise 2. Let 푋 be a set and 퐴, 퐵 ⊂ 푋. Prove that the square 퐴 ∩ 퐵 퐴 퐵 퐴 ∪ 퐵 is both a pullback and pushout in Set. Exercise 3. Let 푅 be a commutative ring. Prove that every 푅-module can be written as a filtered colimit of its finitely generated submodules. Exercise 4. Let 푋 be a set. Give a categorical definition of a topology on 푋 as a subposet of the power set of 푋 (regarded as a poset under inclusion) that is stable under certain categorical constructions. Exercise 5. Let 푋 be a space. Give a categorical description of what it means for a set of open subsets of 푋 to form a basis for the topology on 푋. Exercise 6. Let 퐶 be a category. Prove that if the identity functor id퐶 ∶ 퐶 → 퐶 has a limit, then lim퐶 id퐶 is an initial object of 퐶. Definition. Let 퐶 be a category and 푋 ∈ 퐶. If the coproduct 푋 ⊔ 푋 exists, the codiagonal or fold morphism is the morphism 훻푋 ∶ 푋 ⊔ 푋 → 푋 induced by the identities on 푋 via the universal property of the coproduct. If the product 푋 × 푋 exists, the diagonal morphism 훥푋 ∶ 푋 → 푋 × 푋 is defined dually. Exercise 7. In Set, show that the diagonal 훥푋 ∶ 푋 → 푋 × 푋 is given by 훥푋(푥) = (푥, 푥) for all 푥 ∈ 푋, so 훥푋 embeds 푋 as the diagonal in 푋 × 푋, hence the name. -
Arxiv:2008.00486V2 [Math.CT] 1 Nov 2020
Anticommutativity and the triangular lemma. Michael Hoefnagel Abstract For a variety V, it has been recently shown that binary products com- mute with arbitrary coequalizers locally, i.e., in every fibre of the fibration of points π : Pt(C) → C, if and only if Gumm’s shifting lemma holds on pullbacks in V. In this paper, we establish a similar result connecting the so-called triangular lemma in universal algebra with a certain cat- egorical anticommutativity condition. In particular, we show that this anticommutativity and its local version are Mal’tsev conditions, the local version being equivalent to the triangular lemma on pullbacks. As a corol- lary, every locally anticommutative variety V has directly decomposable congruence classes in the sense of Duda, and the converse holds if V is idempotent. 1 Introduction Recall that a category is said to be pointed if it admits a zero object 0, i.e., an object which is both initial and terminal. For a variety V, being pointed is equivalent to the requirement that the theory of V admit a unique constant. Between any two objects X and Y in a pointed category, there exists a unique morphism 0X,Y from X to Y which factors through the zero object. The pres- ence of these zero morphisms allows for a natural notion of kernel or cokernel of a morphism f : X → Y , namely, as an equalizer or coequalizer of f and 0X,Y , respectively. Every kernel/cokernel is a monomorphism/epimorphism, and a monomorphism/epimorphism is called normal if it is a kernel/cokernel of some morphism. -
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, . -
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. -
Foundations of Algebraic Geometry Problem Set 11
FOUNDATIONS OF ALGEBRAIC GEOMETRY PROBLEM SET 11 RAVI VAKIL This set is due Thursday, February 9, in Jarod Alper’s mailbox. It covers (roughly) classes 25 and 26. Please read all of the problems, and ask me about any statements that you are unsure of, even of the many problems you won’t try. Hand in six solutions. If you are ambitious (and have the time), go for more. Problems marked with “-” count for half a solution. Problems marked with “+” may be harder or more fundamental, but still count for one solution. Try to solve problems on a range of topics. You are encouraged to talk to each other, and to me, about the problems. I’m happy to give hints, and some of these problems require hints! Class 25: 1. Verify that the following definition of “induced reduced subscheme structure” is well- defined. Suppose X is a scheme, and S is a closed subset of X. Then there is a unique reduced closed subscheme Z of X “supported on S”. More precisely, it can be defined by the following universal property: for any morphism from a reduced scheme Y to X, whose image lies in S (as a set), this morphism factors through Z uniquely. Over an affine X = Spec R, we get Spec R/I(S). (For example, if S is the entire underlying set of X, we get Xred.) 2+. Show that open immersions and closed immersions are separated. (Answer: Show that monomorphisms are separated. Open and closed immersions are monomorphisms, by earlier exercises. Alternatively, show this by hand.) 3+. -
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 . -
Categories with Fuzzy Sets and Relations
