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Commutative Algebra Commutative Algebra B Totaro Michaelmas 2011 Contents 9 Homological algebra 26 9.1 Derived functors........................ 27 1 Basics2 1.1 Rings & homomorphisms...................2 10 Integral Extensions 30 1.2 Modules.............................4 11 Noether normalisation and Hilbert’s Nullstellensatz 35 1.3 Prime & maximal ideals....................4 12 Artinian rings 37 2 Affine schemes8 13 Discrete valuation rings and Dedekind domains 39 3 Irreducible closed subsets of Spec(R) 8 13.1 Krull’s Principal Ideal Theorem................ 41 4 Operations on modules9 14 Dimension theory for finitely generated algebras over a field 42 5 Direct limits 11 15 Regular local rings 45 15.1 Miscellaneous questions answered............. 47 6 Tensor products 11 15.2 Regular local rings, concluded................ 48 6.1 Algebras and tensor products................. 13 6.2 Exactness properties of tensor products........... 15 7 Localisation 16 7.1 Special cases of localisation.................. 17 7.2 Local rings............................ 17 7.3 Localisation of modules.................... 19 7.4 Nakayama’s Lemma...................... 21 8 Noetherian rings 22 8.1 Decomposition of irreducible closed subsets........ 24 Notes typed by Zach Norwood. Send any comments/corrections to zn217 at cam. 1 Lectures Commutative algebra is about commutative rings: Z, k[x1,...,xn], etc. 1 7 October2 The philosophy of the subject is to try to think of a commutative ring 7/10 2 10 October3 as a ring of functions on some space. 3 12 October5 4 14 October7 1 Basics 5 17 October9 1.1 Rings & homomorphisms 6 19 October 10 Definition. A ring R is a set with two binary operations, and , such Å ¢ 7 21 October 12 that: 8 24 October 14 (1)( R, ) is an abelian group (with identity 0); 9 26 October 17 Å (2)( R, ) is a monoid (with identity (1); 10 28 October 19 ¢ (3) Addition distributes over multiplication [sic?]: 11 31 October 21 x(y z) x y xz and (y z)x yx zx 12 2 November 23 Å Æ Å Å Æ Å 13 4 November 25 for all x, y,z R. 2 14 7 November 28 Examples. Though we won’t deal with them in this course, here are 15 9 November 30 some examples of noncommutative rings: 16 11 November 32 (1) For a field k, the ring M k of n n matrices over k; and n £ 17 14 November 34 (2) For a field k and a group G, the group ring kG. 18 16 November 37 In this course, a ring means commutative ring unless otherwise stated. 19 18 November 39 20 21 November 40 Examples. The following are examples of rings: 21 23 November 42 (1) fields, such as Q, R, C, F Z/p for p a prime; p Æ 22 25 November 43 (2) the ring Z of integers; 23 28 November 45 (3) the ring k[x1,x2,...,xn] of polynomials with coefficients in a field k; 24 30 November 47 (4) for a topological space X the ring C(X ) of continuous functions X R; ! 2 (5) for a smooth manifold X the set C 1(X ) of smooth functions X R Can check that a homomorphism f satisfies f (0) 0 and f ( x) ! Æ ¡ Æ forms a ring. f (x). (Exercise!) ¡ Remark. We don’t require 0 1 in a ring. If 0 1 in a ring R, then Example. If A is a subring of B then the inclusion map A , B is a ring 6Æ Æ ! (Exercise!) R {0} {1}, and we call this ring the zero ring, R 0. homomorphism. Also, if f : A B and g : B C are ring homomor- Æ Æ Æ ! ! phisms, then so is the composite g f . Rings and homomorphisms form Exercise. 0 x 0 and ( 1) x x for every x R. ± ¢ Æ ¡ ¢ Æ¡ 2 a category. Definition. Let R be a ring. Say x R is a unit (or is invertible) if there 2 Definition. An ideal I in a ring R is an additive subgroup such that for is an element y in R such that x y 1. If so, then y is unique (Exercise!) any x I and y R, x y I. 1 1 Æ and so write y x¡ or y . 2 2 2 Æ Æ x An element x R is a zerodivisor if there is a nonzero element y R Remark. The kernel of any ring homomorphism is an ideal. 