A Note on Fitting ideals

Transcription

1 A Note on Fitting ideals Jonathan A. Huang Abstract This short note is an expository article on the theory of Fitting ideals and its role in Iwasawa theory. 1 Introduction Given a finite abelian group G, a fundamental theorem in algebra is that there exists an isomorphism G = Z/(a 1 ) Z/(a 2 ) Z/(a n ), a 1 a 2 a n, i.e. all finite abelian groups are just direct sums of finite cyclic groups. In this case, the order of the group is computed as G = a 1 a 2 a n. Knowing the order of a group is far from knowing its isomorphism class; however, it is possible to define a sequence of invariants from which the integers a 1, a 2,, a n may be determined. These are the higher Fitting invariants of G: Fit 0 (G) Fit 1 (G)... Fit n (G), Fit i (G) = (a 1 a 2 a i ) Z. Thus, knowing the isomorphism class of G is equivalent to knowing its Fitting ideals. More generally, Fitting ideals Fit i (M) are invariants that can be attached to any finitely generated module M over a commutative ring. These invariants were first defined by H. Fitting; the canonical reference is D. G. Northcott s textbook Finite Free Resolutions []. They are used in various areas of mathematics to study the structure of modules. Fitting ideals are defined as the ideals generated by the subdeterminants of a free presentation of M, and so are most useful when M naturally comes with a presentation. Moreover, the Fit i (M) often appear as invariants which are finer than the annihilator ideal Ann(M) of a module. Fitting ideals contain information about the free rank and the order of the torsion-part of a module. And in certain cases, as indicated by the above example, they may be used to completely determine the isomorphism class of a finitely generated torsion module. More precisely: Theorem 1. Let M be a finitely generated torsion module over a PID R. Then the isomorphism class of M is completely determined by the higher Fitting ideals Fit i (M), i = 0, 1,... n 1, where n is the minimum number of generators of M. 1

2 One area of mathematics where Fitting ideals are used is Iwasawa theory. Let F be an abelian number field, and let G = Gal(F/Q). Then the ideal class group Cl F can be treated as a module over the group ring ZG. Stickelberger s theorem states that a certain ideal, the Stickelberger ideal Θ F, created from Gauss sums, is contained in the annihilator ideal of Cl F ; a more general statement of this fact is called the Herbrand-Ribet theorem. But both are easy consequences of the main conjecture in Iwasawa theory, which states that the zeroth Fitting ideal of the Iwasawa module is generated by the p-adic zeta function. More or less, the idea behind Iwasawa theory is to study algebraic objects of number theoretic importance (such as ideal class groups) by placing them in a p- adic tower. That is, given F there is a Z p -extension F, Γ := Gal(F /F ) = Z p. The Iwasawa module X is defined using the p-parts of the ideal class groups Cl Fn [p] of the intermediate fields F n in the Z p -extension F. X is the main algebraic object of study, and it is a module over the Iwasawa algebra Λ := Z p [[Γ]]. The Iwasawa algebra Λ is a commutative Noetherian ring, and a unique factorization domain. Moreover, Λ = Z p [[T ]], power series with coefficients in Z p. Thus the structure theory of finitely generated modules over Λ behaves much like the structure theory of finitely generated modules over Z[t]. However, the classification occurs only up to finite cokernel, i.e. up to quasi-isomorphism. The main point is that Fitting ideals are invariant under quasi-isomorphism. The purpose of this paper is to give an expository account of Fitting ideals and their use in Iwasawa theory. In Section 2, we introduce the basic definitions and properties of Fitting ideals, as well as provide a proof of Theorem 1. In Section 3, we will survey the use of Fitting ideals in Iwasawa theory, most notably in the statement of the main conjecture by Mazur and Wiles [] and in various refinements of the main conjecture due to Kurihara []. We will also show in certain cases the appropriate analog of Theorem 1 holds in this setting. Finally, the last section is a brief overview of various ways to generalize the definition of Fitting ideals to noncommutative rings, which plays an important role in attempts to state and prove noncommutative versions of the main conjecture and various refinements. 2 Fitting Ideals: Basic Theory Finite Free Presentations. Let M be a finitely generated module over R a Noetherian ring. Then M has a free presentation as follows. Let {a 1,..., a n } be a set of generators for M. Construct R n ϕ M 0 by sending each basis element to a generator. Then (ker ϕ) is a submodule of the free finitely generated module G := R n. Since R is Noetherian, (ker ϕ) is finitely generated. Let {r 1,..., r m } be the generators of (ker ϕ), each r i representing a relation among the generators of M. Construct R m (ker ϕ) 0 by sending each basis element to a generator. Setting F := R m, we have the exact sequence F G M 0, 2

