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## Generators

Previously we proved a theorem due to Gabriel characterizing categories of modules as cocomplete abelian categories with a compact projective generator, where “generator” meant “every object is a colimit of finite direct sums of copies of the object.”

But we also used “generator” to mean “every object is a colimit of copies of the object,” and noted that these conditions are not equivalent: as this MO question discusses, the abelian group $\mathbb{Z}$ satisfies the first condition but not the second. More generally, as Mike Shulman explains here, there are in fact many inequivalent definitions of “generator” in category theory.

The goal of this post is to sort through a few of these definitions, which turn out to be totally ordered in strength, and find additional hypotheses under which they agree. As an application we’ll restate Gabriel’s theorem using weaker definitions of “generator” and give a more explicit description of all of the rings Morita equivalent to a given ring.

Conventions

All categories appearing in this post will again be either “ordinary” ($\text{Set}$-enriched) or “linear” ($\text{Ab}$-enriched). $\widehat{C}$ will again denote either presheaves $C^{op} \to \text{Set}$ or $C^{op} \to \text{Ab}$ depending on whether $C$ is ordinary or linear.

A family of objects $S$ in a category $C$ is an essentially small full subcategory $i : S \hookrightarrow C$. This is partly just a way of saying “set of objects” which is invariant under equivalence, although the morphisms in $S$ will be important too. At a few points in this post we will take coproducts involving every object in $S$. If $S$ is essentially small but not small these coproducts should be understood as ranging over representatives of the isomorphism classes of $S$.

The naming convention for generation conditions is that for some adjective X, a single object is an “X generator” and a family of objects is a “family of X generators.” In both cases this might be shortened to “is X.” We will not use the convention that a family of objects is also an “X generator” because we want to be able to summarize Gabriel’s theorem using the phrase “compact projective generator.”

Two classical definitions

In this section we won’t make any explicit use of linearity: everything applies to linear categories thought of as ordinary categories.

Definition: A family of objects $i : S \hookrightarrow C$ is a family of generators or faithful generators (nonstandard) if the following equivalent conditions hold:

1. The functors $\text{Hom}(s, -), s \in S$ are jointly faithful.
2. The restricted Yoneda embedding $\displaystyle C \ni c \mapsto \text{Hom}(i(-), c) \in \widehat{S}$ is faithful.
3. (If $C$ has coproducts) For all $c \in C$, the natural map $\bigsqcup_{f : s \to c, s \in S} s \twoheadrightarrow c$ is epic.
4. (If $C$ has coproducts) For all $c \in C$, some map $\bigsqcup_i s_i \twoheadrightarrow c, s_i \in S$ is epic.

Explicitly, these conditions mean that if $g_1, g_2 : c \to d$ is a pair of morphisms such that $g_1 \circ f = g_2 \circ f$ for all $f : s \to c, s \in S$, then $g_1 = g_2$.

If $C$ is linear, #2 does not depend on whether we take set-valued or abelian group-valued presheaves. #3 and #4 are presumably the motivation for the term “generator”: if, for example, $G$ is a group and $X \subseteq G$ is a subset of $G$, then $X$ generates $G$ iff the induced map $\bigsqcup_X \mathbb{Z} \to X$ is surjective.

Definition: A functor $F$ is conservative if it reflects isomorphisms: if $F(f)$ is an isomorphism, then $f$ is an isomorphism.

Definition: A family of objects $i : S \hookrightarrow C$ is a family of strong generators if the following equivalent conditions hold:

1. The functors $\text{Hom}(s, -), s \in S$ are jointly faithful and conservative.
2. The restricted Yoneda embedding $C \to \widehat{S}$ is faithful and conservative.

Explicitly, these conditions mean first that the above condition holds and second that if $g : c \to d$ is a morphism such that $\text{Hom}(s, g) : \text{Hom}(s, c) \to \text{Hom}(s, d)$ is an isomorphism for all $s \in S$, then $g$ is an isomorphism.