2 2 such that x y 0. An element x R is nilpotent if there is a n 0 such Æ 2 È that xn 0. Examples. Æ Definition. A ring R is a field if 1 0 in R and every nonzero element (1) The only ideals in a field k are 0 and k. 6Æ of R is invertible. We say R is an integral domain (or just a domain) if (2) Any ideal I R that contains 1 must be all of R. So an ideal is not ⊆ 1 0 in R and the product of any two nonzero elements of R is nonzero. usually a subring. 6Æ A ring R is reduced if the only nilpotent element of R is 0. (3) Z is a PID: i.e., every ideal in Z is of the form (n) {nx : x Z} for Æ 2 some n Z. Examples. 2 (4) If A is a ring of functions on a space X and Y X is a subspace, ⊆ (1) The zero ring is reduced but is not a domain (or a field). then © ª (2) For a positive integer n, the ring Z/n is a field iff it’s a domain, iff n I f A : f (y) 0 y Y Æ 2 Æ 8 2 is prime. Also, Z/n is reduced iff n is a product of distinct primes. is an ideal. In Z/12, for instance, 6 is nilpotent and nonzero, the elements 2 & 3 are zerodivisors but not nilpotent, and 5 is a unit. 10/10 (3) Z and k[x ,...,x ] are domains and not fields if n 1. (See Exam- 1 n ¸ Definition. For ideals I and J in a ring R we define I J to be the ideal ple(s) Sheet 1.) Å (Check: Exercise!) I J : ©x y : x I, y Jª. Å Æ Å 2 2 Definition. A homomorphism from a ring A to a ring B is a function For an ideal I in a ring R, the quotient ring R/I is the quotient abelian f : A B that preserves , , and 1; that is, group with product structure defined by f (x)f (y) : f (x y) (where f is ! Å ¢ Æ the quotient map f : R R/I). • f (x y) f (x) f (y) for all x, y R; Å Æ Å 2 This is well defined since I is an ideal, and f : R R/I is a ring homo- • f (x y) f (x)f (y) for all x, y R; Æ 2 morphism. Usually we use the same name x for an element of R and its • f (1 ) 1 . A Æ B image in R/I. 3 Example. In Q[x] the elements x3 and 5x2 aren’t equal, but in the quo- (3) For a field k, a k[x]-module M is equivalent to a k-vector space with tient ring Q[x]/(x2 5), we do have x3 5x. a k-linear map x : M M. ¡ Æ ! (4) An ideal I in a ring R determines two R-modules. First, an ideal is For a ring homomorphism f : A B the image f (A) im f is a ! Æ exactly an R-submodule of R. But also the quotient ring R/I is an subring of B, and ker f ©a A : f (a) 0ª is an ideal of A. Moreover, Æ 2 Æ R-module. A/ker f and im f are isomorphic as rings: A/ker f im f . Æ» Note: for a positive integer n, the quotient ring Z/(n) is usually called Definition. An R-module homomorphism (or R-linear map) M M 1 ! 2 Z/n for short. is a homomorphism f : M M of abelian groups such that f (r m ) 1 ! 2 1 Æ r f (m ) for every r R, m M . Exercise. For a nonzero ring R, the following are equivalent: 1 2 1 2 1 This definition makes the collection of R-modules (for a fixed R) into (1) R is a field; a category. (2) the only ideals in R are 0 and R; For R-modules M and N the set Hom (M,N) of R-linear maps M R ! (3) any ring homomorphism from R to a nonzero ring is injective. N is an abelian group under pointwise addition: (f g )(m) f (m) g (m) Å Æ Å for m M. Since R is commutative, HomR (M,N) is an R-module: Exercise. Show that in any ring R the set of nilpotent elements forms 2 an ideal, called the nilradical of R, N rad(0) R. Show the quotient (a f )(m) a f (m) for a R, f HomR (M,N), m M. Æ ⊆ ¢ Æ ¢ 2 2 2 R/N is reduced. Definition. An R-submodule of an R-module M is an abelian subgroup N M such that r n N for all r R, n N. 1.2 Modules ⊆ ¢ 2 2 2 For an R-submodule N of M, the quotient R-module M/N is the quo- Definition. A module M over a ring R is an abelian group with a function tient abelian group with the obvious R-module structure: r (f (m)) Æ R M M, written (r,m) r m, satisfying f (r m), if f : M M/N is the quotient map.
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