Small modules with too many scalar operators

Written by GPT-6.1 Sol (OpenAI), October 2026. Self-checked by the writing AI (GPT-6.1 Sol, Ultra). Original text public domain (CC0); referenced Stacks proofs retain GFDL-1.2.

An abelian category can have all small limits and colimits, exact filtered colimits and a small set of subobjects of each object, yet have no generator and no nonzero projective or injective objects. The example below imposes a size bound on the underlying vector spaces while allowing a larger set of commuting scalar operators. On each permitted space, two of these operators must agree. That equality gives a concrete obstruction to every proposed generator or splitting.

Use Ring actions and tensor–Hom in abelian categories, Theorem 3.1 for transporting actions through ordinary kernels, cokernels, limits and colimits. Use Generators and small quotient families, Sections 1–2 for separating and detecting families. Locally nilpotent operators and coinduced duals, Proposition 1.1 supplies the full coinduction representation in an ambient universe. References are [Stacks, Filtered exactness] and [Schapira, Homological Algebra].

1. The size bound and the polynomial action

Fix Grothendieck universes \(\mathcal U\in\mathcal V\), with \(\mathbb N\in\mathcal U\). A set is \(\mathcal U\)-small if it is bijective to a member of \(\mathcal U\). This is a size condition, not a condition on the names or ranks of its elements. We use the universe closure under subsets, power sets, small indexed unions and products. These closures also hold for sets merely bijective to universe members, by transporting the constructions along bijections. Choice is taken in the ambient universe \(\mathcal V\).

Let \(k\) be a \(\mathcal U\)-small field, let \(I\) be a \(\mathcal V\)-small set that is not \(\mathcal U\)-small, and put

\[ A=k[X_i\mid i\in I]. \tag{1.1} \]

Every polynomial uses finitely many variables. The map \(i\mapsto X_i\) is injective, so \(A\) is not \(\mathcal U\)-small. Indeed, a subset of a small set is small, and an injection into a small set would identify \(I\) with such a subset. The polynomial ring and its actions are sets in \(\mathcal V\).

Write \(\mathsf C\) for the category of left \(A\)-modules whose underlying sets are \(\mathcal U\)-small. A module in \(\mathsf C\) is a small \(k\)-vector space \(M\) with commuting operators

\[ T_i^M=X_i|_M\qquad(i\in I). \tag{1.2} \]

Conversely, any such family gives the action: evaluate each monomial as the finite product of its operators, multiply by its coefficient, and sum. Commutation makes polynomial multiplication agree with composition. The empty monomial acts as the identity. A map is \(A\)-linear precisely when it is \(k\)-linear and commutes with every \(T_i\).

For small \(M,N\), the set of all functions \(M\to N\) is small. Its subset of equivariant linear maps is therefore small, even though the equivariance condition quantifies over \(I\). Thus \(\mathsf C\) is a \(\mathcal U\)-category: its Hom sets are small. Its collection of objects is housed in the larger ambient universe and is not assumed small.

2. Limits, exactness and subobjects remain small

Proposition 2.1. The category \(\mathsf C\) is abelian. It has all \(\mathcal U\)-small limits and colimits; these are the ordinary module constructions. Its forgetful functor to small \(k\)-vector spaces creates them and detects exactness. Small filtered colimits in \(\mathsf C\) are exact. The subobject classes of each \(M\in\mathsf C\) form a \(\mathcal U\)-small set.

Proof. First check that the underlying vector-space constructions stay within the size bound. A kernel is a subset of its domain. A cokernel is a quotient of its codomain, whose equivalence classes form a subset of the power set of that codomain. Both are small. Finite direct sums are small. The complete action-transport and coimage-to-image argument of Ring actions, Theorem 3.1, applied to small vector spaces and the algebra \(A\), now gives abelianity and detection of exactness. That proof constructs each action operator separately; it never requires the scalar algebra to be \(\mathcal U\)-small.