Example. Let $C$ be a familiar category of algebraic objects such as groups, rings, or modules over a ring. Then the free object $F$ on one generator is a strong generator: $\text{Hom}(F, -)$ is the forgetful functor, which is faithful by definition, and a morphism of groups, rings, or modules over a ring which is a bijection is an isomorphism. This generalizes to the category of models of any Lawvere theory.

Example. $1 \in \text{Top}$ is a generator but not a strong generator: $\text{Hom}(1, -)$ is the forgetful functor, which is faithful by definition, but a continuous bijection between topological spaces need not be a homeomorphism.

Example. $1 \in \text{CHaus}$ (compact Hausdorff spaces) is a strong generator: $\text{Hom}(1, -)$ is the forgetful functor, which is faithful by definition, and a continuous bijection between compact Hausdorff spaces is a homeomorphism. This generalizes to any category which is monadic over $\text{Set}$.

Recall that a basic property of faithful functors is that they reflect monos and epis. The corresponding property of conservative functors is the following.

Proposition: Let $F : C \to D$ be a conservative functor. Then $F$ reflects any (shapes of) limits or colimits that it preserves.

Proof. By duality it suffices to prove the statement for colimits. Suppose that $J$ is a shape of colimit which $C, D$ have and which $F$ preserves. To say that $F$ reflects colimits of shape $J$ is to say the following: suppose $c$ is a cocone over a diagram $c_j, j \in J$ of shape $J$, so that it is equipped with suitable maps $f_j : c_j \to c$. If the induced maps $F(f_j) : F(c_j) \to F(c)$ exhibit $F(c)$ as the colimit $\text{colim}_j F(c_j)$, then the original maps exhibit $c$ as the colimit $\text{colim}_j c_j$.

Equivalently, the maps $f_j$ and $F(f_j)$ describe maps $\text{colim}_j c_j \to c$ and $\text{colim}_j F(c_j) \to F(c)$, and the condition is that if the latter map is an isomorphism, then so is the former map. But by hypothesis, since $F$ preserves colimits of shape $J$, the former map is also the induced map $F(\text{colim}_j c_j) \to F(c)$, and if this is an isomorphism then, since $F$ is conservative, so is the map $\text{colim}_j c_j \to c$. $\Box$

Corollary: Let $F : C \to D$ be a conservative functor. If $C$ has either equalizers or coequalizers and $F$ preserves them, then $F$ is faithful.

Proof. Let $f, g : c \rightrightarrows d$ be a pair of parallel arrows. Then $f = g$ iff $\text{id}_c : c \to c$ is the coequalizer of $f, g$ iff $\text{id}_d : d \to d$ is the equalizer of $f, g$. If $F$ is conservative and preserves either of these, then it reflects them as well. $\Box$

Corollary: If a category $C$ has equalizers, then we can drop “jointly faithful” from the definition of strong generators: that is, $i : S \hookrightarrow C$ is a family of strong generators iff the functors $\text{Hom}(s, -)$ are jointly conservative.

Proof. The functors $\text{Hom}(s, -)$ preserve any limits that exist in $C$. $\Box$

Example. Let $C$ be the homotopy category of pointed connected CW complexes. Then Whitehead’s theorem asserts that the functors $\pi_n : C \to \text{Set}$ are jointly conservative, and these functors are hom functors $\text{Hom}(S^n, -)$, so they preserve all limits that exist in $C$. However, they are not jointly faithful: for example, there are interesting homotopy classes of morphisms between Eilenberg-MacLane spaces whose homotopy groups live in distinct degrees, and for other examples see this MO question. It follows that $C$ fails to have equalizers.

What can we say about the converse? First, say that a morphism is a fake isomorphism (nonstandard) if it is epic and monic, but not an iso. For example, any continuous bijection which is not a homeomorphism is fake.

Proposition: Let $F : C \to D$ be a faithful functor. If $C$ has no fake isos, then $F$ is conservative.

