# 09/02/2020: Commutative Algebra (M. F. Atiyah, I. G. MacDonald) /home/zack/Dropbox/Library/M. F. Atiyah, I. G. MacDonald/Commutative Algebra (756)/Commutative Algebra - M. F. Atiyah, I. G. MacDonald.pdf Last Annotation: 09/02/2020 ## Highlights - Proposition 1\.1\. There is a one-to-one order-preserving, correspondence between the ideals b of A which contain a, and the ideals b of A/a, given by b = cp-l\(b\)\. (M. F. Atiyah, I. G. MacDonald 11) - Proposition 1\.2\. Let A be a ring '# o\. Then the following are equivalent: i\) A is a field; ii\) the only ideals in A are 0 and \(1\); iii\) every homomorphism of A into a non-zero ring B is injective\. (M. F. Atiyah, I. G. MacDonald 12) - Proof i\) => ii\)\. Let a '# 0 be an ideal in A\. Then a contains a non-zero element x; x is a unit, hence a ;;2 \(x\) = \(1\), hence a = \(1\)\. ii\) => iii\)\. Let 4>: A --+ B\. be a ring homomorphism\. Then Ker \(4\)\) is an ideal '# \(1\) in A, hence Ker \(4\)\) = 0, hence 4> is injective\. iii\) => i\)\. Let x be an element of A which is not a unit\. Then \(x\) '# \(1\), hence B = A/\(x\) is not the zero ring\. Let 4>: A -?B be the natural homomorphism of A onto B, with kernel \(x\)\. By hypothesis, 4> is injective, hence \(x\) = 0, hence x = O\. • (M. F. Atiyah, I. G. MacDonald 12) - Theorem 1\.3\. Every ring A '# 0 has at least one maximal ideal\. \(Remember that "ring" means commutative ring with 1\.\) (M. F. Atiyah, I. G. MacDonald 12) - Corollary 1\.5\. Every non-unit of A is contained in a maximal ideal\. (M. F. Atiyah, I. G. MacDonald 13) - i\) Let A be a ring and m :f\. \(1\) an ideal of A such that every x E A m is a unit in A\. Then A is a local ring and m its maximal ideal\. (M. F. Atiyah, I. G. MacDonald 13) - 1\) A = k[Xb ' \. " x n ], k a field\. Let f E A be an irreducible polynomial\. By unique factorization, the ideal \(f\) is prime\. (M. F. Atiyah, I. G. MacDonald 13) - The ideal m of all polynomials in A = k[Xl, ' , " x n ] with zero constant term is maximal (M. F. Atiyah, I. G. MacDonald 13) - 2\) There exist rings with exactly one maximal ideal, for example fields\. A ring A with exactly one maximal ideal m is called a local ring\. The field k = A/m is called the residue field of A\. (M. F. Atiyah, I. G. MacDonald 13) - Proposition 1\.7\. The set 91 of all nilpotent elements in a ring A is an ideal, and A/91 has no nilpotent element '# O\. (M. F. Atiyah, I. G. MacDonald 14) - Proposition 1\.8\. The nilradical of A is the intersection of all the prime ideals ofA\. (M. F. Atiyah, I. G. MacDonald 14) - fm E P + \(x\), rEp + \(y\) for some m, n\. It follows thatfm+n E p + \(xy\), hence the ideal p + \(xy\) is not in ~ and therefore xy ¢ p\. Hence we have a prime ideal p such thatf ¢ p, so that f¢ 91'\. • (M. F. Atiyah, I. G. MacDonald 14) - The product of two ideals a, 0 in A is the ideal ao generated by all products xy, where x E a and YEO\. It is the set of all finite sums 2: XIYI where each XI E a and each YI EO (M. F. Atiyah, I. G. MacDonald 15) - an \(n > 0\) is the ideal generated by all products XiX:!· •• Xn in which each factor XI belongs to a\. (M. F. Atiyah, I. G. MacDonald 15) - 2\) A = k[Xl, ••\. , x n ], a = \(Xl, \. \.\. , Xn\) = ideal generated by Xl>"" X n• Then am is the set of all polynomials with no terms of degree < m\. (M. F. Atiyah, I. G. MacDonald 15) - Two ideals a, 0 are said to be coprime \(or comaximal\) if a + b = \(I\)\. Thus for coprime ideals we have a \("\) b = ao (M. F. Atiyah, I. G. MacDonald 16) - Clearly two ideals a, b are coprime if and only if there exist x E a and YEO such that x + Y = 1\. (M. F. Atiyah, I. G. MacDonald 16) - Proof i\) by induction on n\. The case n = 2 is dealt with above\. Suppose n > 2 and the result true for al> \.