Kaplansky's theorem on projective modules
In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free;[1] where a not-necessarily-commutative ring is called local if for each element x, either x or 1 − x is a unit element.[2] The theorem can also be formulated so to characterize a local ring (#Characterization of a local ring). For a finite projective module over a commutative local ring, the theorem is an easy consequence of Nakayama's lemma.[3] For the general case, the proof (both the original as well as later one) consists of the following two steps:
- Observe that a projective module over an arbitrary ring is a direct sum of countably generated projective modules.
- Show that a countably generated projective module over a local ring is free (by a "[reminiscence] of the proof of Nakayama's lemma"[4]).
The idea of the proof of the theorem was also later used by Hyman Bass to show big projective modules (under some mild conditions) are free.[5] According to (Anderson Fuller), Kaplansky's theorem "is very likely the inspiration for a major portion of the results" in the theory of semiperfect rings.[1]
Proof
The proof of the theorem is based on two lemmas, both of which concern decompositions of modules and are of independent general interest.
Lemma 1 — [6] Let [math]\displaystyle{ \mathfrak{F} }[/math] denote the family of modules that are direct sums of some countably generated submodules (here modules can be those over a ring, a group or even a set of endomorphisms). If [math]\displaystyle{ M }[/math] is in [math]\displaystyle{ \mathfrak{F} }[/math], then each direct summand of [math]\displaystyle{ M }[/math] is also in [math]\displaystyle{ \mathfrak{F} }[/math].
Proof: Let N be a direct summand; i.e., [math]\displaystyle{ M = N \oplus L }[/math]. Using the assumption, we write [math]\displaystyle{ M = \bigoplus_{i \in I} M_i }[/math] where each [math]\displaystyle{ M_i }[/math] is a countably generated submodule. For each subset [math]\displaystyle{ A \subset I }[/math], we write [math]\displaystyle{ M_A = \bigoplus_{i \in A} M_i, N_A = }[/math] the image of [math]\displaystyle{ M_A }[/math] under the projection [math]\displaystyle{ M \to N \hookrightarrow M }[/math] and [math]\displaystyle{ L_A }[/math] the same way. Now, consider the set of all triples ([math]\displaystyle{ J }[/math], [math]\displaystyle{ B }[/math], [math]\displaystyle{ C }[/math]) consisting of a subset [math]\displaystyle{ J \subset I }[/math] and subsets [math]\displaystyle{ B, C \subset \mathfrak{F} }[/math] such that [math]\displaystyle{ M_J = N_J \oplus L_J }[/math] and [math]\displaystyle{ N_J, L_J }[/math] are the direct sums of the modules in [math]\displaystyle{ B, C }[/math]. We give this set a partial ordering such that [math]\displaystyle{ (J, B, C) \le (J', B', C') }[/math] if and only if [math]\displaystyle{ J \subset J' }[/math], [math]\displaystyle{ B \subset B', C \subset C' }[/math]. By Zorn's lemma, the set contains a maximal element [math]\displaystyle{ (J, B, C) }[/math]. We shall show that [math]\displaystyle{ J = I }[/math]; i.e., [math]\displaystyle{ N = N_J = \bigoplus_{N' \in B} N' \in \mathfrak{F} }[/math]. Suppose otherwise. Then we can inductively construct a sequence of at most countable subsets [math]\displaystyle{ I_1 \subset I_2 \subset \cdots \subset I }[/math] such that [math]\displaystyle{ I_1 \not\subset J }[/math] and for each integer [math]\displaystyle{ n \ge 1 }[/math],
- [math]\displaystyle{ M_{I_n} \subset N_{I_n} + L_{I_n} \subset M_{I_{n+1}} }[/math].