Proof. Since $F$ is faithful, it reflects epis and monos. If $F(f)$ is an iso, then in particular it’s epic and monic, hence so is $f$. By hypothesis, it follows that $f$ is an iso. $\Box$

Corollary: If $i : S \hookrightarrow C$ is a family of objects in a category with no fake isos, then $S$ is faithful iff it is strong.

In general, a category $C$ won’t have fake isos if its epis are well-behaved; for example, whenever epis are regular, a hypothesis we’ll use later as well. This holds in abelian categories but also in $\text{Set}$ and $\text{Grp}$; however, it does not hold in $\text{Ring}$, since for example $\mathbb{Z} \to \mathbb{Q}$ is epic but not regular epic, and it does not hold in $\text{Top}$, since there epis are just surjections but regular epis are quotient maps.

The term “generator” can be interpreted as having something to do with generating a category under colimits. Here are three definitions along those lines. As in the previous section, we won’t make any explicit use of linearity.

Definition: Let $C$ be a cocomplete category. A family of objects $i : S \hookrightarrow C$ is a family of

1. naive generators (nonstandard) if every $c \in C$ is a colimit of objects in $S$,
2. presenting generators (nonstandard) if every $c \in C$ is a coequalizer of a pair of morphisms $\bigsqcup_i s_i \rightrightarrows \bigsqcup_j s_j$ between coproducts of objects in $S$, and
3. iterated generators (nonstandard) if every $c \in C$ is an iterated colimit of objects in $S$.

Mike Shulman uses the terms colimit-dense generator for naive generators and colimit generator for iterated generators respectively, but I find it hard to remember which is which (and in fact I mixed them up while writing this sentence). The idea behind the definition of presenting generators is that a coequalizer of the above form is a presentation by “generators” (the $s_j$) and “relations” (the $s_i$).

Example. $\mathbb{Z} \in \text{Ab}$ is not naive, but it is presenting, iterated, strong, and faithful.

The five definitions we’ve introduced are totally ordered in strength as follows. As above, $i : S \hookrightarrow C$ is a family of objects in a cocomplete category.

Theorem: Naive $\Rightarrow$ presenting $\Rightarrow$ iterated $\Rightarrow$ strong $\Rightarrow$ faithful.

Proof.

Naive $\Rightarrow$ presenting: in a cocomplete category, every colimit can be computed as the coequalizer of a pair of morphisms between two coproducts.

Presenting $\Rightarrow$ iterated: every coequalizer of a pair of morphisms between coproducts is an iterated colimit (iterated twice: once for the coproducts, once for the coequalizers).

Iterated $\Rightarrow$ faithful: Suppose $S$ is iterated. We want to show that every $c \in C$ admits some epi $\bigsqcup_i s_i \twoheadrightarrow c, s_i \in S$. Every $s \in S$ satisfies this condition, and moreover it is closed under coproducts (since coproducts of epis are epis) and coequalizers (since coequalizer projections are epis, and epis are closed under composition), hence it is closed under colimits. So in fact every $c \in C$ admits such an epi, and $S$ is faithful.

Iterated $\Rightarrow$ strong: Suppose $S$ is iterated. By the above, we know that the functors $\text{Hom}(s, -)$ are jointly faithful, so it suffices to show that they are jointly conservative.

Say that a morphism $w : c \to d$ is an $S$-isomorphism if it induces an isomorphism

$\displaystyle \text{Hom}(s, w) : \text{Hom}(s, c) \cong \text{Hom}(s, d)$

for all $s \in S$. If $W$ denotes the collection of all $S$-isomorphisms, our goal is to show that $W$ consists precisely of the isomorphisms.

An object $c \in C$ is $W$-colocal if every $S$-isomorphism $w : d \to d'$ induces an isomorphism $\text{Hom}(c, w) : \text{Hom}(c, d) \cong \text{Hom}(c, d')$. By hypothesis, every $s \in S$ is $W$-colocal. If $c = \text{colim}_j c_j$ is a colimit of $W$-colocal objects and $w : d \to d'$ is an isomorphism, then

$\displaystyle \text{Hom}(c, w) : \text{lim}_j \text{Hom}(c_j, d) \to \text{lim}_j \text{Hom}(c_j, d')$

is an isomorphism since its components are. Hence the collection of $W$-colocal objects is closed under colimits. Since $S$ is iterated, it follows that every $c \in C$ is $W$-colocal.

