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Commutative Algebra Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . 146 8.4 Finite Morphisms and the Fibre Theorem . 147 8.5 Projective Varieties . 150 8.6 Nonsingular Varieties . 153 8.7 Abstract Curves . 156 8.8 The Spectrum of a Ring . 159 ii 1 Preliminaries 1 Preliminaries The contents of this document come from an Algebra IV course taught by Dr. Peter Abra- menko at the University of Virginia in Spring 2016. The main theory of commutative rings are covered, along with applications to algebraic geometry towards the end. Topics include: Integral dependence Noetherian, Artinian and Dedekind rings Valuations and discrete valuation rings Dimension theory Affine and projective varieties Affine algebras Hilbert's Nullstellensatz Morphisms of varieties The commutative algebra may be found in Atiyah and MacDonald's Introduction to Commu- tative Algebra, while the algebraic geometry generally follows the first chapter of Hartshorne's Algebraic Geometry. Several topics in these notes also come from a course on commutative algebra taught by Dr. Craig Huneke at UVA in Spring 2019. The main topics I've included from this course are: associated primes, Cohen's structure theory for complete local rings, some dimension theory and notions of singularity in rings. We will adhere to several conventions throughout these notes: All rings are commutative with unity 1. A will always denote such a ring. All subrings contain 1. In particular, proper ideals are never subrings. For any ring homomorphism f : A ! B, in addition to the homomorphism axioms, we stipulate that f(1A) = 1B. In any integral domain, we assume 1 6= 0. 1.1 Radicals There are three important types of radicals that we may define in a commutative ring A. The first two are defined below. Definition. The nil radical of A is defined as the intersection of all prime ideals of A, written \ N(A) = P: prime ideals P ⊂A We say A is reduced if N(A) = 0. 1 1.1 Radicals 1 Preliminaries Definition. The Jacobson radical of A is the intersection of all maximal ideals of A, written \ J(A) = M: maximal ideals M⊂A Remark. Clearly N(A) ⊆ J(A). Lemma 1.1.1. For any x 2 A, x 2 J(A) if and only if 1 − xy is a unit in A for all y 2 A. Proof. ( =) ) Suppose A(1 − xy) 6= A for some y 2 A. Then A(1 − xy) ⊂ M for a maximal ideal M. By definition, x 2 J(A) ⊆ M and since J(A) is an ideal, xy 2 M as well. This means 1 = xy + (1 − xy) 2 M, a contradiction. Therefore A(1 − xy) = A so there is some a 2 A such that a(1 − xy) = 1; that is, 1 − xy is a unit. ( ) = ) Conversely, suppose x 62 M for some maximal ideal M ⊂ A. Then M + Ax = A by maximality of M, so b + xy = 1 for some b 2 M, or 1 − xy = b 2 M, meaning 1 − xy cannot be a unit in A. Proposition 1.1.2. N(A) = fx 2 A j xn = 0 for some n ≥ 1g. That is, the nil radical consists of all nilpotent elements in A. Proof. (⊇) Suppose xn = 0. Then for any prime ideal P , xn = 0 2 P which implies x 2 P . Thus x 2 N(A). (⊆) If x 2 A is not nilpotent, we will produce a prime ideal not containing x. Let Σ denote the set of ideals I ⊂ A such that xn 62 I for any n ≥ 1. Notice that 0 is an element of Σ since we are assuming x is not nilpotent; thus Σ is nonempty. Therefore we may apply Zorn's Lemma to choose an ideal P 2 Σ which is maximal among this collection of ideals. We claim that P is prime. Clearly P 6= A since A is not an element of Σ. Suppose y; z 2 A but y; z 62 P . Since P is maximal in Σ, the ideals P + Ay and P + Az are both strictly larger ideals than P . Thus P + Ay; P + Az 62 Σ so there exist m; n 2 N such that xm 2 P + Ay and xn 2 P + Az. Then xm+n 2 (P + Ay)(P + Az) ⊆ P + Ayz, but xm+n 62 P by assumption, so we must have yz 62 P . This proves P is prime. n Definition. For an ideal I ⊂ A, the radical of Ipis r(I) = fx 2 A j x 2 I for some n ≥ 1g. Alternate notations for the radical ideal include I and rad(I). Corollary 1.1.3. For any ideal I ⊂ A, r(I) equals the intersection of all prime ideals of A containing I. In particular, r(I) is an ideal. Proof. Consider the quotient ring A¯ = A=I. For x 2 A we have the following equivalent statements: n x 2 r(I) () there exists an n 2 N such that x 2 I n () there exists an n 2 N such that (x + I) = 0 2 A¯ () x + I is nilpotent, i.e. x + I ⊂ N(A¯) () x + I is contained in every prime of A¯ () x lies in every prime ideal of A containing I: \ Therefore r(I) = P as desired. prime ideals P ⊇I 2 1.1 Radicals 1 Preliminaries Remark. By Corollary 1.1.3, N(A) = r(0). Lemma 1.1.4. Suppose I;J ⊂ A are ideals. Then (a) I ⊆ r(I). (b) I = r(I) if and only if A=I is reduced. In particular, every prime ideal is radical. (c) r(r(I)) = r(I). (d) r(Im) = r(I) for all m 2 N. (e) r(IJ) ⊇ r(I)r(J). (f) r(I + J) = r(r(I) + r(J)). (g) r(I) = A if and only if I = A. Proof. (a) is clear from the definition. (b) By Corollary 1.1.3, r(I) equals the intersection of all primes lying over I. Let π : A ! A=I be the quotient map. Then N(A=I) = 0 () π(r(I)) = 0 () r(I) ⊆ ker π = I. By part (a), we always have I ⊆ r(I) so this shows that N(A=I) = 0 () r(I) = I. (c) By part (a), it suffices to show r(r(I)) ⊆ r(I). If x 2 r(r(I)) then xn 2 r(I) for some n ≥ 1, but then (xn)m = xnm 2 I for some m ≥ 1. Thus x 2 r(I). (d) If x 2 r(Im) then xn 2 Im for some n ≥ 1, but Im ⊆ I so this shows x 2 r(I). On the other hand, if x 2 r(I) then xn 2 I for some n ≥ 1. Letting k = maxfn; mg, we see that xk 2 Im. Hence r(I) = r(Im). (e) It suffices to prove this for elements of the form xy 2 r(I)r(J), where x 2 r(I) and y 2 r(J). Then xn 2 I and ym 2 J for some n; m ≥ 1. Without loss of generality, assume n ≥ m. Then (xy)n = xnyn 2 IJ so xy 2 r(IJ). (f) First take z 2 r(I + J), so that zn 2 I + J for some n ≥ 1. Then zn = x + y for x 2 I; y 2 J. Since I ⊆ r(I) and J ⊆ r(J) by part (a), we have zn = x + y 2 I + J ⊆ r(I) + r(J). Therefore z 2 r(r(I) + r(J)). On the other hand, if w 2 r(r(I) + r(J)) then wm 2 r(I) + r(J) for some m ≥ 1. This means wm = x + y for some x 2 r(I); y 2 r(J). In turn this says that xk 2 I and y` 2 J for some k; ` ≥ 1. Using the binomial formula, write k + ` wm(k+`) = (x + y)k+` = xk+` + (k + `)xk+`−1y + ::: + xk+`−qyq + ::: + yk+`: q Note that q ranges from 0 to k + `. When 0 ≤ q ≤ `, the qth term (the first term in the sum being the 0th term) is divisible by xk and so it lies in I.
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