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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 March 23, 2020, v4.03 Abstract These notes collect the basic results in commutative algebra used in the rest of my notes and books. Although most of the material is standard, the notes include a few results, for example, the affine version of Zariski’s main theorem, that are difficult to find in books. (Minor fixes from v4.02.) Contents 1 Rings and algebras......................3 2 Ideals...........................3 3 Noetherian rings.......................9 4 Unique factorization...................... 14 5 Rings of fractions....................... 18 6 Integral dependence...................... 25 7 The going-up and going-down theorems.............. 30 8 Noether’s normalization theorem................. 34 9 Direct and inverse limits.................... 35 10 Tensor Products....................... 38 11 Flatness.......................... 43 12 Finitely generated projective modules............... 51 13 Zariski’s lemma and the Hilbert Nullstellensatz............ 59 14 The spectrum of a ring..................... 63 15 Jacobson rings and max spectra.................. 70 16 Artinian rings........................ 75 17 Quasi-finite algebras and Zariski’s main theorem............ 77 18 Dimension theory for finitely generated k-algebras........... 84 19 Primary decompositions.................... 89 20 Dedekind domains...................... 93 21 Dimension theory for noetherian rings............... 99 22 Regular local rings...................... 103 23 Flatness and fibres...................... 105 24 Completions......................... 108 A Solutions to the exercises..................... 110 References........................... 111 Index............................. 112 c 2009, 2012, 2014, 2017, 2020 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;::: . Throughout, k is a field and kal is an algebraic D f g closure 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. Historical Notes Sometime I’ll add these. For the moment, I refer the reader to Bourbaki AC, Historical Note; Matsumura 1986, Introduction; Nagata 1962, Appendix A2. Acknowledgements I thank the following for providing corrections and comments for earlier versions of these notes: Amir Dzambiˇ c,´ Qi Ge, Florian Herzig, Chun Yin Hui, Keenan Kidwell, Leon Lampret, Junyu Lu, Andrew McLennan, Shu Otsuka, Dmitri Panov, Roy Smith, Moss Sweedler, Bhupendra Nath Tiwari, Wei Xu. 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/. This means that 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. This means that 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 AŒX1;:::;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 A a k-algebra. If 1A 0, then 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. If 1A 0, then the D 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 both an integral domain and an algebra over a field k. If A is finite over k, then it 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 is injective, and hence is surjective if A is finite, which 7! W ! shows that a has an inverse. More generally, if every element a of A is algebraic over k, then kŒa is finite over k, and hence contains an inverse of a; again A is a field. An A-module M is faithful if aM 0, a A, implies a 0. D 2 D Exercises EXERCISE 1.1. Let k be an infinite field, and let f be a nonzero polynomial in kŒX1;:::;Xn. Show that there exist a1;:::;an k such that f .a1;:::;an/ 0. 2 ¤ 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 3This is Bourbaki’s terminology (AC V 1, 1). Finite homomorphisms of rings correspond to finite maps of varieties and schemes. Some authors say “module-finite”. 2 IDEALS 4 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 the only proper ¤ ideal in A is .0/. 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 (regarded as an additive group) forms a ring A=a, and a a a is a homomorphism ' A A=a whose kernel is a. There is a one-to-one 7! C W ! correspondence 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. 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 a 0 or b 0; D H) D D i.e., A=p is an integral domain.
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