But this means that if $w : d \to d'$ is an $S$-isomorphism, then $\text{Hom}(c, w) : \text{Hom}(c, d) \cong \text{Hom}(c, d')$ is an isomorphism for all $c \in C$, and by the Yoneda lemma it follows that $w$ is an isomorphism. Hence $S$ is strong.

Strong $\Rightarrow$ faithful: by definition. $\Box$

So far we’ve seen the following counterexamples to the converses of the above implications:

1. $\mathbb{Z} \in \text{Ab}$ shows that presenting $\not \Rightarrow$ naive.
2. $1 \in \text{Top}$ shows that faithful $\not \Rightarrow$ strong.

According to Mike Shulman, iterated $\Leftrightarrow$ strong under mild hypotheses (in addition to $C$ being cocomplete, it must have finite limits and satisfy a mild smallness condition), but we won’t use this. I expect that iterated $\not \Rightarrow$ presenting in general but don’t know a counterexample. Under additional hypotheses, we have the following.

Theorem: The implications presenting $\Rightarrow$ iterated $\Rightarrow$ strong $\Rightarrow$ faithful can be reversed under the following additional hypotheses:

1. Faithful $\Rightarrow$ strong if $C$ has no fake isos (in particular if $C$ is abelian).
2. Faithful $\Rightarrow$ presenting if every epi in $C$ is regular (in particular if $C$ is abelian).
3. Iterated $\Rightarrow$ presenting if every $s \in S$ is projective.

Proof.

Faithful sometimes $\Rightarrow$ strong: proven previously.

Faithful sometimes $\Rightarrow$ presenting: suppose $S$ is faithful and let

$\displaystyle f : \bigsqcup_i s_i \twoheadrightarrow c, s_i \in S$

be an epi. We would like to use this epi to exhibit $c$ as a coequalizer.

Now suppose that every epi is regular. Then $f$ is regular, hence is the coequalizer of some morphisms $g_1, g_2 : d \rightrightarrows \bigsqcup_i s_i$. By faithfulness, we can find another epi

$\displaystyle h : \bigsqcup_j s_j \twoheadrightarrow d$

and since coequalizers are unchanged by precomposing with an epi, we can replace $g_1, g_2$ with $g_1 \circ h, g_1 \circ h$, and we conclude that

$\displaystyle \bigsqcup_j s_j \rightrightarrows \bigsqcup_i s_i \xrightarrow{f} c$

is a coequalizer diagram. Hence $S$ is presenting.

Iterated sometimes $\Rightarrow$ presenting: suppose $S$ is iterated. Let’s try to imitate the argument used in this post for discussing finite presentations in order to reduce any iterated colimit to a coequalizer of coproducts.

First, a colimit is a coequalizer of coproducts in the usual way. Since a coproduct of colimits can be expressed as a single colimit (take the coproduct of the corresponding diagrams), it suffices to show that we can express a coequalizer of colimits as a coequalizer of coproducts.

Hence let

$\displaystyle f, g : \text{colim}_i s_i \rightrightarrows \text{colim}_j s_j$

be morphisms, where $s_i, s_j \in S$, whose coequalizer $c$ we are trying to express as a coequalizer of coproducts. As before, $\text{colim}_i s_i$ admits an epi from a coproduct $\bigsqcup_i s_i$ and coequalizers are unchanged by precomposing with an epi, so WLOG the above diagram has the form

$\displaystyle f, g : \bigsqcup_i s_i \rightrightarrows \text{colim}_j s_j$.

