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coproduct-preserving representable

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Introduction

In this article, we collect some results about representable functors C(c,):CVC(c, -) \colon C \to V (where VV is some base of hom-enrichment) that preserve coproducts.

When CC is an extensive category regarded as enriched in SetSet, we call cc a connected object, and this terminology matches well one’s intuition about connectedness from familiar cases such as C=TopC = Top, or CC the category of graphs, etc. Some basic results with proofs may be found at connected object, including

  • A connected colimit (i.e., a colimit over a connected diagram of) connected objects is connected.

  • If XX is connected and XYX \to Y is epic, then YY is connected.

For non-extensive categories (e.g., categories of modules), the relation to “connectedness” tends to be less intuitive1. Nevertheless, the concept of (arbitrary) coproduct-preserving representable remains important, and it is useful to collect some basic information.

Absoluteness

In an Ab-enriched category, finite coproducts are absolute colimits, hence are preserved by every representable. Thus, the interest is in representables which preserve infinite coproducts as well (which is what most of this page is about).

At the other extreme from extensive categories, in a SupLat-enriched category (such as SupLat or Rel) arbitrary coproducts are absolute, and hence preserved by every representable.

Module categories

Let Mod R\mathbf{Mod}_R denote the category of right modules over a ring RR, and suppose an object MM induces a coproduct-preserving hom-functor Mod R(M,):Mod RAb\mathbf{Mod}_R(M, -) \colon \mathbf{Mod}_R \to \mathbf{Ab}. Various names for such MM appear in the literature, “compact” and “dually slender” among them. In any event, a question arose on MathOverflow as to whether or to what extent this condition coincides with the condition of being finitely generated, for various rings RR.

The information provided below was mostly culled from the answers given to that question, especially those by Pierre-Yves Gaillard and Fernando Muro. Here and there some minor details and background information have been filled in, and some key results have been slightly rearranged.

Positive results

We begin with some easy preliminary remarks. Given a family of objects {B i} iI\{B_i\}_{i \in I} in an Ab\mathbf{Ab}-enriched category CC and a functor F:CAbF \colon C \to \mathbf{Ab}, there is a canonical arrow

iF(B i)F( iB i)\oplus_i F(B_i) \to F(\oplus_i B_i)

and if this arrow is an isomorphism for every family B iB_i, we say FF preserves coproducts. Turning to the case of representable functors on modules, let

p j: iB iB jp_j \colon \oplus_i B_i \to B_j

be the obvious projection (p ji j=1 B jp_j \circ i_j = 1_{B_j}, else p ji k=0p_j \circ i_k = 0), and given f:M iB if \colon M \to \oplus_i B_i, put

f jp jf.f_j \coloneqq p_j \circ f.

Then Mod R(M,)\mathbf{Mod}_R(M, -) preserves the particular coproduct iB i\oplus_i B_i if for each f:M iB if \colon M \to \oplus_i B_i, we have f j=0f_j = 0 for all but finitely many jj.

Clearly Mod R(R,)\mathbf{Mod}_R(R, -) preserves coproducts, and if FF, GG are coproduct-preserving functors Mod RAb\mathbf{Mod}_R \to \mathbf{Ab}, then so is FGF \oplus G. It follows that

  • Mod R(R n,)\mathbf{Mod}_R(R^n, -) preserves coproducts.
Proposition

If Mod R(M,)\mathbf{Mod}_R(M, -) preserves coproducts and q:MNq \colon M \to N is epic, then Mod R(N,)\mathbf{Mod}_R(N, -) preserves coproducts.

Proof

Given f:N iB if \colon N \to \oplus_i B_i, we have p jfq=0p_j \circ f \circ q = 0 for all but finitely many jj, whence f j=p jf=0f_j = p_j \circ f = 0 for all but finitely many jj since qq is epic.

Combining the two preceding observations, we infer that

  • Mod R(M,)\mathbf{Mod}_R(M, -) preserves coproducts if MM is finitely generated.

Here is a sharper description of coproduct-preserving representables, based on subobject lattices.

Theorem

Mod R(M,)\mathbf{Mod}_R(M, -) preserves coproducts if and only if the union of every countable chain of proper submodules of MM is a proper submodule.