3 a free presentation of M. Note that we may only care that F be free, in which case R Noetherian is unnecessary. But in the case R is Noetherian, there is a presentation of finite rank free modules. It is possible to continue the above process. That is, construct a map from a free module to F by considering generators for the kernel of F G, and so on. The resulting exact sequence F : F n F 1 F 0 M 0 is called a free resolution of M. If F n+1 = 0 (and thus F m = 0 for all m > n), then F is called a finite free resolution of length n. Every module has a free resolution, and a free presentation is just the truncation at the first step. If the module is finitely generated, then F 0 is finitely generated free. If R is Noetherian, then we can ensure that F 1 is finitely generated free. In this case, M is said to be finitely presented. Determinantal Ideals. Suppose that ϕ : F G is a map of free R-modules, R a commutative ring. Consider the induced map F G f g ψ R g (ϕ(f)) Then I(ϕ) := im ψ R. Choosing bases {f i } for F and {g i } for G, this is exactly the ideal generated by the entries of the matrix representing ϕ, i.e. the ideal generated by the coefficients of the ϕ(f i ). In general, we can define the ideal I j ϕ as I(Λ j ϕ). That is, consider the map on exterior powers Λ j ϕ : Λ j F Λ j G This is simply another map of free modules. And since (Λ j G) = Λ j G, it induces a map Λ j F Λ j G ψ R and I j ϕ := im(ψ) R. Again, choosing bases {f i } for F and {g i } for G, this is exactly the ideal generated by the entries of the matrix representing Λ j ϕ, i.e. the ideal generated by the j-minors (j j subdeterminants) of the matrix representing ϕ. The ideals I j ϕ are called determinantal ideals. Fitting Ideals.We are now ready to define the Fitting ideals of a finitely generated module M over a commutative Noetherian ring R. Consider a finite free presentation of M F ϕ G M 0 where F and G are free modules of rank m and n, respectively. Then define the ith Fitting ideal of M to be Fit i (M) = I n i ϕ R. That is, given bases for F and G, Fit i (M) is the ideal generated by the (n i)- minors of the matrix representing ϕ, the free presentation of M. The top Fitting ideal Fit n (M) = I 0 (ϕ) = R since a 0 0 determinant is set to be 1 by convention. 3

4 This definition is independent of the chosen free presentation of M (see [?]). When distinguishing Fitting ideals over different rings, we will denote the Fitting ideal by Fit R i (M). Examples. Before presenting some basic properties of Fitting ideals, here are a few examples. We begin with the initial Fitting ideal Fit 0 ((M). Over a PID, the initial Fitting ideal is the order ord(m) of the module. Example 1. Let G be a finite abelian group. Fit 0 (G) = ord(g) = G, the order of the group. Then, considered as a Z-module, Example 2. The characteristic ideal is an example of the order of a module, and thus of the initial Fitting ideal. The higher Fitting ideals give more information about the structure of a module. Example 3. Consider the torsion module M = R/(a) R/(b). This has a free presentation as a finitely generated R-module: R R a 0 0 b R R Then Fit 0 (M) = (ab), Fit 1 (M) = (a, b) and Fit 2 (M) = R. Note that if R = Z, then Fit 0 (M) is simply the order of the finite abelian group M. Thus Fitting ideals give more information that the order of a group. Example 4. Consider the finite abelian groups of order p 2, A = Z/(p 2 ) and B = Z/(p) Z/(p). Then Fit 0 (A) = (p 2 ) and Fit 0 (B) = (p 2 ), but Fit 1 (A) = Z (by convention) and Fitt 1 (B) = (p, p) = (p). So Fitting ideals can tell finite abelian groups of the same order apart. This is true even if the groups have the same number of generators. Indeed, consider A = Z/(p) Z/(p 3 ) and B = Z/(p 2 ) Z/(p 2 ). Then Fit 1 (A) = (p 2 ) and Fit 1 (B) = (p), and Fit 2 (A) = Fit 2 (B) = Z Example 5. Consider the module M = R n M tors a module over R with free part R n and torsion part M tors. Then the first nonzero Fitting ideal is Fit n (M). In particular, for torsion modules, Fit 0 (M) 0. The notions of order and (free) rank of a module aren t always definable, nor easy to define. In general, Fitting ideals give more information about the structure of a finitely generated module M over R than order or rank. Basic Properties. Here we provide some basic properties of Fitting ideals. All of the proofs can be found in Northcott []. Lemma 2. The Fitting ideals form a nested sequence of ideals 0 Fitt 0 (M) Fitt 1 (M) Fitt n (M) = R. Lemma 3. Let S be a multiplicative set in R, and consider the localization R S. Then F it R i (M) = FitR S i (M S ). That is, Fitting ideals localize well. 4

