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A Primer of Commutative Algebra

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A Primer of Commutative Algebra James S. Milne May 2, 2013, v3.00 Abstract These notes collect the basic results in commutative algebra used in the rest of my notes and books. Contents 1 Rings and algebras......................3 2 Ideals...........................4 3 Noetherian rings.......................9 4 Unique factorization...................... 14 5 Rings of fractions....................... 17 6 Integrality......................... 23 7 Artinian rings........................ 32 8 Direct and inverse limits.................... 34 9 Tensor Products....................... 37 10 Flatness.......................... 41 11 Finitely generated projective modules............... 46 12 Zariski’s lemma and the Hilbert Nullstellensatz............ 53 13 The spectrum of a ring..................... 57 14 Jacobson rings and max spectra.................. 62 15 Quasi-finite algebras and Zariski’s main theorem............ 66 16 Dimension theory for finitely generated k-algebras........... 73 17 Primary decompositions.................... 76 18 Dedekind domains...................... 80 19 Dimension theory for noetherian rings............... 85 20 Regular local rings...................... 89 21 Completions......................... 91 References........................... 93 Index............................. 94 c 2009, 2010, 2011, 2012, 2013 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 CONTENTS 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). Prerequisites A knowledge of the algebra usually taught in advanced undergraduate or first-year graduate courses. References A reference to monnnn is to question nnnn on mathoverflow.net. 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. 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. 1 RINGS AND ALGEBRAS 3 1 Rings and algebras A ring is an integral domain if it is not the zero ring and if ab 0 in the ring implies that D a 0 or b 0. D D 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. ! When A B and x1;:::;xn B, we let AŒx1;:::;xn denote the A-subalgebra of B 2 generated by the xi . 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, i.e, if B is a quotient of a polynomial ring AŒX1;:::;Xn. An A-algebra B is finitely presented if it is the quotient of a polynomial ring kŒX1;:::;Xn by a finitely generated ideal. 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 integral domain and an algebra over a field k. If A is finite over k (more generally, if every element of A is algebraic over k), then A is a field. To see this, let a be a nonzero element of A. Because A is an integral domain, the k-linear map x ax A A 7! W ! is injective, and hence is surjective if A is finite, which shows that a has an inverse. More generally, if a is algebraic over k, then kŒa is 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 . Every sum of orthogonal idempotents is again idempotent. A set e1;:::;en of ¤ f g orthogonal idempotents is complete if e1 en 1. Every set of orthogonal idempotents CC 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 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 4 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 H) 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 if 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 [ For an 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.
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