Draft lesson.

Localizing subcategories and noetherian Serre pieces

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

In a locally noetherian category, an object is assembled from its noetherian subobjects. This makes a localizing class recoverable from a small Serre class of finite pieces. We prove the correspondence, construct its torsion coreflection, and identify the actual quotient and its noetherian objects. Explicit examples distinguish the finite-piece theorem from a claim about arbitrary finitely presented generators or ambient injective objects.

Fix universes and choice. Let $\mathcal A$ be a locally noetherian Grothendieck category and $\mathcal B=\mathcal A^{\mathrm{noeth}}$, with a small skeleton chosen when needed. A localizing subcategory here is a full, isomorphism-closed Serre subcategory closed under small coproducts. Serre means closed under ambient subobjects, quotients and extensions, including zero.

Noetherian generators, filtered injectives and Ext, Sections 1–3 and 7, proves that noetherian, finitely generated and finitely presented objects coincide, that every object is a directed union of noetherian subobjects, and that $\operatorname{Ind}(\mathcal B)\simeq\mathcal A$ exactly. Retain the full small-subobject proof in Generators and small quotient families, Sections 3–4. Serre quotients and local saturation, Sections 1–2 and 8, supplies the exact quotient and its source-restriction/target-quotient fraction formula.

Ind-completion and Serre quotients, Sections 1–7, constructs the exact Ind quotient, its right adjoint, invertible counit, kernel and natural Hom comparisons. The finite-part and exact realization interfaces are proved in Finitely presented generators and coherent Ind reconstruction. The matrix bimodules in Local module categories and Morita equivalence, Sections 1 and 5, are used in the last exercise with their actions and units made explicit.

1. Colimits and the finite-subobject test

Every localizing subcategory $\mathcal T$ is closed under all small colimits formed in $\mathcal A$. Indeed a diagram colimit is the cokernel of the difference of the two maps from the coproduct indexed by arrows to the coproduct indexed by objects: the maps send an arrow's source copy by its transition and by its source inclusion. The cokernel universal property is exactly the compatible-cocone property. Coproduct and quotient closure therefore put that colimit in $\mathcal T$.

Consequently an object $M$ belongs to $\mathcal T$ exactly when every noetherian subobject of $M$ does. Necessity uses Serre closure. For sufficiency, express $M$ as its directed union of noetherian subobjects and apply colimit closure. Intersections and finite sums of such subobjects are still noetherian; finite sums are quotients of finite biproducts. The small-subobject theorem makes the family a small directed index. Thus the test involves an actual small union with its specified inclusions.

2. Construct a localizing class from a Serre class

For a Serre subcategory $\mathcal S\subseteq\mathcal B$, define $\mathcal T_{\mathcal S}= \{M\in\mathcal A\mid\text{every noetherian subobject of }M \text{ belongs to }\mathcal S\}. \tag{2.1}$

Proposition 2.1. This is a localizing subcategory. It is also the smallest localizing subcategory containing $\mathcal S$, and $\mathcal T_{\mathcal S}\cap\mathcal B=\mathcal S. \tag{2.2}$

Proof. Zero and isomorphism closure are immediate. An ambient subobject inherits the test. For a quotient $M\twoheadrightarrow Q$ of an object passing the test, let $P\subset Q$ be noetherian. The inverse image of $P$ is a directed union of noetherian subobjects. Their images have union $P$, by exact filtered colimits. Finite generation of $P$ makes one image all of $P$. Thus $P$ is a quotient of a noetherian subobject of $M$ in $\mathcal S$, and is in $\mathcal S$. This proves quotient closure without assuming an arbitrary inverse image noetherian.

For $0\to M'\to M\to M''\to0$, with both ends passing the test, any noetherian $N\subset M$ has noetherian intersection with $M'$ and noetherian image in $M''$. Both lie in $\mathcal S$, and their short exact intersection/image row makes $N$ an extension in $\mathcal B$. Hence $N\in\mathcal S$, proving extension closure.

Let $N\subset\bigoplus_iM_i$ be noetherian, with every $M_i$ passing the test. Its inclusion factors through a finite partial sum: pull back the partial sums, whose union is the coproduct, and use finite generation. Its component images $P_i\subset M_i$ are noetherian and belong to $\mathcal S$. The inclusion factors monically through the finite sum of these images. Serre closure in $\mathcal B$ puts $N$ in $\mathcal S$. This proves coproduct closure.

For noetherian $M$, its own subobject test includes $M$; conversely all subobjects of an $M\in\mathcal S$ are noetherian and in $\mathcal S$. This is (2.2). Every object in (2.1) is a directed union of objects of $\mathcal S$; any localizing class containing $\mathcal S$ contains that union by Section 1. This proves minimality. $\square$

3. The exhaustive correspondence and the Ind realization

Theorem 3.1. The assignments $\mathcal T\longmapsto\mathcal T\cap\mathcal B, \qquad\mathcal S\longmapsto\mathcal T_{\mathcal S} \tag{3.1}$ are mutually inverse, inclusion-preserving bijections between localizing subcategories of $\mathcal A$ and Serre subcategories of $\mathcal B$. Moreover the realization gives an exact equivalence $\operatorname{Ind}(\mathcal S)\simeq\mathcal T_{\mathcal S} \tag{3.2}$ compatible with their inclusions into $\operatorname{Ind}(\mathcal B)\simeq\mathcal A$. Each localizing subcategory is itself a locally noetherian Grothendieck category, and its noetherian objects are exactly $\mathcal S$.