Categories with Fuzzy Sets and Relations John Harding, Carol Walker, Elbert Walker Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003, USA jhardingfhardy,[email protected] Abstract We define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrich this category with additional monoidal and involutive structure coming from t-norms and negations on the unit interval. We develop the basic properties of this category and consider its relation to other familiar categories. A discussion is made of extending these results to the setting of type-2 fuzzy sets. 1 Introduction A fuzzy set is a map A : X ! I from a set X to the unit interval I. Several authors [2, 6, 7, 20, 22] have considered fuzzy sets as a category, which we will call FSet, where a morphism from A : X ! I to B : Y ! I is a function f : X ! Y that satisfies A(x) ≤ (B◦f)(x) for each x 2 X. Here we continue this path of investigation. The underlying idea is to lift additional structure from the unit interval, such as t-norms, t-conorms, and negations, to provide additional structure on the category. Our eventual aim is to provide a setting where processes used in fuzzy control can be abstractly studied, much in the spirit of recent categorical approaches to processes used in quantum computation [1]. Order preserving structure on I, such as t-norms and conorms, lifts to provide additional covariant structure on FSet. In fact, each t-norm T lifts to provide a symmetric monoidal tensor ⊗T on FSet. -
MAT 615 Topics in Algebraic Geometry Stony Brook University
MAT 615 Topics in Algebraic Geometry Jason Starr Stony Brook University Fall 2018 Problem Set 4 MAT 615 PROBLEM SET 4 Homework Policy. This problem set fills in the details of the Stable Reduction Theorem from lecture, used to complete the proof of the Irreducibility Theorem of Deligne and Mumford for geometric fibers of Mg;n ! Spec Z. Problems. Problem 0.(Variant of the valuative criterion of properness.) Let B be a Noe- therian scheme that is integral, i.e., irreducible and reduced. Let X be a projective B-scheme. Let Z be a nowhere dense closed subset of X (possibly empty). Let U denote the dense open complement of Z. Let V be finite type and separated over B, and let f : U ! V be a proper, surjective morphism. Let V o ⊂ V be an open subset that is dense in every B-fiber. (a) A B-scheme is generically geometrically connected if every irreducible compo- nent of the B-scheme dominates B and if the fiber of the B-scheme over Spec Frac(B) is connected. Show that V o is generically geometrically connected over B if and only if V is generically geometrically connected over B. If U is generically geomet- rically connected over B, prove that V is generically geometrically connected over B, and the converse holds provided that U is generically geometrically connected over V . Finally, prove that U is generically geometrically connected over B if and only if X is generically geometrically connected over B. (b) If X is a projective B-scheme that is generically geometrically connected over B, use Zariski's Connectedness Theorem to prove that every geometric fiber is connected (cf. -
Algebraic Geometry UT Austin, Spring 2016 M390C NOTES: ALGEBRAIC GEOMETRY
Algebraic Geometry UT Austin, Spring 2016 M390C NOTES: ALGEBRAIC GEOMETRY ARUN DEBRAY MAY 5, 2016 These notes were taken in UT Austin’s Math 390c (Algebraic Geometry) class in Spring 2016, taught by David Ben-Zvi. I live-TEXed them using vim, and as such there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Shamil Asgarli, Adrian Clough, Feng Ling, Tom Oldfield, and Souparna Purohit for fixing a few mistakes. Contents 1. The Course Awakens: 1/19/163 2. Attack of the Cones: 1/21/166 3. The Yoneda Chronicles: 1/26/16 10 4. The Yoneda Chronicles, II: 1/28/16 13 5. The Spectrum of a Ring: 2/2/16 16 6. Functoriality of Spec: 2/4/16 20 7. The Zariski Topology: 2/9/16 23 8. Connectedness, Irreducibility, and the Noetherian Condition: 2/11/16 26 9. Revenge of the Sheaf: 2/16/16 29 10. Revenge of the Sheaf, II: 2/18/16 32 11. Locally Ringed Spaces: 2/23/16 34 12. Affine Schemes are Opposite to Rings: 2/25/16 38 13. Examples of Schemes: 3/1/16 40 14. More Examples of Schemes: 3/3/16 44 15. Representation Theory of the Multiplicative Group: 3/8/16 47 16. Projective Schemes and Proj: 3/10/16 50 17. Vector Bundles and Locally Free Sheaves: 3/22/16 52 18. Localization and Quasicoherent Sheaves: 3/24/16 56 19. The Hilbert Scheme of Points: 3/29/16 59 20. Differentials: 3/31/16 62 21. -
Lecture 11 Notes
