Commutative Algebra seminar talk Reeve Garrett January 26, 2015 1 The ideal-adic topology For the entirety of this talk, let R be a commutative unitary ring, I be an ideal of R, and M be an R-module. n 1 Definition 1.1 Consider the family F := fI Mgn=0. The topology for which F is a fundamental system of neighborhoods of 0 2 M is called the I-adic topology (sometimes called the Krull topology). Recall that a collection V of neighborhoods of a point x is called a fundamental system of neighborhoods of x if for any neighborhood U of x there exists a finite sequence V1;V2; :::; Vn of neighborhoods in V such that V1 \ V2 \···\ Vn ⊆ U. Consequently, all open sets of M are unions of arbitrary numbers of sets of the form b + IsM for b 2 M and s 2 N. The I-adic topology makes M a topological R-module, and in the case M = R, R is a topological ring, meaning all the algebraic operations (addition, multiplication, inversion) are continuous - in particular, translation defines a homeomorphism. Observation 1.2 Since by definition each InM is open, each coset x + InM (x2 = InM) is open, so their union, the complement of InM, is open, meaning InM must be closed as well. Thus, if we endow M=InM with the quotient topology, that quotient topology must be the discrete topology. We now establish some basic facts about this topology. Lemma 1.3 [5, (16.2), p. 51] A submodule N of the R-module M is an open set in the I-adic topology iff (N : M) := fr 2 R : rM ⊆ Ng contains some power of I. In that case, N is also closed. Proof. If N is open, then 0 2 N implies InM ⊆ N for some n since F forms a neighborhood base of 0. Conversely, if InM ⊆ N, then since N is a submodule, b + InM ⊆ N for any b 2 N, and N is the union of the open sets b + InM (b 2 N), proving N is open. Moreover, by the same reasoning as the preceding observation, we see N is closed. Theorem 1.4 [5, (16.3), p. 51] The closure N of a submodule N of M in M with the I-adic topology is T n T n n≥0(N + I M). In particular, the set N := n≥0 I M, is the closure of f0g. Proof. Each N + InM is an open set and therefore simultaneously closed by the preceding lemma. T n T n Therefore, N ⊆ n≥0(N + I M). Conversely, let x 2 n≥0(N + I M). Then, x = bn + an with bn 2 N n n and an 2 I M for each n 2 N. Consequently, we see x + I M meets N for any n (with the element x − an). Since the x + InM form a neighborhood base at the point x, we consequently have x 2 N. In order for a topological space to be Hausdorff, points must be closed. In fact, we have an equivalence: Lemma 1.5 [8, p. 252] If the zero of a topological module M is closed set, M is a Hausdorff space. Proof. Let x 6= y 2 M, V be a neighborhood of y − x which does not contain 0 2 M, and U = x − y + V . Then U is a neighborhood of 0 not containing x − y. Let W be another neighborhood of 0 such that W − W ⊆ U (the existence of such a neighborhood follows from the continuity of (x; y) 7! x − y). Then x + W and y + W are disjoint neighborhoods of x and y, and thus since x and y were arbitrary, we have M is Hausdorff. T n Thus, M is Hausdorff in the I-adic topology iff n≥0 I M = 0. 1 Corollary 1.6 [5, (16.4), p. 51] For a submodule N of M, the I-adic topology of M=N is Hausdorff iff N is a closed subset in the I-adic topology of M. T n Proof. The I-adic topology of M=N is Hausdorff iff n≥0(I M + N)=N) = 0 as just noted, or equiva- T n lently, N = n≥0(N + I M), which is equivalent to N being closed by the theorem. For N a submodule of M, the I-adic topology of N might not coincide with the subspace topology of N in M. However, when M is a Noetherian module, these topologies do coincide. To prove this, we need an important lemma: the Artin-Rees lemma. Theorem 1.7 [1, Proposition 4.1.5 and Lemmas 4.1.7 and 4.1.8] If R is a Noetherian ring, M is a finite R-module, and N is a submodule