Varieties Without Minimal Generators

Hans–E. Porst  Jiří Rosický^{∗}

Abstract

While many familiar varieties have a minimal varietal generator, i.e., a regular projective finitely presentable regular generator such that none of its retracts is a regular generator, and even a unique one, we present (a) a variety having no minimal varietal generator at all and (b) a variety having two nonisomorphic minimal varietal generators. Moreover we demonstrate that the same effects can happen with respect to a weaker notion of minimality and are common even in module categories.

Mathematics Subject Classification (1991): Primary 18C05, Secondary 03C05, 08B99

Keywords: Varieties, regular projective regular generator, equational representation of varieties.

Introduction

It is well known that the varieties W Morita equivalent to a given variety V are determined by the varietal generators of V, i.e., by those regular generators G of V which are finitely presentable and regular projective: the Lawvere theory ThW of W is isomorphic to the theory Th_{V}G for some varietal generator G in V (where Th_{V}G is the dual of the full subcategory of V spanned by all finite copowers nG, one for each n ∈ ℕ). Thus, the possible equational representations of V are given by its varietal generators. Given varietal generators G and Q, there exists a retraction r: nG ⟶ Q for some n ∈ ℕ. Given a retraction r: G ⟶ Q (with section s: Q ⟶ G) between non-isomorphic varietal generators G and Q of the variety V the equational representation of V by means of Q is simpler than the representation given by G in the following sense:

1. for each V in V the underlying set hom_{V}(Q, V) is “smaller” than the underlying set hom_{V}(G, V) since hom(r, V) embeds the first into the latter; 2. the theory Th_{V}Q has “fewer” morphisms than Th_{V}G, since the theory-mor-

∗Partially supported by the Grant Agency of the Czech Republic under the grant No.

201/99/0310

phism ū: Th_{V}G ⟶ Th_{V}Q defined by

ū(jG ⟶^{t} kG) = jQ ⟶^{js} jG ⟶^{t} kG ⟶^{kr} kQ

is surjective on morphisms: for each jQ ⟶^{τ} kQ there exists, by projectivity of jG, some jG ⟶^{t} kG with kr ∘ t = τ ∘ jr; then ū(t) = τ (see [8] for details).

In order to find an as simple as possible equational representation of V one, therefore, has to look for a varietal generator G of V which is, in some sense, minimal with respect to the following preorder on (isomorphism classes of) varietal generators:

H ⊑ G iff H is a retract of G.

Definition. A varietal generator G in a variety V is called minimal provided none of its proper^{1} retracts is a regular (hence varietal) generator. G will be called preminimal provided that, for each varietal generator H with H ⊑ G, one also has G ⊑ H.

Note that a varietal generator G is preminimal iff [G] is minimal in the order associated to the preorder ⊑. A minimal varietal generator is preminimal.

In many familiar varieties as, e.g., Ab, Ring, and Bool, the representing objects

of the canonical underlying functors (ℤ, ℤ[X], and [figure: •; •; •; •] respectively) are the

only minimal varietal generators (see [9]). In contrast to these observations — and maybe contradicting algebraic intuition — we will show in this note that

• a variety need not to have a (pre)minimal varietal generator, and

• a variety might have even two non-isomorphic (pre)minimal varietal generators.

We are going to use two completely different methods to produce such varieties. The first uses fundamental results of categorical algebra while the second is based on a result of Bergman [5] and shows that (some of) the varieties in question can even be chosen to be categories of R-modules for suitable rings R.

The following immediate consequence of the definitions will be useful here.

Lemma 1 If P is a preminimal varietal generator in V then either P is minimal or there exists a preminimal varietal generator Q in V such that P and Q are not isomorphic but mutually retracts of each other.

1We call a retract G′ of G proper, if G′ is not isomorphic to G; observe that a retract might be improper even if the retraction under consideraton is not an isomorphism.