5 Lemma 4. Consider a cyclic module M = R/I for I R an ideal. Fit 0 (M) = I and F it i (M) = R for i > 0. Then Lemma 5. Suppose 0 M N P 0 is a short exact sequence of modules. Then we have Fit k (N) Fit i (M) Fit j (P ). i+j=k Lemma 6. Suppose N = M P, that is, the above sequence is split exact. Then we have equality in the above lemma. That is, Fit k (N) = Fit i (M) Fit j (P ). i+j=k Relation to the structure of modules. Computing the Fitting ideals Fit i (M) of a module requires a lot of information about M. Therefore, it may not be surprising that they give a lot of information about the structure of the module and in some cases determine the module up to isomorphism. We formulate and prove a precise statement along these lines. Theorem 7. Let M and N be a finitely generated modules over R a PID. Then M = N if and only if Fit i (M) = Fit i (N) for all i. Proof. Over a PID, there is a rigid structure theorem called the invariant factor decomposition. Given a finitely generated module M, we have M = R k M tors and M tors = R/(a1 ) R/(a 2 ) R/(a n ), a 1 a 2 a n. That is, there is a sequence of ideals (a i ) R, with each principal generator dividing the next. We know from Example 5 that Fitting ideals determine the free rank of a module. Thus it suffices to consider M and N finitely generated torsion modules, in which case the isomorphism class is completely determined by the invariant factors. Recall the definition of ideal quotient. Given two ideals I, J R, the ideal quotient (I : J) is defined to be the ideal (I : J) = {x R xj I} Let a, b R and consider the ideals I = (ab) and J = (a). Then the ideal quotient (I : J) = (b). Let M be a finitely generated torsion modules over R a PID. Then we claim that the Fitting ideals of M suffice to determine the invariant factors of M, from which the theorem follows. But the Fitting ideals are invariant under isomorphism, and M is isomorphic to a direct sum of cyclic modules R/(a i ). In this case, the Fitting ideals are computed as Fit n i (M) = (a 1 )(a 2 ) (a i ) = (a 1 a 2 a i ) R. If this holds, then the invariant factor a i is recovered by taking the ideal quotient of successive Fitting ideals: (a i ) = (Fit n i (M) : Fit n (i 1) (M)) = ((a 1 a 2 a i ) : (a 1 a 2 a i 1 )) so that the invariant factors are determined by the Fitting ideals. We must now simply show that the the Fitting ideals are computed as stated. 5

6 Claim. For M given as above, Fit n i (M) = (a 1 a 2 a i ) We prove this claim using the following lemma: Lemma 8. If M = N R/I, then Fit i (M) = I Fit i (N) + Fit i 1 (N) + + Fit 0 (N) = I Fit i (N) + Fit i 1 (N). Proof. The first equality follows from Lemma 4 and Lemma 6. If J I, then I + J = I. As shown in Lemma 2, the Fitting ideals of N form a nested sequence. Thus the second equality holds. Note that when i = 0, there is only one term. Set N = R/(a 1 ) R/(a 2 ) R/(a n 1 ). Then by Lemma 8 we see that and for i > 0, Fit 0 (M) = (a n ) Fit 0 (N) = (a 1 a 2 a n ) Fit n i (M) = (a n ) Fit n i (M) + Fit (n i) 1 (N) = Fit (n 1) i (N), where the second equality follows from the fact that the invariant factors divide each other, a 1 a 2 a n. Now we recursively apply this formula until N = R/(a 1 ) R/(a 2 ) R/(a i ), so that which completes the proof of the claim. Fit n i (M) = Fit 0 (N) = (a 1 a 2 a i ) Note*. There is a much easier way to prove Claim 2. The Fitting ideals are independent of the chosen free presentation of M. Thus, we can chose one whose matrix is simply a diagonal matrix with entries the invariant factors (a i ). Since they divide each other, the subdeterminants, and thus the Fitting ideals, are as stated. Or in other words, Lemma 6 shows that the Fitting ideals of a sum of cyclic modules is just the sum of the i-fold products of the invariant factors. Since the principal generators are successively divisible, this sum is simply the product of the first i-many invariant factors. However, we would prefer to get away from the dependency on invariant factors. 3 Fitting ideals in Iwasawa theory The main conjecture of Iwasawa theory. Fitting Ideals over Λ. The analog to Theorem 7 for modules over the Iwasawa algebra Λ is Theorem 9. Suppose that M N quasi-isomorphic modules over Λ. Fit Λ i (M) = Fit Λ i (N). Then Fitting ideals and the structure of Λ-modules. 4 Noncommutative Fitting Ideals and Other Generalizations 6