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Commutative Algebra and Algebraic Geometry

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Commutative Algebra and Algebraic Geometry Commutative Algebra and Algebraic Geometry Robert Friedman August 1, 2006 2 i Disclaimer: These are rough notes for a course on commutative algebra and algebraic geometry. I would appreciate all suggestions concerning ty- pos, errors, unclear exposition or obscure exercises, and any other helpful feedback. Contents 1 Introduction to Commutative Rings 1 1.1 Introduction............................ 1 1.2 Primeideals............................ 5 1.3 Localrings............................. 9 1.4 Factorizationinintegraldomains . 10 1.5 Motivatingquestions . 18 1.6 Theprimeandmaximalspectrum . 25 1.7 Gradedringsandprojectivespaces . 31 Exercises................................. 37 2 Modules over Commutative Rings 43 2.1 Basicdefinitions.......................... 43 2.2 Directandinverselimits . 48 2.3 Exactsequences.......................... 54 2.4 Chainconditions ......................... 59 2.5 ExactnesspropertiesofHom. 62 2.6 ModulesoveraPID........................ 67 2.7 Nakayama’slemma ........................ 70 2.8 Thetensorproduct ........................ 73 2.9 Productsofaffinealgebraicsets . 81 2.10Flatness .............................. 84 Exercises................................. 89 3 Localization 98 3.1 Basicdefinitions.......................... 98 3.2 Idealsinalocalization . .101 3.3 Localizationofmodules. .103 3.4 Localpropertiesofringsandmodules . 104 ii CONTENTS iii 3.5 Regularfunctions . .107 3.6 Introductiontosheaves. .112 3.7 Schemesandringedspaces . .118 3.8 Projasascheme .........................123 3.9 Zariskilocalproperties . .126 Exercises.................................127 4 Integral Ring Homomorphisms 135 4.1 Definitionofanintegralhomomorphism . 135 4.2 The going up theorem and dimension . 140 4.3 Thegoingdowntheorem . .144 4.4 Moreondimension . .. .. .146 4.5 Noethernormalization,version1 . 148 4.6 Noethernormalization,version2 . 154 Exercises.................................158 5 Ideals in Noetherian Rings 166 5.1 Irreducible sets and radical ideals . 166 5.2 Associatedprimes. .169 5.3 Primarydecomposition . .174 5.4 Artinianrings ...........................179 5.5 Fractional ideals and invertible modules . 184 5.6 Dedekinddomains . .. .. .192 5.7 Higherdimensions . .199 5.8 Extensions of Dedekind domains . 200 Exercises.................................208 6 Quasiprojective varieties 215 6.1 Projective and quasiprojective varieties . 215 6.2 Localringsandtangentspaces. 216 6.3 Derivations, differentials, and tangent spaces . 225 6.4 Elementaryprojectivegeometry . 231 6.5 Products..............................238 6.6 Grassmannians ..........................244 Exercises.................................245 iv CONTENTS 7 Graded Rings 248 7.1 FiltrationsandtheArtin-Reeslemma . 248 7.2 Hilbertfunctions . .252 7.3 Thedegreeofaprojectivevariety . 256 7.4 Thedimensiontheorem. .264 7.5 Applications of the dimension theorem . 268 7.6 Regularlocalrings . .273 7.7 Completions............................275 7.8 Hensel’s lemma and the implicit function theorem . 285 Exercises.................................291 Chapter 1 Introduction to Commutative Rings 1.1 Introduction Commutative algebra is primarily the study of those rings which most nat- urally arise in algebraic geometry and number theory.For example, let k be a field (typically algebraically closed, and often the field C of complex num- bers). Then (affine) algebraic geometry is to a large extent the study of the ring R = k[x1,...,xn] and associated objects, for example ideals I in R or the corresponding quotients R/I. In number theory, one studies the ring Z 1+√ 3 as well as related rings, for example Z[i], where i = √ 1, or Z[ − ]. There − 2 are other kinds of associated rings, for example the field of quotients Q of Z or of k[x1,...,xn] (which is denoted by k(x1,...,xn), and called the field of rational functions in n variables). More complicated rings derived from the standard ones are for example the ring of p-adic integers Zp (defined for every prime number p), as well as its polynomial analogue, the ring of formal power series in n variables, denoted by k[[x1,...,xn]]. We will discuss the construction of these and other rings later. Although the rings above are very disparate, they have a great deal of structure in common, and one of the goals of commutative algebra is to elucidate this structure. One obvious property that all of the above rings share is that they are commutative, and as its name suggest, commutative algebra is almost exclu- sively concerned with such rings. In fact, from now on we shall always make the assumptions: 1 2 CHAPTER 1. INTRODUCTION TO COMMUTATIVE RINGS Assumption: Every ring R is