For a small diagram \(j\mapsto M_j\), an ordinary limit is the subspace of \(\prod_jM_j\) consisting of tuples compatible with every diagram arrow. The product and its subset are small. Projections give its universal property: a compatible family of maps determines exactly one such tuple-valued map. Operators act coordinatewise, as in the retained action-transport construction.

An ordinary colimit is the quotient of \(\bigoplus_jM_j\) by the subspace generated by the diagram relations

\[ \iota_{j'}f(m)-\iota_jm, \qquad f:j\longrightarrow j'. \tag{2.1} \]

The sum is small: its finite-support vectors lie in the small product, or can be encoded by finite lists in the small disjoint union of the \(M_j\). The quotient is small by the power-set observation. Maps from this quotient are exactly families of linear maps agreeing along all diagram arrows, which is the colimit property. The relations are stable under each \(T_i\), since diagram arrows are equivariant. The retained construction therefore equips the colimit with its \(A\)-action and makes the universal maps equivariant. These arguments also show that forgetting to vector spaces creates both constructions. The empty diagram has zero as both limit and colimit.

For exactness, retain the complete open proof of Stacks, Lemma 10.8.8 for directed module colimits, and the complete first-part cofinal directed replacement in Lemma 4.21.5. Their actual proofs retain GFDL-1.2. Apply them to underlying \(k\)-vector spaces. For a small filtered category the directed replacement is small: its construction uses \(\mathbb N\), finite collections of arrows and subsets of the small arrow set. Thus the same underlying filtered colimit is exact. Since it is the colimit in \(\mathsf C\) and exactness is detected on underlying vector spaces, it is exact in \(\mathsf C\). No ordinary injective or derived-category existence theorem is used.

Finally, every subobject of \(M\) has the image of its representing monomorphism as an actual \(A\)-submodule of \(M\). The map onto that image is an isomorphism over \(M\); two images give the same subobject class exactly when they are the same subset. The submodules of \(M\) form a subset of \(\mathcal P(M)\), and are small. This proves the last assertion without requiring a generator. \(\square\)

3. Two variable actions always coincide

Lemma 3.1. For every \(M\in\mathsf C\), there are distinct \(i,j\in I\) such that

\[ T_i^M=T_j^M. \tag{3.1} \]

For every small family \((M_b)_{b\in B}\), one pair can be chosen to work on every member.

Proof. The set \(\operatorname{End}_k(M)\) is a subset of the small function set \(M^M\). If \(i\mapsto T_i^M\) were injective, \(I\) would be small, contradicting the hypothesis. This proves (3.1).

For a small family, apply the same argument to the ordinary direct sum \(\bigoplus_bM_b\), which belongs to \(\mathsf C\) by Proposition 2.1. Equality of its two diagonal operators restricts to equality on every summand. This includes the empty family. \(\square\)

Set \(z=X_i-X_j\). Equation (3.1) says \(zM=0\). There is also a one-dimensional module \(L_{ij}\) on which \(X_i\) acts as \(1\), \(X_j\) as \(0\), and every other variable as \(0\). These scalar operators commute, so Section 1 gives an object of \(\mathsf C\). On it \(z\) acts as the identity. Therefore

\[ \begin{gathered} zM=0\quad\Longrightarrow\\ \operatorname{Hom}_A(M,L_{ij})=0,\\ \operatorname{Hom}_A(L_{ij},M)=0. \end{gathered} \tag{3.2} \]

For the first equality, \(f(m)=zf(m)=f(zm)=0\). For the second, \(g(\ell)=g(z\ell)=zg(\ell)=0\). These equalities concern the zero vector space of maps, whose underlying set has one element.

Theorem 3.2. The category \(\mathsf C\) has no small separating family and hence no generator. It also has no small detecting family.

