jeffrey daniel kasik carlson: Exercises to Atiyah and Macdonald’s Introduction to Commutative Algebra FIX REFERENCES LINK EXERCISES DO CHAPTER 10 BODY OMISSION: COMPLETION FIX TENSOR HEIGHT CHECK SEQUENCE NOTATION THROUGHOUT CHECK WHICH ASTERISK SOLUTIONS ARE STATED IN THE TEXT 1 Motivation This note is intended to contain full solutions to all exercises in this venerable text,1 as well as proofs of results omitted or left to the reader. It has none of the short text’s pith or elegance, tending rather to the other extreme, citing chapter and verse from the good book and spelling out, step-by-step, things perhaps better left unsaid. It attempts to leave no “i” undotted, no “t” uncrossed, no detail unexplained. We are rather methodical in citing results when we use them, even if they’ve likely long been assimilated by the reader. Having found some solution sets online unclear at points (sometimes due to our own shortcomings, at other times due to theirs), we in this note strive to suffer from the opposite problem. Often we miss the forest for the trees. We prefer to see this not as “pedantic,” but as “thorough.” Sometimes we have included multiple proofs if we have found them, or failed attempts at proof if their failure seems instructive. The work is our own unless explicitly specified otherwise. It is hoped that the prolix and oftentimes plodding nature of these solutions will illuminate more than it conceals. 1 [?] 2 Notation Problems copied from the book and propositions are in italics, definitions emphasized (italic when surrounding text is Roman, and Roman when surrounding text is italic), and headings in bold or italics, following the book, and our solutions and occasional comments in Roman. Solutions that were later supplanted by better ones but still might potentially be worth seeing have usually been included, but as footnotes, decreasing both page count and legibility. Propositions, exercises, theorems, lemmas, and corollaries from the main body of the text will always be cited as “(n.m)” or (n.m.t)”, where n is the chapter number, m the section number, and “t” an optional Roman numeral. For example, Proposition 1.10, part ii) is cited as “(1.10.ii)”. Propositions or assertions proved in these solutions but not stated in the text are numbered with an asterisk, e.g. “Proposition 4.12*” and thereafter “(4.12*)”. Exercises from the “EXERCISES” sections that follow each chapter, on the other hand, we cite with (square) brackets as e.g. “[3.1]” and “[1.2.i]”. Displays in this document are referred back to as e.g. “Eq. 1.1” or “Seq. 2.2”. Ends of proofs for exercises go unmarked, though we will sometimes mark proofs for discrete propositions we prove in the course of doing problems or expanding upon material. Our mathematical notation follows that of the book, with a few exceptions as noted below. For strict set containment, “⊂” is supplanted by “(”, which is preferred for its lack of ambiguity; it is not to be confused with “6⊆”, which means “does not contain.” “3” is a backward “2”, and means “contains the element” (rather than “such that”). The ideal generated by a set of elements is noted by listing them between parentheses: e.g., (x, y) is generated by fx, yg and (xa)a2A is generated by fxa : a 2 Ag. Contrastingly (and disagreeing with the book), we notate sequences and ordered lists with angle brackets: hx, yi is an ordered pair and hxaia2A is a list of elements xa indexed by a set A; in particular hxnin2N is a sequence. Popular algebraic objects like N, Z, Q, R, and C will be denoted in blackboard bold instead of bold, and 0 is included in N. Fq denotes a (“the”) finite field with q elements. “a ¢ A” means that a is an ideal of the ring A. For set complement/exclusion, the symbol “n” replaces “−” on the off chance it might otherwise be confused with subtraction in cases (like topological groups) where both operations are feasible. The set of units of the ring A will be denoted by A× rather than A∗, which is assigned a different meaning