1 Varieties without minimal generators

The varieties in this section are constructed as model categories of finite product sketches. The corresponding facts important for this note are collected as follows: If A is a small category with some specified set of finite products, we denote by ModA the category of all Set-valued functors preserving all the specified finite products. This category is equivalent to a (many-sorted) variety. A^{op} is contained as a full subcategory in ModA via Yoneda embedding. The dual of the closure of A^{op} in ModA under finite coproducts is called the finite product theory FP(A) generated by A; it is again a sketch by all its finite products. FP(A)^{op} is the full subcategory of ModA consisting of the finitely generated free algebras in ModA w.r.t. to the evaluation functor ModA ⟶ Set^{obA}. FP(A) sketches the same variety as A (i.e., ModA ≃ ModFP(A)) and its dual is a regular generator of ModA (see [3, 8.4, Theorem 1]).

A given variety V can have several (non-equivalent) finite product theories. But all these theories have the same Cauchy-completion which is the dual of the full subcategory Proj_{fp}V of V consisting of its finitely presentable regular projectives. Following [7, 11.10], (Proj_{fp}V)^{op} is the only Cauchy-complete finite product theory sketching V (see [1] and [8] for more details). We can collect the just stated facts as follows.

Theorem 1 For each small category A with a specified set of finite products the following hold:

1. There are equivalences

ModA ≃ ModFP(A) ≃ Mod(Proj_{fp}ModA)^{op}

given by extending the models.

2. The subcategory FP(A)^{op} is a regular (even a dense) generator of ModA.

3. If A is Cauchy-complete, so is FP(A) and FP(A) = (Proj_{fp}ModA)^{op}. If, in addition, A has all finite products one even has A = FP(A) = (Proj_{fp}ModA)^{op}.

The resulting one-to-one correspondence between Cauchy-complete categories with finite products and varieties can be extended to a dual biequivalence. Morphisms among Cauchy-complete categories with finite products then are finite product preserving functors while morphisms among varieties are right adjoints preserving filtered colimits and regular epimorphisms.

Theorem 2 ([1]) The passage A ⟶ ModA serves as a dual biequivalence between small Cauchy-complete categories with finite products and varieties.

Remark 1 If, in a Cauchy complete category A with finite coproducts, there exists some object G such that each A-object A is a retract of some finite copower of G, the variety determined by A is even one-sorted. In fact ModA is equivalent to the variety V with Lawvere theory ThV = Th_{A^{op}}G made up by G and its finite powers in A. ThV is, as a category, equivalent to FP(A).

1.1 A variety without preminimal generators

We are going to sketch a variety as follows:

Let ℕ_{i} = {n ∈ ℕ | n ≥ i} for i ∈ ℕ. Denote by μ_{i} and ν_{i} the maps ℕ_{i+1} ⟶ ℕ_{i} with μ_{i}(i + 1) = e_{i}, ν_{i}(i + 1) = o_{i} and μ_{i}(n + 1) = μ_{i}(n) + 2, ν_{i}(n + 1) = ν_{i}(n) + 2, where e_{i} is smallest even number in ℕ_{i} and o_{i} the smallest odd number in ℕ_{i} respectively. Then ℕ_{i+1} ⟶^{μ_{i}} ℕ_{i} ⟵^{ν_{i}} ℕ_{i+1} is a coproduct in Set. There is a codiagonal △_{i}: ℕ_{i} ⟶ ℕ_{i+1} defined by △_{i}μ_{i} = △_{i}ν_{i} = 1 and a symmetry s_{i}: ℕ_{i} ⟶ ℕ_{i} defined by s_{i}μ_{i} = ν_{i} and s_{i}ν_{i} = μ_{i}.

Let F be the subcategory of Set generated by the sets and maps just defined. Then the following facts are easily seen to be true:

Fact 1 For each i ∈ ℕ the diagram

ℕ_{i+1} ⟶^{μ_{i}} ℕ_{i} ⟵^{ν_{i}} ℕ_{i+1}

is a coproduct in F.

Fact 2 For i, j ∈ ℕ, i < j, ℕ_{i} is not a retract of ℕ_{j} in F.

Fact 3 For i, j ∈ ℕ, i ≠ j, the objects ℕ_{i} and ℕ_{j} are not isomorphic in F.

