A Primer of Commutative Algebra James S. Milne May 30, 2009, v2.10 Abstract These notes prove the basic theorems in commutative algebra required for alge- braic geometry and algebraic groups. They assume only a knowledge of the algebra usually taught in advanced undergraduate or first-year graduate courses. Available at www.jmilne.org/math/. Contents 1 Rings and algebras . 2 2 Ideals . 3 3 Noetherian rings . 8 4 Unique factorization . 13 5 Integrality . 15 6 Rings of fractions . 19 7 Direct limits. 24 8 Tensor Products . 25 9 Flatness . 29 10 The Hilbert Nullstellensatz . 33 11 The max spectrum of a ring . 35 12 Dimension theory for finitely generated k-algebras . 43 13 Primary decompositions . 46 14 Artinian rings . 50 15 Dimension theory for noetherian rings . 51 16 Regular local rings . 55 17 Connections with geometry . 57 NOTATIONS AND CONVENTIONS Our convention is that rings have identity elements,1 and homomorphisms of rings respect the identity elements. A unit of a ring is an element admitting an inverse. The units of a 2 ring A form a group, which we denote A. Throughout “ring” means “commutative ring”. c 2009 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. 1An element e of a ring A is an identity element if ea a ae for all elements a of the ring. It is usually D D denoted 1A or just 1. Some authors call this a unit element, but then an element can be a unit without being a unit element. Worse, a unit need not be the unit. 2 This notation differs from that of Bourbaki, who writes A for the multiplicative monoid A r 0 and A f g for the group of units. We shall rarely need the former, and is overused. 1 1 RINGS AND ALGEBRAS 2 Following Bourbaki, we let N 0;1;2;::: . For a field k, kal denotes an algebraic closure D f g of k. X YX is a subset of Y (not necessarily proper). X def YX is defined to be Y , or equals Y by definition. D X YX is isomorphic to Y . X YX and Y are canonically isomorphic (or there is a given or unique isomorphism). ' ACKNOWLEDGEMENTS I thank the following for providing corrections and comments for earlier versions of these notes: Andrew McLennan, Shu Otsuka. 1 Rings and algebras Let A be a ring. A subring of A is a subset that contains 1A and is closed under addition, multiplication, and the formation of negatives. An A-algebra is a ring B together with a homomorphism iB A B.A homomorphism of A-algebras B C is a homomorphism W ! ! of rings ' B C such that '.iB .a// iC .a/ for all a A. W ! D 2 Elements x1;:::;xn of an A-algebra B are said to generate it if every element of B can be expressed as a polynomial in the xi with coefficients in iB .A/, i.e., if the homomorphism of A-algebras AŒX1;:::;Xn B acting as iB on A and sending Xi to xi is surjective. We ! then write B .iB A/Œx1;:::;xn. D A ring homomorphism A B is of finite type, and B is a finitely generated A-algebra, ! if B is generated by a finite set of elements as an A-algebra. A ring homomorphism A B is finite, and B is a finite3 A-algebra, if B is finitely ! generated as an A-module. If A B and B C are finite ring homomorphisms, then so ! ! also is their composite A C . ! Let k be a field, and let A be a k-algebra. When 1A 0, the map k A is injective, ¤ ! and we can identify k with its image, i.e., we can regard k as a subring of A. When 1A 0, D the ring A is the zero ring 0 . f g Let AŒX be the ring of polynomials in the symbol X with coefficients in A. If A is an integral domain, then deg.fg/ deg.f / deg.g/, and so AŒX is also an integral domain; D C moreover, AŒX A . D Let A be an algebra over a field k. If A is an integral domain and finite as a k-algebra, then it is a field because, for each