on p. 107. Bourbaki’s word “quasi-compact” for the condition that every open cover has a finite subcover (not necessarily requiring the space be Hausdorff) we replace with “compact”; this usage seems to hold generally outside of algebraic geometry and most of the topologies we encounter here are not Hausdorff anyway. “ker”, “im”, and “coker” will go uncapitalized. N(A) and R(A) always denote, respectively, the nilradical and the Jacobson radical of the ring A; we just write N and R where no ambiguity is possible. The notation idM : M ! M for the identity map replaces the book’s “1” as slightly more unambiguous; 1 or 1A is instead the unity of the ring A. “Multiplicative submonoid” is preferred to the book’s “multiplicatively closed subset” as indicating that a subset of a ring contains 1 and is closed under the ring’s multiplication. “Zorn’s Lemma,” capitalized, is the proper name of a result discovered by Kazimierz Kuratowski some thirteen years earlier than by Max Zorn. For a map f: A ! B, we can replace the arrow with “ ” when f is surjective, “ ” when it is injective, “,!” when it is an inclusion, ∼ and “−!” when it is an isomorphism. A =∼ B means that there exists some isomorphism between A and B, and X ≈ Y that X and Y are homeomorphic topological spaces. [M] is the isomorphism class of M. For V a map f: A ! B, if U ⊆ A and V ⊆ B are such that f(U) ⊆ V, then f jU is the restricted and corestricted map U ! V. Very occasionally, k, l, m may be cardinals (or homomorphisms), and indices a, b, g may be ordinals. @0 is the cardinality of N. 3 Chapter 1: Rings and Ideals Theorem 1.3. Every ring A 6= 0 has at least one maximal ideal. In order to apply Zorn’s Lemma, it is necessary to prove that if haaia2A is a chain of ideals (meaning, S recall, that for all a, b 2 A we have aa ⊆ ab or ab ⊆ aa) then the union a = a2A aa is an ideal. Indeed, if a, b 2 a, then there are a, b 2 A such that and a 2 aa and b 2 ab. Without loss of generality, suppose aa ⊆ ab. Then a, b 2 ab, so since ab is an ideal we have a − b 2 ab ⊆ a. If x 2 A and a 2 a, then there is a 2 A such that a 2 aa. As aa is an ideal, xa 2 aa ⊆ a. Therefore a is an ideal. Exercise 1.12. i) a ⊆ (a : b). For each a 2 a we have ab ⊆ ab ⊆ a, so a 2 (a : b). ii) (a : b)b ⊆ a. By definition, for x 2 (a : b) we have xb ⊆ a. iii) (a : b) : c = (a : bc) = (a : c) : b. x 2 (a : b) : c () xc ⊆ (a : b) () xcb ⊆ a () x 2 (a : bc); x 2 (a : c) : b () xb ⊆ (a : c) () xbc ⊆ a () x 2 (a : bc). T T iv) i ai : b = i (ai : b). \ \ \ x 2 ai : b () xb ⊆ ai () 8i (xb ⊆ ai) () x 2 (ai : b). i i i T v) a : ∑i bi = i (a : bi). \ x 2 a : ∑ bi () a ⊇ x ∑ bi = ∑ xbi () 8i (xbi ⊆ a) () x 2 (a : bi). i i i i For an A-module M and subsets N ⊆ M and E ⊆ A, define (N : E) := fm 2 M : Em ⊆ Ng; for subsets N, P ⊆ M and E ⊆ A, define (N : P) := fa 2 A : aP ⊆ Ng. Note for future use that then ii) holds equally well for subsets a, b ⊆ M, or b ⊆ A and a ⊆ M; iii) holds for a, b ⊆ M and c ⊆ A; and iv) and v) hold for modules a, ai and modules or ideals b, bi. Exercise 1.13. −i) a ⊆ b =) r(a) ⊆ r(b). If x 2 r(a), for some n > 0 we have xn 2 a ⊆ b, so x 2 r(b). 0) r(an) = r(a) for all n > 0. an ⊆ a, so by part −i) we have r(an) ⊆ r(a). If x 2 r(a), then for some m > 0, xm 2 a. But then xmn 2 an and x 2 r(an). i) r(a) ⊇ a. For each a 2 a we have a1 2 a, so a 2 r(a). 4 Chapter 1: Rings and Ideals ii) rr(a) = r(a). x 2 rr(a) () 9n > 0 xn 2 r(a) () 9n, m > 0 (xn)m = xmn 2 a () x 2 r(a). iii) r(ab) = r(a \ b) = r(a) \ r(b). For the first equality, note (a \ b)2 ⊆ ab ⊆ a \ b, so by parts 0) and −i), r(a \ b) = r(a \ b)2 ⊆ r(ab) ⊆ r(a \ b). For the second, note that if m, n > 0 are such that xm 2 a and xn 2 b, then xmaxfm, ng 2 a \ b, and conversely. iv) r(a) = (1) () a = (1). If r(a) = (1), then 1 2 r(a), so for some n we have 1 = 1n 2 a, and then a = (1). v) r(a + b) = rr(a) + r(b). Since a, b ⊆ a + b, by part −i) we have r(a), r(b) ⊆ r(a + b), so r(a) + r(b) ⊆ r(a + b). By parts −i), and ii), we see rr(a) + r(b) ⊆ rr(a + b) = r(a + b).