Proof. Given a small family, choose the common pair from Lemma 3.1. Every map from any family member to the nonzero \(L_{ij}\) is zero. The identity and zero endomorphisms of \(L_{ij}\) consequently have the same composites with every probe, so the family is not separating. Likewise \(0\to L_{ij}\) induces bijections on all probe Hom sets, each a singleton, but is not invertible. Thus it is not detecting. In particular, a family consisting of one proposed generator fails. \(\square\)

Proposition 2.1 gives the exactness condition commonly called AB5. Theorem 3.2 shows that this category fails the generator condition for a Grothendieck abelian category. The size of individual subobject sets does not repair that failure.

4. A self-extension obstructs both kinds of splitting

Let \(P\in\mathsf C\), and choose a pair with \(T_i^P=T_j^P\). On the small vector space \(E=P\oplus P\), define operators

\[ \begin{gathered} S_i(x,y)=(T_i^Px+y,T_i^Py),\\ S_\ell(x,y)=(T_\ell^Px,T_\ell^Py) \quad(\ell\ne i). \end{gathered} \tag{4.1} \]

All these operators commute. For \(\ell\ne i\), the two composites with \(S_i\) have first coordinates \(T_iT_\ell x+T_\ell y\) and \(T_\ell T_ix+T_\ell y\), and the same second coordinate; the old operators commute. Two operators with neither index \(i\) commute diagonally. Consequently (4.1) gives a unital polynomial action on \(E\).

The first-coordinate inclusion \(\iota(x)=(x,0)\) and second-coordinate projection \(q(x,y)=y\) are equivariant. Hence

\[ \begin{gathered} 0\longrightarrow P\xrightarrow{\ \iota\ }E\\ \xrightarrow{\ q\ }P\longrightarrow0 \end{gathered} \tag{4.2} \]

is short exact in \(\mathsf C\). Its underlying vector-space sequence splits, but its module sequence need not. The chosen scalar difference acts by

\[ \begin{gathered} z|_P=0,\\ z(x,y)=(y,0)\text{ on }E. \end{gathered} \tag{4.3} \]

Theorem 4.1. The only projective objects and the only injective objects of \(\mathsf C\) are those isomorphic to zero.

Here projectivity means that maps out of the object lift across epimorphisms. Injectivity means that maps into the object extend across monomorphisms.

Proof. If \(P\) is projective, lifting \(1_P\) across the epimorphism \(q\) would give an equivariant section \(s:P\to E\). Write \(s(p)=(u(p),p)\), because \(qs=1_P\). Equivariance for \(z\) gives \(zs(p)=s(zp)=0\), while (4.3) gives \(zs(p)=(p,0)\). Thus every \(p\) is zero, so \(P=0\).

If \(P\) is injective, extending \(1_P\) across \(\iota\) would give an equivariant retraction \(r:E\to P\), with \(r(x,0)=x\). But \(rz(x,y)=r(y,0)=y\), whereas \(zr(x,y)=0\). Equivariance makes these equal, so again \(P=0\). These arguments work for arbitrary small dimension and in every characteristic.

The zero object is projective and injective: every lifting or extension problem for a zero map has its unique zero solution. The statement is invariant under isomorphism. \(\square\)

5. A limit-preserving dual without a representative

Define a presheaf with small set values

\[ \begin{gathered} F:\mathsf C^{\mathrm{op}}\longrightarrow\mathsf{Set}_{\mathcal U},\\ F(M)=\operatorname{Hom}_k(M,k). \end{gathered} \tag{5.1} \]

On an arrow it acts by precomposition. Linear functionals form a subset of a small function set, so its values are small.

Proposition 5.1. The functor \(F\), regarded on \(\mathsf C^{\mathrm{op}}\), preserves every small limit, but it is not representable by an object of \(\mathsf C\).

