Some Finiteness Conditions in Algebra

SOME FINITENESS CONDITIONS IN ALGEBRA

ANDREW BAKER

Throughout, R will be a commutative and unital ring. However, much of the theory also works for non-commutative rings. Recall that the Jacobson radical of R is the intersection of all the maximal ideals of R, \ J(R) = m. maximal m/R This is an ideal of R which is characterized as the largest ideal I/R for which ∀t ∈ I ∀s ∈ R 1 + st ∈ R×.

1. Nakayama’s lemma Theorem 1.1 (Nakayama’s lemma). Let M be a ﬁnitely generated R-module and I/R. Suppose that IM = M. (a) There is an element t ∈ I for which (1 + t)M = 0. (b) If I ⊆ J(R), then M = 0. Proof. See [4] for the commutative case and also [5] for (b) in the general case. ¤

Corollary 1.2. Suppose that I ⊆ J(R) and N 6 M with M/N ﬁnitely generated. If N + IM = M, then N = M. Proof. We have IM/N = M/N, hence M/N = 0. ¤

Here is an important sample application of Nakayama’s lemma and its corollary. Proposition 1.3. Suppose that M is a ﬁnitely generated R-module and that F is a ﬁnitely generated free R-module for which there is an isomorphism F/ J(R)F =∼ M/ J(R)M. Then for any factorization

F / F/ J(R)F

ϕ ∼= M / M/ J(R)M the map ϕ: F −→ M is an epimorphism. Proof. We have im ϕ + J(R)M = M. ¤

This means that construction of ‘minimal’ free covers can be done by working modulo J(R). For example, if R is a local ring with unique maximal ideal m, then M/mM is a ﬁnite dimensional vector space over the ﬁeld R/m. Given a basis for M/mM, there is a corresponding isomorphism

∼ ∼ Rd/mRd −→= (R/m)d −→= M/mM which lifts (non-uniquely) to a homomorphism Rd −→ M which is an epimorphism by Proposition 1.3. If M is projective over R, this implies that M is free of rank equal to dimR/m M/mM. Here is another application. Proposition 1.4. Let S be a commutative ring. Suppose that M is a ﬁnitely generated S-module and ϕ: F −→ M is an S-module epimorphism. Then ϕ is an isomorphism.

Date: 21/04/2009. 1 2 ANDREW BAKER

Proof. Let R = S[X] and view M as the R-module obtained by with extending the existing action of S by taking X · m = ϕ(m). Consider the ideal RX/R, and notice that (RX)M = RϕM = M. Hence for some f(X) ∈ R, (1 − Xf(X))M = (1 − f(X)X)M = 0, from which it follows that ϕ is invertible with inverse f(ϕ). ¤

2. Noetherian and coherence conditions Let R be a commutative ring. Definition 2.1. Let M be an R-module. (a) M is finitely generated if for some n ∈ N there is a short exact sequence Rn −→ M → 0. (b) M is finitely presented if for some m, n ∈ N there is a short exact sequence Rm −→ Rn −→ M → 0, i.e., there is a short exact sequence Rn −→π M → 0. with ker π finitely generated. An important result is the following. Proposition 2.2. Suppose that M is finitely presented and that there is an exact sequence 0 → K −→ P −→ M → 0 with P finitely generated projective. Then K is finitely generated. Proof. This requires Schanuel’s Lemma which says that if for an R-module N there are exact sequences 0 → K −→ P −→ N → 0, 0 → L −→ Q −→ N → 0, with P and Q projective, then P ⊕ L =∼ K ⊕ Q. Given this result, let 0 → ker π −→ Rn −→π M → 0 be short exact with ker π finitely generated. Then P ⊕ ker π =∼ K ⊕ Rn. But since P ⊕ ker π is finitely generated, so is K ⊕ Rn. Therefore K is finitely generated. ¤

Recall that an R-module M is Noetherian if for every increasing chain of submodules Nk 6 M,

