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A Primer of Commutative Algebra

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A Primer of Commutative Algebra A Primer of Commutative Algebra James S. Milne April 29, 2012, v2.23 Abstract These notes prove the basic theorems in commutative algebra required for algebraic geometry and algebraic groups. They assume only a knowledge of the algebra usually taught in advanced undergraduate or first-year graduate courses. Contents 1 Rings and algebras......................2 2 Ideals...........................3 3 Noetherian rings.......................8 4 Unique factorization...................... 13 5 Integrality......................... 16 6 Rings of fractions....................... 21 7 Direct limits......................... 25 8 Tensor Products....................... 26 9 Flatness.......................... 30 10 Finitely generated projective modules............... 35 11 The Hilbert Nullstellensatz................... 42 12 The max spectrum of a ring................... 45 13 Dimension theory for finitely generated k-algebras........... 52 14 Primary decompositions.................... 56 15 Artinian rings........................ 59 16 Dimension theory for noetherian rings............... 61 17 Regular local rings...................... 65 18 Connections with geometry................... 67 References........................... 73 Index............................. 75 c 2009, 2010, 2011, 2012 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Available at www.jmilne.org/math/. 1 1 RINGS AND ALGEBRAS 2 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 ring 2 A form a group, which we denote by A. Throughout “ring” means “commutative ring”. 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: Florian Herzig, Chun Yin Hui, Keenan Kidwell, Leon Lampret, Andrew McLennan, Shu Otsuka, Bhupendra Nath Tiwari. 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 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. 2This notation differs from that of Bourbaki, who writes A for the multiplicative monoid A 0 and A X f g for the group of units. We shall rarely need the former, and is overused. 3This is Bourbaki’s terminology (AC V 1, 1). Finite homomorphisms of rings correspond to finite maps of varieties and schemes. Some other authors say “module-finite”. 2 IDEALS 3 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 idempotents D — they are called the trivial idempotents. Idempotents e1;:::;en are orthogonal if ei ej 0 D for i j . Any sum of orthogonal idempotents is again idempotent. A set e1;:::;en of ¤ f g orthogonal idempotents is complete if e1 en 1. Any set of orthogonal idempotents C C D e1;:::;en can be made into a complete set of orthogonal idempotents by adding the f g 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 form P ri si with ri A, si S. The ideal generated by the empty set is the zero ideal 0 . When 2 2 f g S a;b;::: , we write .a;b;:::/ 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. The f C j 2 2 g C 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 every 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 [ 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 For every 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 i.e., A=p is an integral domain. Note that if p is prime and a1 an p, then at least one 2 of the ai p (because either a1 p or a2 an p; if the latter, then either a2 p or 2 2 2 2 a3 an p; etc.).
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