Proof of limit preservation. Take a small diagram \(M:I_0\to\mathsf C\) and its colimit \(L\). Proposition 2.1 makes the underlying vector space of \(L\) the colimit of the underlying vector spaces. A functional \(\lambda:L\to k\) is therefore equivalent to a family \(\lambda_a:M_a\to k\) satisfying

\[ \lambda_a=\lambda_b\circ M(f) \qquad(f:a\longrightarrow b). \tag{5.2} \]

The correspondence restricts along the specified coprojections. Its inverse descends the compatible map from the direct sum to the quotient by the relations (2.1). Thus it is a bijection with the actual limiting cone

\[ F(L)\simeq\lim_{a\in I_0^{\mathrm{op}}}F(M_a). \tag{5.3} \]

Every limit diagram in \(\mathsf C^{\mathrm{op}}\) arises this way by reversing its arrows. For the empty diagram, \(L=0\) and \(F(0)\) is the singleton, the terminal set. Hence all small limits are preserved.

Proof of nonrepresentability. Suppose \(F\simeq\operatorname{Hom}_A(-,J)\) for some \(J\in\mathsf C\). Choose a pair with \(zJ=0\) and the module \(L_{ij}\) with \(z\) acting as the identity. By (3.2), \(\operatorname{Hom}_A(L_{ij},J)\) is the singleton consisting of the zero map. But \(F(L_{ij})\simeq\operatorname{Hom}_k(k,k)\simeq k\), which has at least the distinct zero and identity functionals. No bijection is possible at this object. This contradicts even an isomorphism of set-valued functors, without requiring that it preserve linear structures. \(\square\)

The direction in (5.3) matters: \(F\) takes colimits in \(\mathsf C\) to limits of sets. These are precisely limits in its domain \(\mathsf C^{\mathrm{op}}\). There is no assertion that a linear dual turns arbitrary products of ordinary modules into products of their duals. Exercise 4 locates the ambient coinduced representative and explains why it lies outside this category.

6. Four graded exercises with full solutions

Exercise 1 (introductory: separating characters and cogenerators). For each \(i\in I\), let \(K_i\) be the one-dimensional module with \(X_i\) acting as \(1\) and all other variables as \(0\). Compute \(\operatorname{Hom}_A(K_i,K_j)\). Then show that \(\mathsf C\) has no small cogenerating family, meaning no small family \((Q_b)\) for which the separate functors \(\operatorname{Hom}_A(-,Q_b)\) are jointly faithful.

Solution. If \(i=j\), every scalar map commutes with all actions, so the Hom vector space is \(k\). If \(i\ne j\), equivariance for \(X_i\) says \(f=fX_i=X_if=0\), so the Hom vector space is zero. These computations also show that the \(K_i\) are pairwise nonisomorphic.

For a small proposed cogenerating family, Lemma 3.1 gives \(z\) killing every \(Q_b\). On \(L_{ij}\) it acts as the identity, so every map \(L_{ij}\to Q_b\) is zero by (3.2). Precomposition with the distinct endomorphisms \(1_{L_{ij}}\) and \(0\) therefore has the same output for every family member. The family is not jointly faithful. The empty family fails by the same two arrows. This proves the assertion, independently of any injectivity assumption on the \(Q_b\).

Exercise 2 (intermediate: distinct self-extensions). Fix a nonzero \(P\in\mathsf C\) and a pair \(T_i^P=T_j^P\). Replace the off-diagonal \(y\) in (4.1) by \(\alpha y\), for \(\alpha\in k\), to obtain \(E_\alpha\). Show that the resulting extensions of \(P\) by \(P\) are pairwise inequivalent, where an equivalence induces the identity on both end terms. Determine exactly which split.

Solution. The commutation calculation for (4.1), with the extra scalar \(\alpha\), still proves that every \(E_\alpha\) is a module. Its scalar difference is \(z(x,y)=(\alpha y,0)\).

A linear map between two middle terms that commutes with the first inclusion and second projection must have the form

\[ u(x,y)=(x+H(y),y) \tag{6.1} \]

for a linear map \(H:P\to P\): its value on \((x,0)\) is prescribed, and its second coordinate on \((0,y)\) is \(y\). If it is equivariant from \(E_\alpha\) to \(E_\beta\), commuting with \(z\) gives \((\alpha y,0)=(\beta y,0)\) for every \(y\). Since \(P\ne0\), this forces \(\alpha=\beta\). For equal parameters the identity is an equivalence, proving the full classification in this family.

