Universal Additive Groups, Direct Limits and Inverse Limits

A category C is a ⋆-category if there is a commutative and associative binary operation ⋆: obj(C) × obj(C) →obj(C); that is, (i) If A ∼= A′ and B ∼= B′, where A, A′, B, B′ ∈obj(C), then A ⋆B ∼= A′ ⋆B′. (ii) there is an equivalence A ⋆B ∼= B ⋆A for all A, B ∈obj(C); (iii) there is an equivalence A ⋆(B ⋆C) ∼= (A ⋆B) ⋆C for all A, B, C ∈obj(C). Any category having finite products or finite coproducts is a ⋆-category.

Let R be a commutative ring, and let C be a subcategory of RMod. Two R-modules A and B are called stably isomorphic in C if there exists a module C ∈obj(C) with A ⊕C ∼= B ⊕C. With this terminology, Proposition 7.77 says that two modules determine the same element of a universal additive groups if and only if they are stably isomorphic. It is clear that isomorphic modules are stably isomorphic; the next example shows that the converse need not hold.

A sequence in a category C of modules, M = M0 ⊇M1 ⊇M2 ⊇· · · ⊇Mn = {0}, is called a composition series of M if each of its factor modules Qi = Mi−1/Mi is a simple module in obj(C). We say that a category C of modules is a canonical-block–Holder category if: (i) Each object M has a composition series; (ii) For every two composition series M = M0 ⊇M1 ⊇M2 ⊇· · · ⊇Mn = {0} and M = M′ 0 ⊇M′ 1 ⊇M′ 2 ⊇· · · ⊇M′ m = {0}, we have m = n and a permutation σ ∈Sn such that Q′ j ∼= Qσ j for all j, where Qi = Mi−1/Mi and Q′ j = M′ j−1/M′ j. Define the length ℓ(M) of a module M in a canonical-block–Holder category to be the number n of terms in a composition series. If the simple factor modules of a composition series are Q1, . . . , Qn, we define jh(M) = Q1 ⊕· · · ⊕Qn. A composition series may have several isomorphic factor modules, and jh(M) records their multiplicity. universal additive groups

Let I be a partially ordered set, and let {Mi, ψ j i } be an inverse system of R-modules over I. The inverse limit (also called projective limit or limit) is an R-module lim ←−Mi and a family of R-maps {αi : lim ←−Mi →Mi : i ∈I}, such that (i) ψ j i α j = αi whenever i ⪯j; (ii) for every module X having maps fi : X →Mi satisfying ψ j i f j = fi for all i ⪯j, there exists a unique map θ : X →lim ←−Mi making the following diagram commute: lim ←−Mi α j αi ∫ X θ fi f j Mi M j ψ j i The notation lim ←−Mi for an inverse limit is deficient in that it does not display the maps of the corresponding inverse system (and lim ←−Mi does depend on them). However, this is standard practice. As with any object defined as a solution to a universal mapping problem, the inverse limit of an inverse system is unique (to isomorphism) if it exists. Limits

Let I be a partially ordered set. Thus, we can consider direct systems involving objects and morphisms in any category C as being a (covariant) functor Limits F : PO(I) →C. For example, it makes sense to consider direct systems of commutative rings.

A directed set is a partially ordered set I such that, for every i, j ∈I, there is k ∈I with i ⪯k and j ⪯k.

A functor F : C →D is an equivalence if there is a functor G : D →C such that the composites GF and FG are naturally equivalent to the identity functors 1C and 1D, respectively. module-category equivalence theory proves that if R and S are commutative rings, then equivalence of their module categories implies R ∼= S.

A ring R is an additive abelian group equipped with a multiplication R × R →R, denoted by (a, b) ↦ab, such that, for all a, b, c ∈R, (i) a(bc) = (ab)c; (ii) a(b + c) = ab + ac and (b + c)a = ba + ca; (iii) there is 1 ∈R such that, for all a ∈R, 1a = a = a1. Here are some examples of rings that are not commutative.

Let C be a ⋆-category. (i) If x ∈K0(C), then x = [A] −[B] for A, B ∈obj(C). (ii) If A, B ∈obj(C), then [A] = [B] in K0(C) if and only if there exists C ∈obj(C) with A ⋆C ∼= B ⋆C.

If C is the category of all finite abelian groups, then K0(C) is a free abelian group with a basis B consisting of all the cyclic primary groups.

Let R be a commutative ring and let C be the category of all finitely generated R-modules. If M ∈obj(C) and M = M0 ⊇M1 ⊇M2 ⊇· · · ⊇Mn = {0} has factor modules Qi = Mi−1/Mi, then (M) = (Q1) + · · · + (Qn) in K ′(C).

Let R be a commutative ring and let C be the category of all finitely generated R-modules. If A, B ∈obj(C), then (A) = (B) in K ′(C) if and only if there are C,U, V ∈obj(C) and exact sequences 0 →U →A ⊕C →V →0 and 0 →U →B ⊕C →V →0.

If 0 →A →B →C →0 is an exact sequence in a canonical-block–Holder category, then jh(B) ∼= jh(A) ⊕jh(C).