Fact 4 F is a Cauchy complete category.

F^{op}, together with the products specified in Fact 1 now sketches a (possibly many-sorted) variety V.

Proposition 1 V is a one-sorted variety without preminimal varietal generators.

Proof. Let us observe first that each object ℕ_{i} is — considered as a V-object — a varietal generator of V. Clearly, each ℕ_{i} belongs to Proj_{fp}V. ℕ_{0} is a varietal generator of V. Indeed, since FP(F^{op})^{op} is a regular generator of V, for each Vobject V there exists a regular epimorphism ∐_{J} ℕ_{i_{j}} ⟶^{q} V. Now, for each k ∈ ℕ, by definition of F there exists a retraction r_{k}: ℕ_{0} ⟶ ℕ_{k}. Hence there results a regular epimorphism

J · ℕ_{0} \underrightarrow{∐_{J} r_{i_{j}}} ∐_{J} ℕ_{i_{j}} ⟶^{q} V.

Thus, ℕ_{0} is a regular generator of V, and V is one-sorted.

But then also each ℕ_{i} is a regular (hence varietal) generator; in fact each regular quotient X·ℕ_{0} → V is a regular quotient Y ℕ_{i} → V since, for each i ∈ ℕ, ℕ_{0} = 2^{i}ℕ_{i}.

Assume now that G is a preminimal varietal generator. By Theorem 1 and Fact

4 G = ∐_{j=1}^{k} ℕ_{i_{j}}. With m = min{n_{i_{j}}} and n = max{n_{i_{j}}} + 1 there are retractions

G → ℕ_{m} and G → ℕ_{n}. Since G is preminimal G has to be a retract of ℕ_{n}. It follows that ℕ_{m} is a retract of ℕ_{n} which contradicts Fact 2 since m < n. ◇

1.2 A universal variety with two non-isomorphic preminimal generators

We define a Cauchy complete category as follows.

Let F be the free category over the graph

[figure: Q_{0}; Q_{1}; σ; ρ; s; r]

subject to the equations

rs = 1_{Q_{1}} and ρσ = 1_{Q_{0}}.

Lemma 2 Let v be a morphism in F, v not an identity. Then v is of precisely one of the following types:

Type 1 a composition of finitely many copies of r and ρ;

Type 2 a composition of finitely many copies of s and σ;

Type 3 a composition w_{2} ∘ w_{1}, where w_{1} is of type 1 and w_{2} is of type 2.

Proof. Since none of the morphisms r, s, ρ, σ is an isomorphism the obove types exclude each other.

Assume v is neither of type 1 nor of type 2. v can be written as a composition of r, s, ρ, σ containing no sequence rs and ρσ. Decompose v as

v = w_{n} ∘ … ∘ w_{1}

such that each w_{i} is a composition of either retractions r and ρ or sections s and σ and, for each i < n, if w_{i} consists of sections (retractions), w_{i+1} consists of retractions (sections).

Now only one of the segments w_{i} consists of sections only. For, if w_{i} = s ∘ …, w_{i+1} must be of the form w_{i+1} = … ∘ r since the composition w_{i+1} ∘ w_{i} is defined. But this is impossible since v contains no sequence rs. Similarly for w_{i} = σ ∘ . . ..

Hence v = w_{2} ∘ w_{1} where w_{1} consists of retractions and w_{2} consists of sections only. ◇

Fact 5 Q_{0} and Q_{1} are not isomorphic in F.

Proof. None of the types of morphisms listed in the Lemma allows for an isomorphism. ◇

Fact 6 F is Cauchy complete.

Proof. Any idempotent v—say v: Q_{0} → Q_{0}—can, by the Lemma, be decomposed as v = w_{2} ∘ w_{1}.

Case 1: cod w_{1} = dom w_{2} = Q_{0} Then \begin{aligned}w_{1} & = & ρr … ρr & = & (ρr)^{n} \\ w_{2} & = & sσ … sσ & = & (sσ)^{m} \end{aligned}

Now v^{2} = v means

(sσ)^{m}(ρr)^{n}(sσ)^{m}(ρr)^{n} = (sσ)^{m}(ρr)^{n}

which implies (since sections are monomorphisms and retractions epimorphisms respectively)

(ρr)^{n}(sσ)^{m} = 1.