Now suppose that every $s \in S$ is projective. Then projective objects are closed under coproducts (assuming the axiom of choice: pick a lift for each object), so $\bigsqcup_{i \in I} s_i$ is projective, and the coequalizer projection $\bigsqcup_j s_j \twoheadrightarrow \text{colim}_j s_j$ is an epi, so $f, g$ lift to $\bigsqcup_j s_j$, and as before we find that we can reduce the above computation to the computation of a single coequalizer. Equivalently, WLOG the above diagram has the form

$\displaystyle f, g : \bigsqcup_i s_i \rightrightarrows \bigsqcup_j s_j$

(although there may be more $s_i$ than there were before). Hence $S$ is presenting.

Corollary: If $i : S \hookrightarrow C$ is a family of objects in an abelian category, then strong $\Leftrightarrow$ faithful. If $C$ is cocomplete, then presenting $\Leftrightarrow$ iterated $\Leftrightarrow$ strong $\Leftrightarrow$ faithful.

Now, in the previous post on tiny objects we proved various facts using naive generators, but an inspection of the proofs involved show that they go through for presenting generators (possibly even iterated generators, but we don’t need this; the point is that if a functor preserves colimits then it preserves iterated colimits), and using the above corollary we deduce the following.

Corollary (Freyd): If $i : S \hookrightarrow C$ is a family of objects in a cocomplete abelian category, then the restricted Yoneda embedding $C \to \widehat{S}$ is an equivalence of categories iff $S$ is a family of compact projective (faithful) generators.

Corollary (Gabriel): If $s \in C$ is an object in a cocomplete abelian category, then

$\displaystyle C \ni c \mapsto \text{Hom}(s, c) \in \text{Mod}(\text{End}(s))$

is an equivalence of categories iff $s$ is a compact projective (faithful) generator.

One last definition for abelian categories

Let $C$ be a linear category. In this setting the condition that $i : S \hookrightarrow C$ is a family of (faithful) generators can be rephrased as the condition that the restricted Yoneda embedding $C \to \widehat{S}$ reflects zero morphisms in the sense that if $f : c \to d$ is a morphism in $C$ such that

$\displaystyle \text{Hom}(s, f) : \text{Hom}(s, c) \to \text{Hom}(s, d)$

is a zero morphism for all $s \in S$, then $f = 0$.

If $C$ is abelian, then a morphism is zero iff its image is zero, which suggests the following definition.

Definition: Let $F : C \to D$ be a linear functor between linear categories. $F$ is weakly faithful (nonstandard) if it reflects zero objects: $F(c) = 0$ implies $c = 0$.

Definition: Let $C$ be a linear category. A family of objects $i : S \hookrightarrow C$ is a family of weak generators if the restricted Yoneda embedding $C \to \widehat{S}$ is weakly faithful.

More explicitly, this condition means that if $c \in C$ is a nonzero object then there is some nonzero morphism $f : s \to c, s \in S$.

Proposition: Let $F : C \to D$ be a linear functor between linear categories. If $F$ is faithful, then it is weakly faithful. The converse holds if $F$ is exact and every morphism in $C$ has an epi-mono factorization (in particular if $C$ is abelian).

By an epi-mono factorization of a morphism $f : c \to d$ we just mean a factorization $f = me$ where $m$ is monic and $e$ is epic. In an abelian category the image factorization accomplishes this. More generally, in a category with either 1) kernel pairs and coequalizers or 2) cokernel pairs and equalizers the regular coimage or regular image factorizations, respectively, accomplish this, so this is a very mild hypothesis.

Proof. $\Rightarrow$: An object $c \in C$ is a zero object iff the identity $\text{id}_c : c \to c$ is a zero morphism. Since $F$ is faithful, it reflects zero morphisms.

$\Leftarrow$: let $f : c \to d$ be a morphism. By hypothesis, $f$ has an epi-mono factorization $f = me$ where $m, e$ fit into a diagram

$\displaystyle c \twoheadrightarrow i \hookrightarrow d$.

Since $F$ is exact, it preserves epis and monos, and hence we get an induced epi-mono factorization $F(f) = F(m) F(e)$.