Let C be a category of modules in which every module M ∈obj(C) has a composition series. Then C is a canonical-block–Holder category if and only if K ′(C) is a free abelian group with basis the set B′ of all (S) as S varies over all nonisomorphic simple modules in obj(C).

gives jh(S) ⊕jh(C) ∼= jh(U) ⊕jh(V ) ∼= jh(T ) ⊕jh(C). By Lemma 7.85, we may cancel the simple summands one by one until we are left with S ∼= T , a contradiction. A similar argument shows that if S is a simple module, then universal additive groups (S) ̸= 0. Finally, let us show that every element in K ′(C) has a unique expression as a linear combination of elements in B′. Suppose there are positive integers mi and n j so that i mi(Si) − j n j(Tj) = 0, (1) where the Si and Tj are simple modules in obj(C) and Si ̸∼= Tj for all i, j. If we denote the direct sum of mi copies of Si by mi Si, then Eq. (1) gives ( i mi Si ) = ( j n jTj ) . By Proposition 7.84, there are modules C,U, V and exact sequences 0 →U →C ⊕ i mi Si →V →0 and 0 →U →C ⊕ j n jTj →V →0, and Lemma 7.86 gives jh ( i mi Si ) ∼= jh ( j n jTj ) . By Lemma 7.85, some Si occurs on the right-hand side, contradicting Si ̸∼= Tj for all i, j. Therefore, Eq. (1) cannot occur. • Remark. A module M is called indecomposable if there do not exist nonzero modules A and B with M ∼= A⊕B. ◀ Compare the next corollary with Proposition 7.79.

The inverse limit of any inverse system {Mi, ψ j i } of R-modules over a partially ordered index set I exists.

If {Mi, ψ j i } is an inverse system, then Hom(A, lim ←−Mi) ∼= lim ←−Hom(A, Mi) for every module A.

The direct limit of any direct system {Mi, ϕi j} of R-modules over a partially ordered index set I exists.

Let {Mi, ϕi j} be a direct system of left R-modules over a directed index set I, and let λi : Mi → Mi be the ith injection, so that lim −→Mi = ( Mi)/S, where S = λ jϕi jmi −λimi : mi ∈Mi and i ⪯j . (i) Each element of lim −→Mi has a representative of the form λimi + S (instead of i λimi + S). (ii) λimi + S = 0 if and only if ϕi t (mi) = 0 for some t ⪰i.

Let I be a directed set, and let {Ai, αi j}, {Bi, βi j}, and {Ci, γ i j } be direct systems over I. If r : {Ai, αi j} →{Bi, βi j} and s : {Bi, βi j} →{Ci, γ i j } are transformations, and if 0 →Ai ri→Bi si→Ci →0 is exact for each i ∈I, then there is an exact sequence 0 →lim −→Ai ⃗r→lim −→Bi ⃗s→lim −→Ci →0. Remark. The hypothesis that I be directed enters the proof only in showing that ⃗r is an injection. ◀

If R is a commutative ring, then HomR(R, M) is an R-module, and the R-isomorphisms ϕM : HomR(R, M) →M, given by ϕM( f ) = f (1), comprise a natural equivalence ϕ : HomR(R, ) →1R, the identity functor on RMod. Remark.

Let (F, G) be an adjoint pair of functors, where F : C →D and G : D →C. Then F preserves all direct limits and G preserves all inverse limits. Remark. (i) There is no restriction on the index sets of the limits; in particular, they need not be directed. (ii) A more precise statement is that if lim −→Ci exists in C, then lim −→FCi exists in D, and lim −→FCi ∼= F(lim −→Ci). ◀

(ii) If R is a commutative ring and F is a free R-module of infinite rank, then R ⊕F ∼= R ⊕R ⊕F. universal additive groups Thus, R and R ⊕R are nonisomorphic modules that are stably isomorphic in RMod. Because of examples of this type, we usually restrict ourselves to subcategories C of RMod consisting of finitely generated modules. Swan, in which stable isomorphism of finitely generated projective modules does not imply isomorphism. Let R = R[x1, . . . , xn]/(1 − i x2 i ) [the coordinate ring of the real (n −1)-sphere]. Regard Rn as n × 1 column vectors, and let X = (x1, . . . , xn)t ∈Rn, where bar denotes coset mod (1 − i x2 i ) in R. Define λ: R →Rn by λ: r ↦r X, and define ϕ : Rn →R by ϕ(Y) = XtY. Note that the composite ϕλ: R →R is the identity, for ϕλ(r) = ϕ(r X) = Xtr X = r, because Xt X = i x2 i = 1. It follows that the exact sequence 0 →R λ −→Rn nat −→Rn/ im λ →0 splits. Thus, if P = Rn/ im λ, then R ⊕Rn−1 ∼= Rn ∼= R ⊕P, and P is stably isomorphic to the free R-module Rn−1 (of course, P is a projective Rmodule). Using topology, Swan proved that P is a free R-module if and only if n = 1, 2, 4 or 8. If n = 3, for example, then P is not isomorphic to Rn−1. ◀

(i) If k is any commutative ring, then Matn(k), all n × n matrices with entries in k, is a ring under matrix multiplication and matrix addition; it is commutative if and only if n = 1.