Consequently n = m since otherwise r or s would be an isomorphism, contradicting Fact 5. Thus the decomposition w_{2} ∘ w_{1} is a splitting of v.

Case 2: cod w_{1} = dom w_{2} = Q_{1} Then \begin{aligned}w_{1} & = & rρ … ρr & = & (rρ)^{n}r \\ w_{2} & = & sσ … σs & = & s(σs)^{m} \end{aligned}

Here v^{2} = v means

s(σs)^{m}(ρr)^{n}rs(σs)^{m}(ρr)^{n}r = s(σs)^{m}(ρr)^{n}r

and one can conclude as above that m = n since rs = 1. Again, w_{2} ∘ w_{1} is a splitting of v. ◇

Remark 2 Instead of proving Facts 5 and 6 directly one might alternatively use the general methods developed in [10].

F (without any specified products) sketches a (possibly many-sorted) variety W with unary operations only; i.e. W = Set^{F}.

Let F̂ = FP(F). Then the following holds where we implicitely refer to Theorem 1. Note in particular that the varietal generators of W (if they exist) are precisely those F̂-objects G such that every F̂-object is a retract of some finite copower mG. Note also that the coproducts in F̂ are coproducts in W and that F̂, hence W, contains F as a full subcategory.

Fact 7 Every object of F̂, different from the initial one, is a varietal generator of W; in particular W is a one-sorted variety.

Proof. Let G = ∐_{I} Q_{i}, H = ∐_{j} Q_{j} be in F̂, hence Q_{i}, Q_{j} ∈ {Q_{0}, Q_{1}}, with I ≠ ∅. Choose i_{0} ∈ I and, for each k ∈ I ∪ J, a retraction

Q_{k} ⟶^{s_{k}} Q_{i_{0}} ⟶^{r_{k}} Q_{k} = 1

with r_{i_{0}} = 1. This is possible by the definition of F. Then one gets retractions

∐_{J} Q_{i_{0}} \underrightarrow{∐_{J} r_{j}} ∐_{J} Q_{j} = H

and

∐_{I} Q_{i} ⟶^{[r_{i}]=r} Q_{i_{0}}

(defined by r ∘ μ_{i} = r_{i} for i ∈ I; observe rμ_{i_{0}} = r_{i_{0}} = 1). Thus,

∐_{J} ( ∐_{I} Q_{i}) ⟶^{J·r} ∐_{J} Q_{i_{0}} \underrightarrow{∐_{J} r_{j}} H

is a retraction. ◇

Fact 8 Every object G of F̂, different from the initial one, has a retract Q in F̂, different from the initial one, which is not isomorphic to G.

Proof. Let G = ∐_{I} Q_{i} be in F̂, I ≠ ∅. By the proof of Fact 7 Q_{0} or Q_{1} is a retract of G which certainly is not isomorphic to G provided that |I| > 1 according to the construction of F̂. Finally, the case |I| = 1 is settled by Fact 5 since Q_{0} is a retract of Q_{1} and Q_{1} is a retract of Q_{0}. ◇

We summarize these results as

Proposition 2 W is a finitary one-sorted variety which has no minimal but precisely two non-isomorphic preminimal varietal generators.

Remark 3 In view of Lemma 1 the variety W constructed above has the smallest possible number of preminimal varietal generators which are not minimal.

This variety is, moreover, universal in the following sense: Let W′ be a variety with two non-isomorphic preminimal varietal generators P_{0}, P_{1}. By Lemma 1 there exist retractions P_{1} ⟶^{s̄} P_{0} ⟶^{r̄} P_{1} = id and P_{0} ⟶^{σ̄} P_{1} ⟶^{ϱ̄} P_{0} = id. Hence there exists a unique functor S: F ⟶ W′ sending Q_{i} to P_{i} and the morphisms in our starting graph to the corresponding barred ones. It is not difficult to see that S in fact embeds F into the canonical algebraic theory of W′. Thus, F is a subtheory of the canonical algebraic theory of W′ for each variety W′ with two non-isomorphic preminimal varietal generators; in other words (see Theorem 2): W is a quotient of each such variety.