commutative, with a unity 1. (Note that the zero ring 0 is allowed.) Every ring homomorphism ϕ: R S is required to satisfy ϕ(1) = 1. In other words, ϕ takes the unique unity→ in R to the corresponding unity in S. In particular, if R′ is a subring of R, then we require that 1 R′. ∈ Thus, for example, if R1 and R2 are rings, then the Cartesian product R R is also a ring, with unity (1, 1), and the projections π : R R R 1 × 2 i 1 × 2 → i are ring homomorphisms in the above sense. However, in case R1 and R2 are nonzero, the subsets R1 0 and 0 R2 are not subrings in the sense described above. In more×{ technical} { terms,} × R R is a product in the 1 × 2 category of commutative rings (with unity), but not a coproduct. We shall describe the coproduct later. Quite often, rings come in pairs. Thus, implicit in the definition of k[x1,...,xn] is the ring homomorphism k k[x1,...,xn]; likewise, there are the obvious homomorphisms Z Z[i] or→Z Q or Z Z/nZ. Given → → → a ring homomorphism ϕ: R S, not necessarily injective, we call S an R-algebra, and refer (if there is→ any ambiguity) to the given homomorphism ϕ as the structural homomorphism. Morphisms, subalgebras, and quotients are required to be compatible with the given structural homomorphism. For example, if I is an ideal in R, then R/I is an R-algebra via the natural projection R R/I. → Another standard example of an R-algebra is the ring R[x1,...,xn] of polynomials in several variables with coefficients in R, with the obvious in- clusion R R[x ,...,x ]. Here, R[x ,...,x ] is the set of all expressions → 1 n 1 n r xa1 xan , a1,...,an 1 ··· n a1,...,an 0 X≥ where r R and r = 0 for only finitely many indices a ,...,a . a1,...,an ∈ a1,...,an 6 1 n More formally, we can think of R[x1,...,xn] as the set of all “multi-sequences” in R, indexed by n-tuples of nonnegative integers, and such that only finitely many terms are nonzero. Addition of polynomials is defined componentwise: r xa1 xan + s xa1 xan = (r +s )xa1 xan . a1,...,an 1 ··· n a1,...,an 1 ··· n a1,...,an a1,...,an 1 ··· n X X X Multiplication is defined so as to insure that (xa1 xan )(xb1 xbn )= xa1+b1 xan+bn . 1 ··· n 1 ··· n 1 ··· n 1.1. INTRODUCTION 3 This can be written efficiently with vector notation: if ~a =(a1,...,an) is an n-tuple of nonnegative integers, and we denote xa1 xan by x~a, then 1 ··· n ~a ~a ~b+~c r~ax s~ax = r~bs~cx . ! ! X~a X~a X~a ~b+X~c=~a If, in the definition of R[x1,...,xn], we omit the requirement that only finitely many coefficients ra1,...,an are nonzero, we obtain a larger ring containing R and R[x1,...,xn], the ring of formal power series R[[x1,...,xn]]. The R-algebra R[x1,...,xn] has the following universal property with re- spect to R-algebras: if S is an R-algebra, to give an R-algebra homomorphism ϕ: R[x ,...,x ] S is equivalent to specifying n elements α ,...,α S. 1 n → 1 n ∈ In one direction, the αi are just given by ϕ(xi), so that ϕ determines n el- ements α ,...,α S. Conversely, given n elements α ,...,α S, the 1 n ∈ 1 n ∈ evaluation homomorphism ev : R[x ,...,x ] S defined by α1,...,αn 1 n → evα1,...,αn (f(x1,...,xn)) = f(α1,...,αn) defines a homomorphism ϕ: R[x ,...,x ] S, and these constructions are 1 n → clearly inverse. In general, if an R-algebra S is the quotient of R[x1,...,xn] (in the sense of R-algebras), then we say that S is a finitely generated R- algebra. If in addition there exists a surjection R[x ,...,x ] S whose 1 n → kernel is a finitely generated ideal of R[x1,...,xn], then we say that S is a finitely presented R-algebra. Both algebraic geometry and algebraic number theory are very much relative theories, often concerned not just with a single ring R but rather with a ring R and an R-algebra S, and we shall try to emphasize this aspect a much as possible. The set of ideals I in a ring R, and in particular the sets of prime and maximal ideals, are among the most important objects of study in algebra. They can be generalized to modules: roughly speaking, a module over a ring R is the formal analogy of a vector space over a field k: Definition 1.1.1. An R-module M consists of an abelian group M (with group operation denoted by addition), together with a function R M M, whose value at (r, m) is denoted by r m or rm, and is usually referred× → to as · multiplication, satisfying: (i) For all r, s R and m M, r(sm)=(rs)m; ∈ ∈ 4 CHAPTER 1. INTRODUCTION TO COMMUTATIVE RINGS (ii) For all r ,r R and m M, (r + r )m = r m + r m; 1 2 ∈ ∈ 1 2 1 2 (iii) For all r R and m , m M, r(m + m )= rm + rm ; ∈ 1 2 ∈ 1 2 1 2 (iv) For all m M, 1 m = m. ∈ · A submodule N of an R-module module M is defined in the obvious way. If N is a submodule of an R-module module M, then there is a natural structure of an R-module on the abelian group M/N, and we will refer to this structure as the quotient R-module.
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