Math 210C Lecture 11 Notes Daniel Raban April 24, 2019 1 Additive and Abelian Categories 1.1 Additive categories Definition 1.1. An additive category C is a category such that 1. For A; B 2 C, HomC(A; B) is an abelian group under some operation + such that for any diagram f g1 A B C h D g2 in C, h ◦ (g1 + g2) ◦ f = h ◦ g1 ◦ g + h ◦ g2 ◦ f. 2. C has a zero object 0. 3. C admits finite coproducts. Example 1.1. Ab is an additive category. Example 1.2. Let R be a ring. R-Mod is an additive category. Example 1.3. The full subcategory of R-Mod of finitely generated R-modules is additive. Example 1.4. The category of topological abelian groups with continuous homomor- phisms is additive. Lemma 1.1. An additive category C admite finite product. In fact, for A1;A2 2 C, there ∼ are natural isomorphisms A1×A2 = A1qA2. For the inclusion morphisms ιi : Ai ! A1qA2 and projection morphisms pi : Ai × A2 ! Ai, we have pi ◦ ιi = idAi , pi ◦ ιj = 0 for i 6= j, and ι1 ◦ p1 + ι2 ◦ p2 = idA1qA2 . Proof. We have the ιis. define pi : A1 q A2 ! Ai by ( idAi i = j pi ◦ ιj = ; 0 i 6= j 1 which exists by the universal property of the coproduct: A1 p1 idA1 0 A1 q A2 ι1 ι2 A1 A2 Then (ι1 ◦ p1 + ι2 ◦ p2) ◦ ι1 = ι1 =) ι1 ◦ p1 + ι2 ◦ p2 = idA1qA2 by the universal property of the coproduct. Given B 2 C and gi : B ! Ai, set = ι1 ◦g1 +ι2 ◦g2 : B ! A1 qA2. -
Properties of Morphisms
Properties of morphisms Milan Lopuhaä May 1, 2015 1 Introduction This talk will be about some properties of morphisms of schemes. One of such properties was already discussed by Johan: Definition 1.1. Let f : X ! Y be a morphism of schemes. f is called separated if the image of the diagonal morphism ∆: X ! X ×Y X is closed. This property is stable under base change and composition; this will be the case for many other properties we will discuss. 2 Finiteness properties Let f : A ! B be a morphism of commutative rings. f is said to be of finite type if B is finitely generated as an A-algebra. If this is the case, ∼ then B = A[X1;X2;:::;Xn]/I for some integer n and some ideal I. This motivates the following definition: Definition 2.1. Let f : X ! Y be a morphism of schemes. f is called lo- cally of finite type if Y can be covered by a collections of affine opens fUigi2I −1 f g such that every f (Ui) can be covered by affine opens Vij j2Ji such that ] every f : OY (Ui) !OX (Vij) is of finite type. If all Ji can be chosen to be finite, then f is said to be of finite type. The ‘geometric’ intuition of ‘of finite type’ is that it is the algebraic equivalent of finite-dimensionality. For example, smooth reduced schemes of finite type over C correspond to (finite-dimensional) smooth complex manifolds. 1 Over non-Noetherian schemes, this is usually not a very useful finiteness property, since the ideal I above may not be finitely generated; in this case, we still need infinitely many relations to describe B. -
The Geometry of Moduli Spaces of Sheaves
The Geometry of Moduli Spaces of Sheaves Daniel Huybrechts and Manfred Lehn UniversitÈatGH Essen UniversitÈatBielefeld Fachbereich 6 Mathematik SFB 343 FakultÈatfÈur Mathematik UniversitÈatsstraûe 3 Postfach 100131 D-45117 Essen D-33501 Bielefeld Germany Germany [email protected] [email protected] v Preface The topic of this book is the theory of semistable coherent sheaves on a smooth algebraic surface and of moduli spaces of such sheaves. The content ranges from the de®nition of a semistable sheaf and its basic properties over the construction of moduli spaces to the bira- tional geometry of these moduli spaces. The book is intended for readers with some back- ground in Algebraic Geometry, as for example provided by Hartshorne's text book [98]. There are at least three good reasons to study moduli spaces of sheaves on surfaces. Firstly, they provide examples of higher dimensional algebraic varieties with a rich and interesting geometry. In fact, in some regions in the classi®cation of higher dimensional varieties the only known examples are moduli spaces of sheaves on a surface. The study of moduli spaces therefore sheds light on some aspects of higher dimensional algebraic geometry. Secondly, moduli spaces are varieties naturally attached to any surface. The understanding of their properties gives answers to problems concerning the geometry of the surface, e.g. Chow group, linear systems, etc. From the mid-eighties till the mid-nineties most of the work on moduli spaces of sheaves on a surface was motivated by Donaldson's ground breaking re- sults on the relation between certain intersection numbers on the moduli spaces and the dif- ferentiable structure of the four-manifold underlying the surface.