2 Module categories with non-isomorphic (pre-)minimal generators

In this section we show that varieties with non-isomorphic preminimal (or even minimal) varietal generators can even appear as categories R-Mod of left R-modules for some ring R.

Concerning varietal generators in categories of modules the following is obvious:

Lemma 3 A varietal generator G in R-Mod is

• minimal iff none of its proper direct summands is a varietal generator;

• preminimal iff each varietal generator H has G as a direct summand, provided H is a direct summand of G. (G = H ⊕ H′ ⇒ H = G ⊕ G′)

Now denote, for a given ring R, by P(R) its monoid of projectives, i.e., the monoid of isomorphism classes |P | of finitely generated projective left R-modules P with the operation |P | + |Q| = |P ⊕ Q|. P(R) is always conical, (i.e., x + y = 0 ⇒ x = 0 = y) and has an order-unit, i.e., an element e ≠ 0 such that for each x there is y and n ∈ ℕ such that x + y = ne (order-units correspond to varietal generators).

Our construction is based on the following result of Bergman:

Theorem 3 ([5, 6.2, 6.4]) For any conical commutative monoid M with an orderunit there exists a ring R_{M} such that P(R_{M}) and M are isomorphic. This ring is even hereditary (i.e., submodules of projective R_{M}-modules are projective) provided that M is finitely generated.

Since finite sums of varietal generators are varietal generators and, for each module category R-Mod, the varietal generators are precisely the order units of the monoid P(R), the order units of P(R) form a subsemigroup V(R). The following is then a direct consequence of Lemma 3.

Lemma 4 If, for some ring R, the monoid V(R) is a non-trivial group, the category R-Mod has no minimal varietal generator and cardV(R) non-isomorphic preminimal varietal generators.

In order to produce conical monoids with order units the following observation is useful: for any commutative monoid M, let M^{∗} be the monoid obtained by adding a new zero-element 0, i.e., M^{∗} = M ∪ {0} and m + 0 = 0 + m = m for all m ∈ M. Then M^{∗} is conical and order units of M remain to be order units in M^{∗}. If in particular M is a group, each element of M is an order unit in M^{∗}.

Proposition 3 For any cardinal number κ > 1 there exists a ring R_{κ} such that the variety R_{κ}-Mod has no minimal varietal generator, but κ non-isomorphic preminimal ones.

Proof. Chose an Abelian group M of cardinality κ. Then R_{M^{∗}} has the required properties by Lemma 4. ◇

Proposition 4 There exists a ring R such that the variety R-Mod has two nonisomorphic minimal varietal generators.

Proof. Let M be the free commutative monoid with two generators p, g satisfying p^{2} = g^{2}. Then M consists of elements 1, p^{n}g^{m}, n, m ∈ ℕ. M is conical and both, p and g are order units because

\begin{aligned}(p^{n}g^{m})g^{m} & = & p^{n}g^{2m} & = & p^{n}p^{2m} & = & p^{n+2m} \\ p^{n}(p^{n}g^{m}) & = & p^{2n}g^{m} & = & g^{2n}g^{m} & = & g^{2n+m} \end{aligned}

Moreover p ∤ g and g ∤ p. Hence R_{M}-Mod has two non-isomorphic minimal varietal generators corresponding to p and g. ◇

3 When minimal generators do exist

Here we present sufficient conditions for the existence of minimal varietal generators.

Proposition 5 Let V be a variety where finitely generated objects are finitely presentable or regular projective objects are closed under directed colimits. Then V has a minimal varietal generator.