If $F(f) = F(m) F(e) = 0$, then by cancelling $F(m)$ on the left and cancelling $F(e)$ on the right we conclude that $\text{id}_i = 0$, so $F(i) = 0$. Since $F$ reflects zero objects, it follows that $i = 0$, so $f = me = 0$. Hence $F$ reflects zero morphisms as desired. $\Box$

I expect that weak $\not \Rightarrow$ faithful in general but don’t know a counterexample.

Corollary: If $i : S \hookrightarrow C$ is a family of objects in a cocomplete linear category, then naive $\Rightarrow$ presenting $\Rightarrow$ iterated $\Rightarrow$ strong $\Rightarrow$ faithful $\Rightarrow$ weak.

Corollary: If $i : S \hookrightarrow C$ is a family of projective objects in an abelian category, then strong $\Leftrightarrow$ faithful $\Leftrightarrow$ weak. If $C$ is cocomplete, then presenting $\Leftrightarrow$ iterated $\Leftrightarrow$ strong $\Leftrightarrow$ faithful $\Leftrightarrow$ weak.

Corollary (Freyd): If $i : S \hookrightarrow C$ is a family of objects in a cocomplete abelian category, then the restricted Yoneda embedding $C \to \widehat{S}$ is an equivalence of categories iff $S$ is a family of compact projective weak generators.

Corollary (Gabriel): If $s \in C$ is an object in a cocomplete abelian category, then

$\displaystyle C \ni c \mapsto \text{Hom}(s, c) \in \text{Mod}(\text{End}(s))$

is an equivalence of categories iff $s$ is a compact projective weak generator.

More explicit Morita equivalences

Let $R$ be a ring. We now know that every Morita equivalence $\text{Mod}(R) \cong \text{Mod}(S)$ comes from a module $P \in \text{Mod}(R)$ which is a compact projective weak generator, where $S \cong \text{End}(P)$ and the equivalence is given by

$\displaystyle \text{Mod}(R) \ni M \mapsto \text{Hom}(P, M) \in \text{Mod}(S)$.

How explicit can we make this? $P$ is compact projective iff it is a finitely presented projective module, or equivalently a retract of a finite free module. This means that we can explicitly describe $P$ by writing down an idempotent

$\displaystyle e \in \text{End}(R^n) \cong M_n(R)$

such that $P = e R^n$ is the splitting of $e$; from here it follows that $S$ can be explicitly described as

$\displaystyle S = e M_n(R) e$.

This description exhibits $S$ as a “non-unital subring” of $M_n(R)$: the inclusion respects multiplication and addition but not the unit, and in fact the unit in $S$ is $e$.

(Edit, 10/28/20: It was pointed out by Matthew Daws on math.SE that the argument that was previously here was incorrect.) It remains to give an explicit criterion for when $P$ is a weak generator. If $M \neq 0$, we want to determine when we can always find a nonzero map $f : P \to M$. The data of such a map is equivalent to the data of an $n$-tuple $(m_1, \dots m_n) \in M^n$ such that $e(m_1, \dots m_n) = (m_1, \dots m_n)$. The fixed points of $e$ are precisely its image so a nonzero such tuple exists iff $e$ acts by a nonzero endomorphism on $M^n$, so now we want to know when $e$ always acts by a nonzero endomorphism on $M^n$. If $M \neq 0$ then the action of $M_n(R)$ on $M^n$ factors through a proper two-sided ideal of $M_n(R)$, so we want to understand these.

Proposition: Every two-sided ideal of $M_n(R)$ has the form $M_n(I)$ where $I$ is a two-sided ideal of $R$.

Proof. This can be done using some explicit calculations with matrices but there is a slicker proof using Morita equivalence. The two-sided ideals of $M_n(R)$ are precisely the submodules of $M_n(R)$ regarded as an $(M_n(R), M_n(R))$-bimodule in the usual way, or equivalently as an $M_n(R^{op} \otimes R)$-module. Submodules are a categorical notion (corresponding to monomorphisms), so they are invariant under Morita equivalence. $M_n(R^{op} \otimes R)$ is Morita equivalent to $R^{op} \otimes R$, with the equivalence sending $M_n(R)$ to $R$ regarded as an $R^{op} \otimes R$-module, or equivalently an $(R, R)$-module in the usual way. And the two-sided ideals of $R$ are precisely the submodules of $R$ regarded as an $(R, R)$-bimodule. $\Box$

It follows that every proper two-sided ideal $M_n(I)$ of $M_n(R)$ is the kernel of some action of $M_n(R)$ on some $M^n$. We conclude the following.