Proof

(As adapted from Gaillard’s answer.) Let M 0M 1M_0\subset M_1\subset\cdots be a chain of proper submodules of MM whose union is MM, and put Q n=M/M nQ_n = M/M_n. Since for each mMm \in M we have q n(m)=0q_n(m) = 0 for all but finitely many nn, the map

q=q n:M nQ n,q = \langle q_n \rangle \colon M \to \prod_n Q_n,

corresponding to the tuple of quotient maps q n:MQ nq_n \colon M \to Q_n, factors through the inclusion nQ n nQ n\oplus_n Q_n \hookrightarrow \prod_n Q_n. However, since each q nq_n is nonzero, qq does not belong to the subgroup

nMod R(M,Q n) nMod R(M,Q n)Mod R(M, nQ n)\oplus_n \mathbf{Mod}_R(M, Q_n) \hookrightarrow \prod_n \mathbf{Mod}_R(M, Q_n) \cong \mathbf{Mod}_R(M, \prod_n Q_n)

and thus the canonical map nMod R(M,Q n)Mod R(M, nQ n)\oplus_n \mathbf{Mod}_R(M, Q_n) \to \mathbf{Mod}_R(M, \oplus_n Q_n) is not an isomorphism.

In the other direction, if Mod R(M,)\mathbf{Mod}_R(M, -) does not preserve coproducts, then we can find some map

f:M iIB if \colon M \to \oplus_{i\in I} B_i

not belonging to the subgroup iIMod R(M,B i)Mod R(M, iIB i)\oplus_{i \in I} \mathbf{Mod}_R(M, B_i) \hookrightarrow \mathbf{Mod}_R(M, \oplus_{i \in I} B_i). This means that infinitely many components f i:MB if_i \colon M \to B_i are nonzero. Choose a countable subset NIN \subset I such that f n:MB nf_n \colon M \to B_n is nonzero for every nNn \in N, and put

M n knker(f k).M_n \coloneqq \bigcap_{k \geq n} \ker(f_k).

Each M nM_n is a proper submodule of MM, and the M nM_n form a nondecreasing chain, but the union of the M nM_n is MM (because for each mMm \in M, only finitely many f n(m)f_n(m) can be nonzero).

Theorem

Let RR be a Noetherian ring, and suppose MNM \to N is a monomorphism of RR-modules. Then if Mod R(N,)\mathbf{Mod}_R(N, -) is coproduct-preserving, so is Mod R(M,)\mathbf{Mod}_R(M, -).

Proof

(As adapted from Muro’s answer.) Consider a family B iB_i of modules, and a map f:M iB if \colon M \to \oplus_i B_i. Since there are enough injectives, there exists an embedding i j:B jE ji_j \colon B_j \to E_j in an injective module, for each jj. Next, as explained here, the Noetherian assumption allows us to infer that iE i\oplus_i E_i is injective. Thus, there exists gg such that the diagram

M N f g jB j ji j jE j\array{ M & \to & N \\ ^\mathllap{f} \downarrow & & \downarrow^\mathrlap{g} \\ \oplus_j B_j & \underset{\oplus_j i_j}{\to} & \oplus_j E_j }

commutes. Because Mod R(N,)\mathbf{Mod}_R(N, -) preserves coproducts, we have g j=0g_j = 0 for all but finitely many jj. Since the diagram

M N f j g j B j i j E j\array{ M & \to & N \\ ^\mathllap{f_j} \downarrow & & \downarrow^\mathrlap{g_j} \\ B_j & \underset{i_j}{\to} & E_j }

commutes and i ji_j is injective, we see f j=0f_j = 0 for all but finitely many jj, whence Mod R(M,)\mathbf{Mod}_R(M, -) preserves coproducts.

Corollary

Let RR be Noetherian. If Mod R(M,)\mathbf{Mod}_R(M, -) preserves coproducts, then MM is finitely generated.

Proof

(Combining Gaillard’s and Muro’s answers.) We prove the contrapositive. Suppose MM is not finitely generated. Then we can find a strictly increasing sequence of submodules of MM:

M 1M 2M.M_1 \hookrightarrow M_2 \hookrightarrow \ldots \hookrightarrow M.

Let MM' be the union of the M iM_i. By Theorem 1, the representable Mod R(M,)\mathbf{Mod}_R(M', -) does not preserve coproducts. By Theorem 2, we infer that Mod R(M,)\mathbf{Mod}_R(M, -) does not preserve coproducts.

  • Remark: A ring for which finitely generated modules coincide with modules MM such that Mod R(M,)\mathbf{Mod}_R(M, -) is coproduct-preserving is called steady. Thus, Noetherian rings are steady. Cf. Martin Brandenburg’s answer.

Negative results

Next, we construct an example of a ring RR and an RR-module MM such that Mod R(M,)\mathbf{Mod}_R(M, -) preserves coproducts but MM is not finitely generated.