nonzero a A, the k-linear map x ax A A is 2 7! W ! injective, and hence is surjective, which shows that a has an inverse. If A is an integral domain and each element of A is algebraic over k, then for each a A, kŒa is an integral 2 domain finite over k, and hence contains an inverse of a; again A is a field. PRODUCTS AND IDEMPOTENTS An element e of a ring A is idempotent if e2 e. For example, 0 and 1 are both idempo- D tents — they are called the trivial idempotents. Idempotents e1;:::;en are orthogonal if ei ej 0 ej ei for i j . Any sum of orthogonal idempotents is again idempotent. A set D D ¤ 3This is Bourbaki’s terminology. Finite homomorphisms of rings correspond to finite maps of varieties and schemes. Some other authors say “module-finite”. 2 IDEALS 3 e1;:::;en of orthogonal idempotents is complete if e1 en 1. Any set of orthogo- f g CC D nal idempotents e1;:::;en can be made into a complete set of orthogonal idempotents by f g adding the idempotent e 1 .e1 en/. D C C If A A1 An (direct product of rings), then the elements D i ei .0;:::;1;:::;0/; 1 i n; D Ä Ä form a complete set of orthogonal idempotents in A. Conversely, if e1;:::;en is a com- 4 f g plete set of orthogonal idempotents in A, then Aei becomes a ring with the addition and multiplication induced by that of A, and A Ae1 Aen. ' 2 Ideals Let A be a ring. An ideal a in A is a subset such that a is a subgroup of A regarded as a group under addition; ˘ a a, r A ra a: ˘ 2 2 ) 2 The ideal generated by a subset S of A is the intersection of all ideals a containing S — it is easy to verify that this is in fact an ideal, and that it consists of all finite sums of the P form ri si with ri A, si S. The ideal generated by the empty set is the zero ideal 0 . 2 2 f g When S s1;s2;::: , we write .s1;s2;:::/ for the ideal it generates. D f g An ideal is principal if it is generated by a single element. Such an ideal .a/ is proper if and only a is not a unit. Thus a ring A is a field if and only if 1A 0 and A contains no ¤ nonzero proper ideals. Let a and b be ideals in A. The set a b a a; b b is an ideal, denoted a b. f C j 2 2 g C The ideal generated by ab a a; b b is denoted by ab. Clearly ab consists of all finite P f j 2 2 g sums ai bi with ai a and bi b, and if a .a1;:::;am/ and b .b1;:::;bn/, then 2 2 D D ab .a1b1;:::;ai bj ;:::;ambn/. Note that ab aA a and ab Ab b, and so D D D ab a b: (1) \ The kernel of a homomorphism A B is an ideal in A. Conversely, for any ideal a in ! a ring A, the set of cosets of a in A forms a ring A=a, and a a a is a homomorphism 7! C ' A A=a whose kernel is a. There is a one-to-one correspondence W ! b '.b/ ideals of A containing a 7! ideals of A=a : (2) 1 f g '! .b/ b f g [ For any ideal b of A, ' 1'.b/ a b. D C The ideals of A B are all of the form a b with a and b ideals in A and B. To see this, note that if c is an ideal in A B and .a;b/ c, then .a;0/ .1;0/.a;b/ c and 2 D 2 .0;b/ .0;1/.a;b/ c. Therefore, c a b with D 2 D a a .a;0/ c ; b b .0;b/ c : D f j 2 g D f j 2 g An ideal p in A is prime if p A and ab p a p or b p. Thus p is prime if and ¤ 2 ) 2 2 only if the quotient ring A=p is nonzero and has the property that ab 0; b 0 a 0; D ¤ ) D 4 But Aei is not a subring of A if n 1 because its identity element is ei 1 : However, the map a ¤ ¤ A 7! aei A Aei realizes Aei as a quotient of A. W ! 2 IDEALS 4 i.e., A=p is an integral domain. Note that if p is prime and a1 an p, then either a1 p 2 2 or a2 an p; if the latter, then either a2 p or a3 an p; continuing in this fashion, 2 2 2 we find that at least one of the ai p. 2 An ideal m in A is maximal if it is a maximal element of the set of proper ideals in A.