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Supplementary Notes on Commutative Algebra1 Supplementary Notes on Commutative Algebra1 April 11, 2017 1These notes are meant as a supplement to the Lectures on Commutative Algebra, that are avail- able online at http://www.math.iitb.ac.in/∼srg/Lecnotes/AfsPuneLecNotes.pdf and are primarily meant for the students of MA 526 at IIT Bombay in Spring 2017. The contents in these notes are mainly drawn from parts of Chapters 2 and 3 of Combinatorial Topology and Algebra, RMS Lecture Notes Series No. 18, Ramanujan Math. Soc., 2012. Contents 1 Graded Rings and Modules 3 1.1 Graded Rings . 3 1.2 Graded Modules . 5 1.3 Primary Decomposition in Graded Modules . 7 2 Hilbert Functions 10 2.1 Hilbert Function of a graded module . 10 2.2 Hilbert-Samuel polynomial of a local ring . 12 3 Dimension Theory 14 3.1 The dimension of a local ring . 14 3.2 The Dimension of an Affine Algebra . 18 3.3 The Dimension of a Graded Ring . 20 References 23 2 Chapter 1 Graded Rings and Modules This chapter gives an exposition of some rudimentary aspects of graded rings and mod- ules. It may be noted that exercises are embedded within the text and not listed separately at the end of the chapter. 1.1 Graded Rings In this section, we shall study some basic facts about graded rings. The study extends to graded modules, which are discussed in the next section. The notions and results in the first two sections will allow us to discuss, in Section 1.3, some special properties of associated primes and primary decomposition in the graded situation. The prototype of a graded ring is the polynomial ring A[X1;:::;Xn] over a ring A. Before giving the general definition, we point out that the set of all nonnegative integers is denoted by N. Definition: A ring R is said to be N-graded, or simply graded, if it has additive subgroups Rd, for d 2 N, such that R = ⊕d2NRd and RdRe ⊆ Rd+e, for all d; e 2 N. The family fRdgd2N th is called an N-grading of R, and the subgroup Rd is called the d graded component of R. Let R = ⊕d2NRd be a graded ring. Elements of Rd are said to be homogeneous of degree d a 2 R a = P a a 2 R . Every can be uniquely written as d2N d with d d such that all except finitely many ad’s are 0; we call ad’s to be the homogeneous components of a. Given a homogeneous element b 2 R, we sometimes write deg(b) to denote its degree. An ideal of R generated by homogeneous elements is called a homogeneous ideal. Note that an ideal is homogeneous iff it contains the homogeneous components of each of its elements. If I is a homogeneous ideal of R, then R=I has the induced N-grading given by (R=I)d = Rd=(I \ Rd). A subring S of R is said to be graded if S = ⊕d2N(S \ Rd). Note that R0 is 0 0 a graded subring of R and each Rd is an R0-module. If R = ⊕d2NRd is any graded ring, then a homomorphism φ : R ! R0 is called a graded ring homomorphism, or a homomorphism 0 of graded rings, if φ(Rd) ⊆ Rd for all d 2 N; note that in this case the kernel of φ is a homogeneous ideal of R and the image of φ is a graded subring of R0. Examples: 1. Let A be a ring and I be a homogeneous ideal of A[X1;:::;Xn]. Then R = th A[X1;:::;Xn]=I is a graded ring, with its d graded component being given by Rd = 3 A[X1;:::;Xn]d=Id, where Id = I \ A[X1;:::;Xn]d. Note that in this case R as well as Rd are f: g:A-algebras, and in particular, A-modules. Graded rings of this type are usually called graded A-algebras. def d d+1 2. Let A be a ring and I be any ideal of A such that I 6= A. Then grI (A) = ⊕d2NI =I is a graded ring. It is called the associated graded ring of A w.r.t. I. (3.1) Exercise: Let R = ⊕d2NRd be a gradedp ring, and I;J be homogeneous ideals of R. Then show that I + J; IJ; I \ J; (I : J) and I are homogeneous ideals. (3.2) Exercise: Let R = ⊕d2NRd be a graded ring, and I be a homogeneous ideal of R. Show that I is prime iff I is prime in the graded sense, that is, I 6= R and a; b homogeneous elements of R and ab 2 I ) a 2 I or b 2 I: ∗ Given an ideal I of a graded ring R = ⊕d2NRd, we shall denote by I the largest homogeneous ideal contained in I. Note that I∗ is precisely the ideal generated by the homogeneous elements in I. ∗ (3.3) Lemma. If R = ⊕d2NRd is a graded ring and p is a prime ideal of R, then p is a prime ideal of R. Proof: Follows from (3.2). 