Proof. Assume that V does not have a minimal varietal generator. Then there is a descending chain

P_{0} ⟶^{r_{0}} P_{1} ⟶^{r_{1}} P_{2} ⟶ …

of finitely presentable regular projectives and proper retractions. Let P_{ω} be a colimit of this chain. Since the colimit component p_{0} : P_{0} ⟶ P_{ω} is a regular epimorphism (recall that the colimiting cone (p_{i}) is jointly surjective), P_{ω} is finitely generated. If it is finitely presentable, some of the p_{i}: P_{i} ⟶ P_{ω} has a right inverse s; hence p_{i} = p_{i} ∘ s ∘ p_{i}. Due to the construction of directed colimits in varieties, for each a ∈ P_{i}, there is some j > i such that r_{ij}(a) = r_{ij}(sp_{i}(a)) where r_{ij} = r_{j−1} ∘ … ∘ r_{i}. Since P_{i} is finitely generated one even has, for a suitable j, r_{ij} = r_{ij} ∘ sp_{i} = r_{ij} ∘ sp_{j} ∘ r_{ij}, which implies r_{ij} ∘ sp_{j} = 1. Therefore p_{j} is an isomorphism. Hence r_{j} is an isomorphism, which is impossible.

If P_{ω} is regular projective then it is again finitely presentable because finitely generated regular projectives are finitely presentable. In fact, a finitely generated regular projective P admits is a retraction P ⟶^{s} F ⟶^{r} P = id with a finitely generated free object F and, hence a coequalizer diagram

[figure: F; 1; sr; F; P; r]

Thus P is finitely presentable. ◇

Corollary 1 Let R be either a left Noetherian ring or a left perfect ring. Then R-Mod has a minimal varietal generator.

Proof. If R is left Noetherian (i.e., left ideals of R are finitely generated) then any finitely generated flat left R-module is projective ([6, 11.31]). Hence any finitely generated directed colimit of projective modules is projective ([6, 11.32]). By the proof of Proposition 5, R-Mod has a minimal varietal generator.

If R is left perfect (i.e., satisfies d.c.c. on principal right ideals), every flat left module is projective (see [4]). Hence projective left R-modules are closed under directed colimits. ◇

Corollary 2 ([9]) The categories Grp of groups, Bool of Boolean algebras and cRng of commutative unital rings have a (unique) minimal varietal generator.

Proof. For groups this follows from the fact that projective groups are free. The free group on one generator is the unique minimal varietal generator in Grp. In the remaining categories under consideration the finitely generated objects are finitely presentable: in Bool every finitely generated algebra is finite; in cRng it is a quotient of ℤ[X_{1}, … X_{n}] for some natural number n. But each congruence on ℤ[X_{1}, … X_{n}] is finitely generated. For the unicity arguments see [9]. ◇

References

[1] J. Adámek, F.W. Lawvere and J. Rosický. On a duality between varieties and algebraic theories. Preprint 1999.

[2] J. Adámek and J. Rosicky. Locally presentable and accessible categories. Cambridge University Press, 1994.

[3] M Barr and C. Wells. Toposes, Triples and Theories. Springer-Verlag, New York 1985.

[4] H. Bass. Finitistic dimension and a homological generalization of semiprimary rings. Trans. Amer. Math. Soc. 95 (1960), 466–488.

[5] G. M. Bergman. Coproducts and some universal ring constructions. Trans. Amer. Math. Soc. 200 (1974), 33–88.

[6] C. Faith Algebra: Rings, Modules and Categories. Springer-Verlag, Berlin 1973.

[7] P. Gabriel und F. Ulmer. Lokal präsentierbare Kategorien. Lecture Notes in Math. 221. Springer, Berlin - New York, 1971.

[8] Hans-E. Porst. Equivalence of varieties. Preprint 1998.

[9] Hans-E. Porst. Minimal Generators in Varieties. To appear in Appl. Categorical Structures.

[10] L. Schröder. Kompositionsgraphen und freie Erweiterungen von Kategorien. PhD-Dissertation, Bremen 1999.

University of Bremen  Masaryk University
P.O. Box 330440  Janáčkovo nám 2a
28334 Bremen, Germany  66295 Brno, Czech Republic

 porst@math.uni-bremen.de                     rosicky@math.muni.cz