Let [math]\displaystyle{ I' = \bigcup_0^\infty I_n }[/math] and [math]\displaystyle{ J' = J \cup I' }[/math]. We claim:
- [math]\displaystyle{ M_{J'} = N_{J'} \oplus L_{J'}. }[/math]
The inclusion [math]\displaystyle{ \subset }[/math] is trivial. Conversely, [math]\displaystyle{ N_{J'} }[/math] is the image of [math]\displaystyle{ N_J + L_J + M_{I'} \subset N_J + M_{I'} }[/math] and so [math]\displaystyle{ N_{J'} \subset M_{J'} }[/math]. The same is also true for [math]\displaystyle{ L_{J'} }[/math]. Hence, the claim is valid.
Now, [math]\displaystyle{ N_J }[/math] is a direct summand of [math]\displaystyle{ M }[/math] (since it is a summand of [math]\displaystyle{ M_J }[/math], which is a summand of [math]\displaystyle{ M }[/math]); i.e., [math]\displaystyle{ N_J \oplus M' = M }[/math] for some [math]\displaystyle{ M' }[/math]. Then, by modular law, [math]\displaystyle{ N_{J'} = N_J \oplus (M' \cap N_{J'}) }[/math]. Set [math]\displaystyle{ \widetilde{N_J} = M' \cap N_{J'} }[/math]. Define [math]\displaystyle{ \widetilde{L_J} }[/math] in the same way. Then, using the early claim, we have:
- [math]\displaystyle{ M_{J'} = M_J \oplus \widetilde{N_J} \oplus \widetilde{L_J}, }[/math]
which implies that
- [math]\displaystyle{ \widetilde{N_J} \oplus \widetilde{L_J} \simeq M_{J'} / M_J \simeq M_{J' - J} }[/math]
is countably generated as [math]\displaystyle{ J' - J \subset I' }[/math]. This contradicts the maximality of [math]\displaystyle{ (J, B, C) }[/math]. [math]\displaystyle{ \square }[/math]
Lemma 2 — If [math]\displaystyle{ M_i, i \in I }[/math] are countably generated modules with local endomorphism rings and if [math]\displaystyle{ N }[/math] is a countably generated module that is a direct summand of [math]\displaystyle{ \bigoplus_{i \in I} M_i }[/math], then [math]\displaystyle{ N }[/math] is isomorphic to [math]\displaystyle{ \bigoplus_{i \in I'} M_i }[/math] for some at most countable subset [math]\displaystyle{ I' \subset I }[/math].
Proof:[7] Let [math]\displaystyle{ \mathcal{G} }[/math] denote the family of modules that are isomorphic to modules of the form [math]\displaystyle{ \bigoplus_{i \in F} M_i }[/math] for some finite subset [math]\displaystyle{ F \subset I }[/math]. The assertion is then implied by the following claim:
- Given an element [math]\displaystyle{ x \in N }[/math], there exists an [math]\displaystyle{ H \in \mathcal{G} }[/math] that contains x and is a direct summand of N.
Indeed, assume the claim is valid. Then choose a sequence [math]\displaystyle{ x_1, x_2, \dots }[/math] in N that is a generating set. Then using the claim, write [math]\displaystyle{ N = H_1 \oplus N_1 }[/math] where [math]\displaystyle{ x_1 \in H_1 \in \mathcal{G} }[/math]. Then we write [math]\displaystyle{ x_2 = y + z }[/math] where [math]\displaystyle{ y \in H_1, z \in N_1 }[/math]. We then decompose [math]\displaystyle{ N_1 = H_2 \oplus N_2 }[/math] with [math]\displaystyle{ z \in H_2 \in \mathcal{G} }[/math]. Note [math]\displaystyle{ \{ x_1, x_2 \} \subset H_1 \oplus H_2 }[/math]. Repeating this argument, in the end, we have: [math]\displaystyle{ \{ x_1, x_2, \dots \} \subset \bigoplus_0^\infty H_n }[/math]; i.e., [math]\displaystyle{ N = \bigoplus_0^\infty H_n }[/math]. Hence, the proof reduces to proving the claim and the claim is a straightforward consequence of Azumaya's theorem (see the linked article for the argument). [math]\displaystyle{ \square }[/math]
Proof of the theorem: Let [math]\displaystyle{ N }[/math] be a projective module over a local ring. Then, by definition, it is a direct summand of some free module [math]\displaystyle{ F }[/math]. This [math]\displaystyle{ F }[/math] is in the family [math]\displaystyle{ \mathfrak{F} }[/math] in Lemma 1; thus, [math]\displaystyle{ N }[/math] is a direct sum of countably generated submodules, each a direct summand of F and thus projective. Hence, without loss of generality, we can assume [math]\displaystyle{ N }[/math] is countably generated. Then Lemma 2 gives the theorem. [math]\displaystyle{ \square }[/math]
Characterization of a local ring
Kaplansky's theorem can be stated in such a way to give a characterization of a local ring. A direct summand is said to be maximal if it has an indecomposable complement.