Theorem: $P = e R^n$ is a weak generator iff $e$ is a full idempotent, meaning that it is not contained in any proper two-sided ideal of $M_n(R)$, or equivalently, that $M_n(R) e M_n(R) = M_n(R)$. In turn this condition is satisfied iff $e \not \equiv 0 \bmod I$ for all proper two-sided ideals $I$ of $R$.

The condition that $e \equiv 0 \bmod I$ is very strong: since $e^2 = e$, it follows by induction that $e \equiv 0 \bmod I^n$ for all $n$, hence that $e \equiv 0 \bmod \bigcap_n I^n$. If $R$ is a commutative Noetherian domain and $I$ is a proper ideal, then by the Krull intersection theorem, $\bigcap_n I^n = (0)$, hence $e = 0$.

Corollary: If $R$ is a commutative Noetherian domain, then $e$ is full iff $e \neq 0$, so the rings Morita equivalent to $R$ are precisely the rings $\text{End}(P)$ where $P$ is a nonzero finitely presented projective module.

Geometrically this has the following interpretation. If $R$ is commutative, then $P = e R^n$ defines an algebraic vector bundle over the affine scheme $\text{Spec } R$, and the condition that $e \not \equiv 0 \bmod I$ is the condition that the restriction $P/I = e (R/I)^n$ of $P$ to affine closed subschemes $\text{Spec } R/I$ is nonzero. Idempotence implies that if the coefficients of $e$ vanish when restricted to $\text{Spec } R/I$, then they vanish to infinite order. The Krull intersection theorem implies that if this happens when $R$ is a Noetherian domain, then $e = 0$, so $P = 0$. (Compare to the statement that if a meromorphic function on a connected open subset of $\mathbb{C}$ vanishes to infinite order at a point, then it vanishes identically.)

If we don’t assume that $R$ is a domain, then what can happen is that $R$ has a nontrivial decomposition as a product $R_1 \times R_2$, so $\text{Spec } R$ has a nontrivial decomposition as a coproduct $\text{Spec } R_1 \sqcup \text{Spec } R_2$, and a vector bundle over $\text{Spec } R$ may be nonvanishing over one component but not the other. Algebraically this corresponds to the intersection $\cap_n I^n$ being a nontrivial idempotent ideal in $R$, such as one generated by a nontrivial idempotent.

### 5 Responses

1. […] the big categories as categories of modules are “bases” for them. As we learned previously, a cocomplete abelian category has a “basis” in this sense iff it has a family of tiny […]

2. […] this is that there are a lot of non-equivalent notions of a generator of a category. In fact, in this post, Qiaochu Yuan presents at least nine different definitions and analyzes their relations in […]

3. On fake isomorphisms: the standard terminology for such a thing is “bimorphism”, which is probably to be avoided. A category with no bimorphisms is usually called “balanced”.

On families of objects: although thinking of them as full subcategories gets around the problem of invariance under equivalence, it seems to me that the coproduct operation you describe is evil, or at least non-functorial. After all, it involves splitting a surjection. I don’t really know how to fix this – maybe it would be better to think of families of objects as sets after all.

• Yeah, “bimorphism” is pretty terrible, and I’m not a huge fan of “balanced” either, especially since it also refers to some monoidal thing.

Yes, the thing I actually want to do is more like remembering the structure of $S$ as an “essentially small large setoid” (a class with an equivalence relation, here isomorphism, which has a set’s worth of equivalence classes) and take the coproduct over $S$ with this structure in mind (so identifying isomorphic objects in the coproduct). But it seemed like too much of a distraction to spell this out explicitly. Maybe I should’ve used sets of objects after all.