2 (3.4) Lemma. Let R = ⊕d2NRd be a graded ring. Then 1 2 R0. 1 = P a a 2 R b 2 R b = P ba Proof: Write d2N d with d d. Then for any e, d2N d. Equating the degree e components, we find b = ba0. This implies that ca0 = 1 for all c 2 R. Hence 1 = a0 2 R0. 2 Given a graded ring R = ⊕d2NRd, we define R+ = ⊕d>0Rd. Note that R+ is a homoge- neous ideal of R and R=R+ ' R0. Thus if R0 is a field, then R+ is a homogeneous maximal ideal of R; moreover, R+ is also maximal among all homogeneous ideals of R other than R, and so it is the maximal homogeneous ideal of R. (3.5) Lemma. Let R = ⊕d2NRd be a graded ring and x1; : : : ; xn be any homogeneous ele- ments of positive degree in R. Then R+ = (x1; : : : ; xn) iff R = R0[x1; : : : ; xn]. In particular, R is noetherian iff R0 is noetherian and R is a f: g:R0-algebra. Proof: Suppose R+ = (x1; : : : ; xn). We show by induction on d that each x 2 Rd is in R0[x1; : : : ; xn]. The case of d = 0 is clear. Suppose d > 0. Write x = a1x1 + ··· + anxn, where a1; : : : ; an 2 R. Since x is homogeneous, we may assume, without loss of generality, that each ai is homogeneous. Then for 1 ≤ i ≤ n, we must have deg(ai) = d − deg(xi) < d, and so, by induction hypothesis, ai 2 R0[x1; : : : ; xn]. It follows that R = R0[x1; : : : ; xn]. Conversely, if R = R0[x1; : : : ; xn], then it is evident that R+ = (x1; : : : ; xn). The second as- sertion in the Lemma follows from the first one by noting that f: g: algebras over noetherian rings are noetherian. 2 (3.6) Exercise: Let R = ⊕d2NRd be a graded ring. Show that R is noetherian iff it satisfies a.c.c. on homogeneous ideals. 4 (3.7) Exercise: Let A be a ring and I = (a1; : : : ; ar). If a¯1;:::; a¯r denote the images of 2 a1; : : : ; ar mod I , then show that grI (A) = (A=I)[¯a1;:::; a¯r] ' (A=I)[X1;:::;Xr]=J, for some homogeneous ideal J of (A=I)[X1;:::;Xr]. Deduce that if (A=I) is noetherian and I is f: g:, then grI (A) is noetherian. (3.8) Graded Noether Normalisation Lemma. Let k be an infinite field and R = k[x1; : : : ; xn] be a graded k-algebra such that R0 = k and deg(xi) = m for 1 ≤ i ≤ n. Then there exist homogeneous elements θ1; : : : ; θd of degree m in R such that θ1; : : : ; θd are algebraically independent over k and R is integral over the graded subring k[θ1; : : : ; θd]. In particular, R is a finite k[θ1; : : : ; θd]-module. Proof: If k is an infinite field, then in (4.11) and (4.13) of Chapter 1, the elements θ1; : : : ; θd can be chosen to be k-linear combinations of x1; : : : ; xn. The result follows. 1.2 Graded Modules The notion of a graded ring extends readily to that of graded modules. For simplicity, we initially consider N-graded modules, but point out toward the end of this section that many of the notions and results below extend easily to Γ-graded modules where Γ is an s s arbitrary additive monoid (in particular, Γ can be Z, Z , or N , where s is any positive integer). Definition: Let R = ⊕d2NRd be a graded ring, and M be an R-module. Then M is said to be N-graded or simply, graded, if it contains additive subgroups Md, for d 2 N, such that M = ⊕d2NMd, and RdMe ⊆ Md+e, for all d; e 2 N. Such a family fMdgd2N is called an th N-grading of M, and Md the d graded component of M. Let R = ⊕d2NRd be a graded ring and M = ⊕d2NMd be a graded R-module. By a graded submodule of M we mean a submodule N of M such that N = ⊕d2N(N \ Md). Note that if N is a graded submodule of M, then M=N is a graded R-module with the induced 0 0 grading given by (M=N)d = Md=(N \Md). If M = ⊕d2NMd is any graded R-module, then by graded module homomorphism, or a homomorphism of graded modules, we mean a R-module 0 0 homomorphism φ : M ! M such that φ(Md) ⊆ Md, for all d 2 N.
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