Theorem — [8] Let R be a ring. Then the following are equivalent.
- R is a local ring.
- Every projective module over R is free and has an indecomposable decomposition [math]\displaystyle{ M = \bigoplus_{i \in I} M_i }[/math] such that for each maximal direct summand L of M, there is a decomposition [math]\displaystyle{ M = \Big(\bigoplus_{j \in J} M_j\Big) \bigoplus L }[/math] for some subset [math]\displaystyle{ J \subset I }[/math].
The implication [math]\displaystyle{ 1. \Rightarrow 2. }[/math] is exactly (usual) Kaplansky's theorem and Azumaya's theorem. The converse [math]\displaystyle{ 2. \Rightarrow 1. }[/math] follows from the following general fact, which is interesting itself:
- A ring R is local [math]\displaystyle{ \Leftrightarrow }[/math] for each nonzero proper direct summand M of [math]\displaystyle{ R^2 = R \times R }[/math], either [math]\displaystyle{ R^2 = (0 \times R) \oplus M }[/math] or [math]\displaystyle{ R^2 = (R \times 0) \oplus M }[/math].
[math]\displaystyle{ (\Rightarrow) }[/math] is by Azumaya's theorem as in the proof of [math]\displaystyle{ 1. \Rightarrow 2. }[/math]. Conversely, suppose [math]\displaystyle{ R^2 }[/math] has the above property and that an element x in R is given. Consider the linear map [math]\displaystyle{ \sigma:R^2 \to R, \, \sigma(a, b) = a - b }[/math]. Set [math]\displaystyle{ y = x - 1 }[/math]. Then [math]\displaystyle{ \sigma(x, y) = 1 }[/math], which is to say [math]\displaystyle{ \eta: R \to R^2, a \mapsto (ax, ay) }[/math] splits and the image [math]\displaystyle{ M }[/math] is a direct summand of [math]\displaystyle{ R^2 }[/math]. It follows easily from that the assumption that either x or -y is a unit element. [math]\displaystyle{ \square }[/math]
See also
Notes
- ↑ 1.0 1.1 Anderson & Fuller 1992, Corollary 26.7.
- ↑ Anderson & Fuller 1992, Proposition 15.15.
- ↑ Matsumura 1989, Theorem 2.5.
- ↑ Lam 2000, Part 1. § 1.
- ↑ Bass 1963
- ↑ Anderson & Fuller 1992, Theorem 26.1.
- ↑ Anderson & Fuller 1992, Proof of Theorem 26.5.
- ↑ Anderson & Fuller 1992, Exercise 26.3.
References
- Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, 13 (2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3
- Bass, Hyman (February 28, 1963). "Big projective modules are free". Illinois Journal of Mathematics (University of Illinois at Champagne-Urbana) 7 (1): 24–31. doi:10.1215/ijm/1255637479. https://typeset.io/papers/big-projective-modules-are-free-54r4kq564h.
- Kaplansky, Irving (1958), "Projective modules", Ann. of Math., 2 68 (2): 372–377, doi:10.2307/1970252
- Lam, T.Y. (2000). "Bass's work in ring theory and projective modules". arXiv:math/0002217. MR1732042
- Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6
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