\.\. , an l> and let 0 = n~;l a j = n~\.;1 a j • Since a j + an = \(1\) \(1 \.;;:; i \.;;:; n 1\) we have equations Xj + Yj = 1 \(Xj E aj, Yj E an\) and therefore n n-1 Xj = n n-1 \(1 Yj\) == 1 \(mod an\)\. j= 1 j =1 Hence an + 0 = \(I\) and so (M. F. Atiyah, I. G. MacDonald 16) - The union a u 0 of ideals is not in general an ideal\. (M. F. Atiyah, I. G. MacDonald 16) - Their direct product n A =nAj i= 1 is the set of all sequences x = \(Xl>' \.\. , x n \) with Xj E Aj \(1 \.;;:; i \.;;:; n\) and componentwise addition and m\lltiplication\. (M. F. Atiyah, I. G. MacDonald 16) - If a, 0 are ideals in a rIng A, their Ideal quotient is \(a:o\) = {xEA:xo S; a} (M. F. Atiyah, I. G. MacDonald 17) - Example\. If A = Z, a = \(m\), 0 = \(n\), where say m = Dp pltp, n = Dp pVP, then \(a:o\) = \(q\)whereq = DppYp and yp = max \(J\.Lp lip, 0\) = J\.Lp min \(J\.Lp, lip\)\. Hence q = m/\(m, n\), where \(m, n\) is the h\.c\.f\. of m and n\. (M. F. Atiyah, I. G. MacDonald 17) - If a is any ideal of A, the radical of a is rea\) = {x E A :x n E a for some n > O}\. (M. F. Atiyah, I. G. MacDonald 17) - Exercise 1\.13\. i\) rea\) ~ a ii\) r\(r\(a\)\) = rea\) iii\) r\(ab\) = rea n b\) = rea\) n reb\) iv\) rea\) = \(1\) <0> a = {I\) v\) rea + b\) = r\(r\(a\) + reb»~ vi\) if p is prime, r\(pn\) = pior all n > O\. (M. F. Atiyah, I. G. MacDonald 18) - Proposition 1\.14\. The radical of an ideal a is the intersection of the prime ideals which contain a\. (M. F. Atiyah, I. G. MacDonald 18) - Proposition 1\.15\. D = set of zero-divisors of A = UX,\.o r\(Ann \(x»\. (M. F. Atiyah, I. G. MacDonald 18) - Proposition 1\.16\. Let a, b be ideals in a ring A such that r\(a\), reb\) are coprime\. Then a, b are coprime\. (M. F. Atiyah, I. G. MacDonald 18) - We define the extension a e of a to be the ideal Bf\(a\) generated by f\(a\) in B: explicitly, a e is the set of all sums 2: yJ\(Xi\) where Xi E a, Yi E B\. (M. F. Atiyah, I. G. MacDonald 18) - If a is an ideal in A, the setf\(a\) is not necessarily an ideal in B (M. F. Atiyah, I. G. MacDonald 18) - Proposition 1\.17\. i\) a s; aec , b ;:> bC' ; ii\) b C = b C' c , a e = aece ; iii\) If C is the set of contracted ideals in A and if E is the set of extended ideals in B, then C = {a/ aee = a}, E = {b / bee = b}, and a f-+ ae is a bijective map of C onto E, whose inverse is b f-+ be\. (M. F. Atiyah, I. G. MacDonald 19) - Proof i\) is trivial, and ii\) follows from i\)\. (M. F. Atiyah, I. G. MacDonald 19) - i\) \(2\)" = \(1 + i\)2\), the square of a prime ideal in Z[i]; ii\) If p == 1 \(mod 4\) then \(p\)e is the product of two distinct prime ideals \(for example, \(5\)e = \(2 + i\)\(2 i\)\); iii\) If p == 3 \(mod 4\) then \(p\)e is prime in Z[i]\. (M. F. Atiyah, I. G. MacDonald 19) - A = k[x] where k is a field; an A-module is a k-vector space with a linear transformation\. (M. F. Atiyah, I. G. MacDonald 26) - Homomorphisms u: M' ~ M and v: N ~ N" induce mappings u: Hom \(M, N\) ~ Hom \(M', N\) and v: Hom \(M, N\) ~ Hom \(M, N"\) defined as follows: u\(f\) =f 0 u, v\(f\) = v 0 f (M. F. Atiyah, I. G. MacDonald 27) - For any module M there is a natural isomorphism Hom \(A, M\) ~ M: any A-module homomorphism f: A ~ M is uniquely determined by f{l\), which can be any element of M\. (M. F. Atiyah, I. G. MacDonald 27) - A-module homomorphism \(or is A-linear\) if f\(x + y\) = f\(x\) + fey\) f\(ax\) = aI\(x\) (M. F. Atiyah, I. G. MacDonald 27) - If M' is a submodule of M such that M' ~ Ker \(f\), thenfgives rise to a homomorphism]: M/M' -+ N, defined as follows: if x E M/M' is