Lemma

A module MM is finitely generated if and only if the union of a totally ordered family of proper submodules of MM is a proper submodule.

Proof

The following proof is a practically verbatim transcription from Gaillard’s answer, modulo some notational changes. Assume that MM is not finitely generated. Let PP be the set of those submodules NN of MM such that M/NM/N is not finitely generated, ordered by inclusion. Clearly the poset PP is nonempty and has no maximal element. By (the contrapositive of) Zorn’s Lemma, there is a nonempty totally ordered subset TPT \hookrightarrow P which has no upper bound. Letting UU be the union of the submodules occurring in TT, we see that M/UM/U is finitely generated. There is thus a finitely generated submodule FF of MM which generates MM modulo UU. Then the collection

{N+F:NT}\{N+F: N \in T\}

is a totally ordered set of proper submodules whose union is MM.

Thus, our task is to construct a ring RR and an RR-module MM such that every countable chain of proper submodules of MM is bounded above by a proper submodule of MM (cf. Theorem 2), but admitting an uncountable chain of proper submodules whose union is MM (so that MM is not finitely generated, by the preceding lemma 1). The solution as presented below is essentially from the answers of Gaillard and Brandenburg, with a few extra glosses.

The task is more or less straightforward if we just remember that valuation rings can model arbitrarily complicated rates of growth, i.e., in the present case, we just want to build a valuation field with uncountable supply of rates of growth (or of degrees of infinite/infinitesimal elements). Thus, consider for example the free abelian group generated by the first uncountable ordinal, and make this a totally ordered group GG by imposing the lexicographic order. This is the value group of a valuation field KK whose elements are Hahn series, formally described as functions

f:Gf \colon G \to \mathbb{R}

whose support is well-ordered when considered as a subset of the opposite order G opG^{op}. (More suggestively, we think of ff as a formal series

gGa gx g\sum_{g \in G} a_g x^g

where a g=f(g)a_g = f(g) and xx is an indeterminate viewed as a generic infinite element, with obvious rules for adding and multiplying. The well-ordering condition is used to ensure that the rules for addition and multiplication are well-founded, and the value v(f)v(f) of such a series is the least gG opg \in G^{op} lying in the support of ff. That is to say, the greatest gGg \in G that indexes a non-zero coefficient a ga_g.)

Now let RR be the valuation ring consisting of bounded elements of KK, i.e., those f:Gf \colon G \to \mathbb{R} in KK where f(g)=0f(g) = 0 whenever gg is greater than the identity element of GG. Then KK is the field of fractions of RR, and we may regard KK as an RR-module.

For each tω 1t \in \omega_1, let R tR_t be the RR-submodule of KK consisting of all those f:Gf \colon G \to \mathbb{R} in KK such that v(f)tv(f) \leq t. These R tR_t are “principal fractional ideals”, and they form a system of proper submodules which is cofinal in the lattice of RR-submodules of KK, ordered by inclusion.

By cofinality, any countable chain of submodules of KK is bounded above by some R tR_t, so that Mod R(K,)\mathbf{Mod}_R(K, -) preserves coproducts. However, the union of all the R tR_t is KK, so that KK is not finitely generated.

References

  • Sasha (mathoverflow.net/users/2095), “Sums-compact” objects = f.g. objects in categories of modules?, Question 59282 (version: 2011-11-19) (link)
  • Fernando Muro (mathoverflow.net/users/12166), “Sums-compact” objects = f.g. objects in categories of modules?, Answer 59320 (version: 2011-03-23) (link)
  • Pierre-Yves Gaillard (mathoverflow.net/users/461), “Sums-compact” objects = f.g. objects in categories of modules?, Answer 81333 (version: 2011-11-29) (link)
  • Martin Brandenburg (mathoverflow.net/users/2841), “Sums-compact” objects = f.g. objects in categories of modules?, Answer 81623 (version: 2011-11-22) (link)

Footnote


  1. The divide between extensive categories and categories of modules is somewhat analogous to the divide between the classical particles and quantum particles. In the classical picture, if an elementary particle (which we can think of as “connected”) is in a state described by a union U+VU + V, then it is either in state UU or in state VV. Whereas in the quantum picture, one has to consider superpositions UVU \oplus V of states UU, VV, and the usual sort of classical logic of disjunctions breaks down. Notice that classical logic (meaning here, non-quantum logic) is largely derived from our experience with extensive categories such as toposes.

Revised on February 24, 2013 19:55:53 by Todd Trimble (67.81.93.26)