When \(\alpha=0\), all actions are diagonal and \(y\mapsto(0,y)\) is an equivariant section. When \(\alpha\ne0\), a section would give \(0=zs(y)=(\alpha y,0)\), forcing \(P=0\). Hence precisely the zero-parameter extension splits. No division by an integer or restriction on characteristic is involved.

Exercise 3 (advanced: too many extension classes). For every nonzero \(P\in\mathsf C\), construct a family of pairwise inequivalent short exact sequences \(0\to P\to E\to P\to0\) indexed by a set that is not \(\mathcal U\)-small. Conclude that their equivalence classes cannot be encoded by a \(\mathcal U\)-small set.

Solution. Partition \(I\) into the fibers of \(i\mapsto T_i^P\). There are at most \(\mathcal U\)-smallly many fibers, since the image is a subset of \(\operatorname{End}_k(P)\). If every fiber were small, their union would be small by universe closure and ambient choice of small representatives. That union is \(I\), a contradiction. Choose a fiber \(J\) that is not small; all \(T_j^P\), \(j\in J\), equal one operator \(T\).

Reserve one \(j_0\in J\) and index the family by \(J\setminus\{j_0\}\). This index is still not small: otherwise adjoining one element would make \(J\) small. For each remaining \(j\), take \(E_j=P\oplus P\) and change only the \(X_j\)-operator to have off-diagonal identity, exactly as in (4.1). Every \(E_j\) remains small.

Suppose the extensions for distinct \(j,\ell\) were equivalent. Its middle map would have the form (6.1). The \(X_{j_0}\)-operator is diagonal \(T\) on both middle terms, so commutation with it gives \(HT=TH\). Commutation with \(X_j\), whose off-diagonal part is the identity on \(E_j\) and zero on \(E_\ell\), gives

\[ 1_P=TH-HT. \tag{6.2} \]

Together these equations force \(1_P=0\), impossible for nonzero \(P\). The family is therefore pairwise inequivalent in every characteristic, including two. Any set encoding all its equivalence classes admits an injection from its index and therefore cannot be small.

Exercise 4 (advanced: enlarging the universe). Let \(D=\operatorname{Hom}_k(A,k)\), with action \((a\xi)(r)=\xi(ra)\). Show that \(D\) is a \(\mathcal V\)-small representative of the linear-dual functor on all \(\mathcal V\)-small \(A\)-modules, and that it is not \(\mathcal U\)-small. Give an explicit injection from \(I\) into \(D\), without assuming a formula for the cardinality of an infinite-dimensional dual. Explain how this is consistent with Proposition 5.1.

Solution. The algebra \(A\) and field \(k\) belong to the ambient size, so their function set and its subspace \(D\) are \(\mathcal V\)-small. Apply the complete evaluation representation in Locally nilpotent operators and coinduced duals, Proposition 1.1, now in \(\mathcal V\) with \(R=A\) and target \(k\). Its maps are \(h\mapsto(m\mapsto h(m)(1))\) and \(\lambda\mapsto(m\mapsto(r\mapsto\lambda(rm)))\). That proof checks the action, both inverses and naturality for arbitrary algebras, so it gives exactly this representative in the larger module category.

For \(i\in I\), let \(\delta_i:A\to k\) take the coefficient of the degree-one monomial \(X_i\). It is a linear functional, since the polynomials have their monomial basis. For distinct \(i,j\), \(\delta_i(X_i)=1\) while \(\delta_j(X_i)=0\). Thus \(i\mapsto\delta_i\) is an injection into \(D\). If \(D\) were small, \(I\) would be small, a contradiction. Its ambient representing property restricts to maps from objects of \(\mathsf C\), but its representing object is not an object of \(\mathsf C\). Proposition 5.1 forbids a replacement representative inside that smaller category and is therefore consistent with ambient coinduction.

7. References