the image of x EM; then]\(x\) = f\(x\)\. The kernel of]is Ker \(f\)/M'\. The homomorphism] is said to be induced by f In particular, taking M' = Ker \(I\), we have an isomorphism of A-modules M/Ker \(f\) ~ 1m \(I\)\. (M. F. Atiyah, I. G. MacDonald 28) - Proposition 2\.1\. i\) If L :2 M :2 N are A-modules, then \(L/ N\)/\(M/ N\) ~ L/ M\. ii\) If Mh M2 are submodules of M, then \(Ml + M 2 \)/M1 ~ M 2/\(M 1 \('\) M2\)' (M. F. Atiyah, I. G. MacDonald 28) - The composite homomorphism M2 -+ Ml + M2 -+ \(Ml + M 2\)/M1 is surjective, and its kernel is Ml \('\) M 2 ; hence \(ii\)\. (M. F. Atiyah, I. G. MacDonald 28) - Define 0: L/ N -+ L/ M by O\(x + N\) = x + M\. (M. F. Atiyah, I. G. MacDonald 28) - We cannot in general define the product of two submodules, but we can define the product aM, where a is an ideal and M an A-module (M. F. Atiyah, I. G. MacDonald 28) - The cokernel off is Coker \(f\) = N/Im \(J\) (M. F. Atiyah, I. G. MacDonald 28) - we define \(N:P\) to be the set of all a E A such that aP ~ N; it is an ideal of A (M. F. Atiyah, I. G. MacDonald 28) - \(0: M\) is the set of all a E A such that aM = 0; this ideal is called the annihilator of M and is also denoted by Ann \(M\)\. (M. F. Atiyah, I. G. MacDonald 28) - An A-module is faithful if Ann \(M\)=O (M. F. Atiyah, I. G. MacDonald 29) - we can define their direct sum EE\eI M j ; its elements are families \(Xi\)ieI such that Xj E Mi for each i E I and almost all XI are 0\. If we drop the restriction on the number of non-zero x's we have the direct product OjeI Mi (M. F. Atiyah, I. G. MacDonald 29) - Proposition 2\.4\. Let M be afinitely generated A-module, let a be an ideal of A, and let ep be an A-module endomorphism of M such that ep\(M\) S; a M\. Then ep satisfies an equation of the form where the aj are in a\. (M. F. Atiyah, I. G. MacDonald 30) - Proposition 2\.3\. M is a finitely generated A-module ~ M is isomorphic to a quotient of An for some integer n > O\. (M. F. Atiyah, I. G. MacDonald 30) - Proof (M. F. Atiyah, I. G. MacDonald 30) - Corollary 2\.5\. Let M be a finitely generated A-module and let a be an ideal of A such that aM = M\. Then there exists X == l\(mod a\) such that xM = O\. (M. F. Atiyah, I. G. MacDonald 30) - Proposition 2\.6\. \(Nakayama's lemma\)\. Let M be a finitely generated A-module and a an ideal of A contained in the Jacobson radical 9t of A\. Then aM = M implies M = O\. (M. F. Atiyah, I. G. MacDonald 30) - MlmM is annihilated by m, hence is naturally an Aim-module, (M. F. Atiyah, I. G. MacDonald 31) - Proposition 2\.8\. Let x, \(1 ~ i ~ n\) be elements of M whose images in MlmM form a basis of this vector space\. Then the x, generate M\. (M. F. Atiyah, I. G. MacDonald 31) - o --+ M' \.4\. M is exact <:> f is injective; (M. F. Atiyah, I. G. MacDonald 31) - M ~ M" --+ 0 is exact <:> g is surjective; (M. F. Atiyah, I. G. MacDonald 31) - 0--+ M' \.4\. M 4 M" --+ 0 is exact <:> f is injective, g is surjective and g induces an isomorphism of Coker if\) = Mlf\(M'\) onto M"\. (M. F. Atiyah, I. G. MacDonald 31) - <=> for all A-modules N, the sequence 0--+ Hom \(M", N\) ~ Hom \(M, N\) ~ Hom \(M', N\) (M. F. Atiyah, I. G. MacDonald 31) - <=> for all A~modules M, the sequence \(5\) Then the sequence \(5\) is 0--+ Hom \(M, N'\) ~ Hom \(M, N\) ~ Hom \(M, N"\) (M. F. Atiyah, I. G. MacDonald 32) - The boundary homomorphism d is defined as follows (M. F. Atiyah, I. G. MacDonald 32) - The function A is additive if, for each short exact sequence \(3\) in which all the terms belong to C, we have A\(M'\) '\\(M\) + A\(M"\) = O\. (M. F. Atiyah, I. G. MacDonald 32) - Proposition 2\.11\. Let 0 ~ Mo ~ Ml~"'~ Mn ~ 0 