· · · ⊆ Nk ⊆ Nk+1 ⊆ · · ·

there is a k0 such that

Nk = Nk0 (k > k0). For such a module, every submodule is finitely generated. The ring R is said to be Noetherian if it is a Noetherian R-module. For such a ring, every finitely generated R-module M is Noetherian, hence it is finitely presented. It is now easy to deduce that M admits a free resolution

· · · −→ Fn −→ Fn−1 −→ · · · −→ F0 −→ M → 0,

where each Fn is ﬁnitely generated and free. If R is also local with maximal ideal m, we can use Nakayama’s lemma to show that such a resolution can be chosen to be minimal in the sense that the boundary maps factor as

Fn −→ mFn−1 ⊆ Fn−1, and therefore the complex

· · · −→ R/m ⊗R Fn −→ R/m ⊗R Fn−1 −→ · · · −→ R/m ⊗R F0 → 0 has trivial diﬀerentials and so its homology has

Hn(R/m ⊗R F∗) = R/m ⊗R Fn. SOME FINITENESS CONDITIONS IN ALGEBRA 3

For a more general ring R, a resolution is minimal if the boundary maps factor as

Fn −→ J(R)Fn−1 ⊆ Fn−1, and such minimal resolutions exist for finitely generated modules if R/ J(R) is semi-simple. What about more general conditions on a ring forcing similar results? Definition 2.3. An R-module M is pseudo-coherent if all its finitely generated submodules are finitely presented. A finitely generated pseudo-coherent module M is coherent. The ring R is coherent if it is a coherent R-module. So an R-module M is coherent if and only if it and all its finitely generated submodules are finitely presented. It is standard that a finitely generated module over a Noetherian ring is coherent. But there are more examples.

Theorem 2.4 (see [3, theorem 2.3.3]). Let {Rα}α∈A be a directed filtered system of coherent rings and suppose that R = colimα Rα is flat over each Rα. Then R is coherent. In particular, this happens if whenever α 4 β, Rβ is flat over Rα. Example 2.5. If S is a Noetherian ring, then the infinitely generated polynomial ring

S[x1, x2,...] = S[xi : i > 1] = colim S[x1, . . . , xn] n is coherent but not Noetherian. Example 2.6. Any localization of a coherent ring R is coherent. In particular, if p /R is a prime ideal, Rp is coherent. Example 2.7. If R is coherent and I/R is a finitely generated ideal, then R/I is coherent. Example 2.8. If R is semi-hereditary then every countably generated polynomial algebra over R is coherent. However, in general, if R is coherent, the polynomial ring R[x] need not be coherent. Coherence satisfies some useful properties. Proposition 2.9. Let 0 → M 0 −→ M −→ M 00 → 0 be an exact sequence of R-modules. If two out of M 0,M,M 00 are coherent, the third is. Proposition 2.10. Let ϕ: M −→ N be a homomorphism between coherent modules. Then ker ϕ, im ϕ and coker ϕ are all coherent. Proposition 2.11. If the following triangle of homomorphisms of R-modules is exact ϕ / L `@ M @@ || @@ || @@ || ψ @ ~|| θ N then if two out of L, M, N are coherent so is the third. We also have Theorem 2.12. The ring R is coherent if and only if every finitely presented R-module is coherent. Over a coherent ring R, the coherent modules admit resolutions by finitely generated free modules. For a coherent local ring there are minimal resolutions of coherent modules; the proof is similar to that in the Noetherian case.

References [1] N. Bourbaki, Eléments´ de Mathematique, Fasc. XXVII: AlgèbreCommutative, Hermann (1961). [2] J. M. Cohen, Coherent graded rings and the non-existence of spaces of finite stable homotopy type, Comment. Math. 44 (1969), 217–228. [3] S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics 1371 (1989). [4] T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag (1999). [5] , A First Course in Noncommutative Rings, Second edition, Graduate Texts in Mathematics 131, Springer- Verlag (2001). [6] H. Matsumura, Commutative Ring Theory, Cambridge University Press (1989). [7] J-P. Serre, Faisceaux algébriquescohérents, Ann. of Math. 61 (1955), 197–278.

Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland. E-mail address: [email protected] URL: http://www.maths.gla.ac.uk/∼ajb