of M, then there is an integer k depending only on R, M, N, and I such that InM \ N = In−k(IkM \ N). Proof. See the cited reference. Theorem 1.8 [5, (16.5), p. 52] If M is a Noetherian module, then for any submodule N of M, the I-adic topology of N coincides with the topology of N as a subspace of M with the I-adic topology. Proof. By [5, (3.11), p. 12], for any Noetherian module M over a ring R we have R=(0 : M) is a Noetherian ring and M is a finite R=(0 : M)-module. Modding out (0 : M) if necessary we may thus assume R is Noetherian. Obviously, InN ⊆ InM \ N, so by the Artin-Rees lemma, for some k > 0 and for n > k, InM \ N = In−k(IkM \ N) ⊆ In−kN, thus proving the assertion. Remark 1.9 By similar reasoning to the above proof, we see if I and J are two ideals such that for some natural numbers m and n Im ⊆ J and J n ⊆ I, the I-adic topology and J-adic topology coincide. This is a helpful observation that allows us in several cases reduce to the case where I is R's maximal ideal (if R is local) or Jacobson radical (if R is semi-local). For the remainder of the section, we will assume M is Hausdorff. In this case, we have a well-defined arithmetic function w : M ! N [ f1g, defined by w(x) = supfk j x 2 IkMg if x 6= 0 and w(0) = 1. From this arithmetic function, we get a metric d defined by d(x; y) = e−w(x−y), which induces the I-adic topology. In fact, d is an ultrametric (that is, a metric with a \strong triangle inequality" d(x; z) ≤ supfd(x; y); d(y; z)g) because w(a − b) ≥ minfw(a); w(b)g. In the case M = R and R is a domain, the quotient field K with the same ultrametric also is a topological ring. Of course, with a metric space, it's natural to consider Cauchy sequences and their convergence, and thus we arrive at one formulation of the I-adic completion, arguably the most natural formulation. 2 The ideal-adic completion Given a commutative unitary ring R, ideal I, and R-module M, the I-adic completion of M is often developed from 2 seemingly different perspectives. The first of these, through Cauchy sequences, is perhaps the most T n elementary and natural. Interestingly, we don't need the hypothesis that n≥0 I M = 0 for the notion of Cauchy sequence to make sense. N(n) Definition 2.1 A Cauchy sequence in M is a sequence fxng 2 M N such that xn − xn+i 2 I M for all i ≥ 0, where N(n) ! 1 as n ! 1. m Notation 2.2 We write xn ! 0 as n ! 1 if for each m 2 N there exists N 2 N such that xn 2 I M for each n ≥ N. In this case, we say fxng is a null sequence. Definition 2.3 A limit of a sequence fxng is any element y of M such that fxn − yg is a null sequence. 2 0 0 T n Remark 2.4 If fxng has a limit y, then y is also a limit of the sequence iff y − y 2 n≥0 I M, and we say y and y0 are equivalent. Definition 2.5 The collection of all equivalence classes of limits of Cauchy sequences of M is called the I-adic completion of M, denoted Mc. Equivalently, if we define two Cauchy sequences fxng and fyng to be m equivalent if xn − yn ! 0 as n ! 1, i.e. for all m 2 N there exists N(m) 2 N such that xn + I M = m yn + I M for n ≥ N(m), we may define Mc as the set of equivalence classes of Cauchy sequences in M according to this equivalence relation. Of course, intuitively, we wish to identify M as a dense subspace of Mc, and since Mc has a natural R-module structure via the latter definition, the easiest way to do so is to assign via a map φ : M ! Mc T n assigning x 2 M to the constant sequence fxg. However, the kernel of this map is n≥0 I M, so we see φ is injective iff M is Hausdorff. In this case, because the ring and module operations of R and M are uniformly continuous, it follows that Rb is a topological ring and Mc is a topological Rb-module and R-module by extending the given operations uniquely (algebraic identities are preserved through passage to limits).