be an exact sequence of A-modules in which all the modules M, and the kernels of all the homomorphisms belong to C\. Then for any additive function A on C we have n L \(-l\)lA\(M,\) = O\. (M. F. Atiyah, I. G. MacDonald 33) - A-bilinear if for each x EM the mappingy 1--+ f\(x, y\) of N into P is A-linear, and for each yEN the mapping x 1--+ f\(x, y\) of Minto P is A-linear\. (M. F. Atiyah, I. G. MacDonald 33) - construct an A-module T, called the tensor product (M. F. Atiyah, I. G. MacDonald 33) - Proof (M. F. Atiyah, I. G. MacDonald 33) - In particular, if M and N are finitely generated, so is M 181 N\. (M. F. Atiyah, I. G. MacDonald 34) - these three conditions determine the ring S -1 A up to isomorphism\. (M. F. Atiyah, I. G. MacDonald 46) - s E S => g\(s\) is a unit in B; (M. F. Atiyah, I. G. MacDonald 46) - g: A -+ B (M. F. Atiyah, I. G. MacDonald 46) - Corollary 3\.2\. (M. F. Atiyah, I. G. MacDonald 46) - g\(a\) = 0 => as = ofor some SE S; (M. F. Atiyah, I. G. MacDonald 46) - The process of passing from A to All is called localization at \.p\. (M. F. Atiyah, I. G. MacDonald 47) - Hence m is the only maximal ideal in All; in other words, All is a local ring\. (M. F. Atiyah, I. G. MacDonald 47) - 8 = A \.p is multiplicatively closed (M. F. Atiyah, I. G. MacDonald 47) - \(in fact A P is multiplicatively closed ~ \.p is prime\)\. (M. F. Atiyah, I. G. MacDonald 47) - Every element of B is of the form g\(a\)g\(s\) -1 (M. F. Atiyah, I. G. MacDonald 47) - hen there is a unique isomorphism h: 81 A ~ B such that g = h 0 f (M. F. Atiyah, I. G. MacDonald 47) - We write All for 8 -lA (M. F. Atiyah, I. G. MacDonald 47) - 81 A ess of passing from A to ° is the zero ring ~ E 8 (M. F. Atiyah, I. G. MacDonald 47) - 8 = {rn},,~o\. We write A, for 81A (M. F. Atiyah, I. G. MacDonald 47) - Let Kbe a field\. A discrete valuation on Kis a mapping v of K\* onto Z \(where K\* = K {O} is the multiplicative group of K\) such that 1\) v\(xy\) = vex\) + v\(y\), i\.e\., v is a homomorphism; 2\) vex + y\) ~ min\(v\(x\),v\(y»\)\. (M. F. Atiyah, I. G. MacDonald 103) - An integral domain A is a discrete valuation ring if there is a discrete valuation v of its field of fractions K such that A is the valuation ring of v\. (M. F. Atiyah, I. G. MacDonald 103) - The set consisting of 0 and all x E K\* such that vex\) ~ 0 is a ring, called the valuation ring of v\. It is a valuation ring of the field K (M. F. Atiyah, I. G. MacDonald 103) - A ring satisfying the conditions of \(9\.3\) is called a Dedekind domain\. (M. F. Atiyah, I. G. MacDonald 104) - i\) A is integrally closed; ii\) Every primary ideal in A is a prime power; iii\) Every local ring All \(\.\):> of 0\) is a discrete valuation ring\. (M. F. Atiyah, I. G. MacDonald 104) - Theorem 9\.3\. Let A be a Noetherian domain of dimension one\. Then the following are equivalent: (M. F. Atiyah, I. G. MacDonald 104) - Corollary 9\.4\. In a Dedekind domain every non-zero ideal has a unique factorization as a product of prime ideals\. (M. F. Atiyah, I. G. MacDonald 104) - Theorem 9\.5\. The ring of integers in an algebraic number field K is a Dedekind domain\. (M. F. Atiyah, I. G. MacDonald 105) - An A-submodule M of K °in A\. In particular, the (M. F. Atiyah, I. G. MacDonald 105) - Let A be an integral domain, K its field of fractions\. is a fractional ideal of A if xM s;; A for some x "# ons\. An A "# ° in A (M. F. Atiyah, I. G. MacDonald 105) - Any element U E K generates a fractional ideal, denoted by \(u\) or Au, and called principal\. If M is a fractional ideal, the set of all x E K such that x M s;; A is denoted by \(A: M\)\. (M. F. Atiyah, I. G. MacDonald 105)