Proof. For a localizing $\mathcal T$, its intersection with $\mathcal B$ is Serre in $\mathcal B$, since all subquotients and finite extensions remain noetherian in $\mathcal A$. Section 1 says exactly that $\mathcal T=\mathcal T_{\mathcal T\cap\mathcal B}$. Equation (2.2) proves the other composite identity. Both constructions preserve inclusion, so these are the stated bijections, including equality tests.

The full exact Ind embedding from the Serre-quotient prerequisite realizes $\operatorname{Ind}(\mathcal S)$ inside $\operatorname{Ind}(\mathcal B)$. Its objects are filtered colimits of objects of $\mathcal S$; these belong to $\mathcal T_{\mathcal S}$ by colimit closure. Conversely the directed noetherian-subobject presentation of every object of $\mathcal T_{\mathcal S}$ is an actual Ind presentation using $\mathcal S$. The same realization, not a different abstract equivalence, gives (3.2). Full faithfulness retains all its Hom maps.

The subcategory inherits its abelian structure from Serre closure, its small colimits by Section 1, and exact filtered colimits from $\mathcal A$. A small skeleton of $\mathcal S$ generates: its noetherian subobject maps jointly cover each object, hence the evaluation coproduct is epic. Each such generator is noetherian in the subcategory, since all its ambient subobjects already belong to the subcategory. Conversely every internally noetherian object is finitely generated there and is a union of its $\mathcal S$-subobjects. One member must be the object, so it belongs to $\mathcal S$. This verifies the full Grothendieck and finite-object assertions. $\square$

4. The torsion coreflection and its colimit comparison

Theorem 4.1. Every localizing $\mathcal T$ has a largest subobject $\tau M\subset M$ in $\mathcal T$. The functor $\tau\colon\mathcal A\to\mathcal T$ is additive, left exact and right adjoint to the exact inclusion. Its counit is that specified inclusion. In the present locally noetherian setting, the canonical comparison $\operatorname{colim}_i\tau M_i \longrightarrow\tau\bigl(\operatorname{colim}_iM_i\bigr) \tag{4.1}$ is invertible for every small filtered diagram. Also $\tau(M/\tau M)=0$.

Proof. The $\mathcal T$-subobjects of $M$ form a small set. Take their sum, the image of their inclusion maps from the coproduct. It belongs to $\mathcal T$ and contains all of them, proving maximality. The image under $f\colon M\to N$ of $\tau M$ is in $\mathcal T$, so factors uniquely through $\tau N$. These factors preserve identity, composition and addition by monic cancellation, making the counits natural.

For $T\in\mathcal T$, every arrow $T\to M$ has a $\mathcal T$-image and therefore factors uniquely through $\tau M$. This gives inverse natural Hom bijections, proving the adjunction and its triangle identities. For $N\subset M$, maximality and Serre closure give $\tau N=N\cap\tau M$. The same formula applied to the kernel of any arrow proves left exactness. If a torsion subobject of $M/\tau M$ existed, its inverse image would be an extension of it by $\tau M$ inside $\mathcal T$. Maximality would force the quotient subobject to be zero. Thus $\tau(M/\tau M)=0$.

For (4.1), exact filtered colimits make $H=\operatorname{colim}\tau M_i\to M_\infty$ monic, and colimit closure makes its image torsion. It is contained in $\tau M_\infty$. A noetherian subobject $N\subset\tau M_\infty$ is finitely presented, so its map into $M_\infty$ factors through some $M_i$. The image of that factor is a quotient of $N$, hence belongs to $\mathcal T$, and the factor goes through $\tau M_i$. Therefore the given subobject $N$ lies in $H$. Such $N$'s have union $\tau M_\infty$, so the containment is equality. These inclusions are precisely the comparison induced by the stage counits, proving both invertibility and its naturality. $\square$

An object $F$ has $\tau F=0$ exactly when $\operatorname{Hom}(T,F)=0$ for every $T\in\mathcal T$, by the adjunction. Every object thus has the natural row $0\to\tau M\to M\to M/\tau M\to0$ with torsion first term and torsion-free quotient. Here torsion-free refers to this Hom condition, not necessarily to ordinary multiplication by nonzero scalars.

5. The quotient and its noetherian objects

Write $Q_0\colon\mathcal B\to\mathcal B/\mathcal S$ for the ordinary exact quotient.

Lemma 5.1. Every subobject of $Q_0B$ is the image of $Q_0P\hookrightarrow Q_0B$ for a subobject $P\subset B$. Every object of $\mathcal B/\mathcal S$ is noetherian.

Proof. Represent a monomorphism $Q_0Y\to Q_0B$ by an ordinary map $f\colon Y'\to B/B_0$, where $Y/Y'$ and $B_0$ belong to $\mathcal S$, using the complete fraction formula in the prerequisite. Set $J=\operatorname{im}f$ and let $P\subset B$ be its inverse image. Exactness makes $Q_0J\hookrightarrow Q_0(B/B_0)$ the image of the represented monomorphism. The map $Q_0P\to Q_0J$ is invertible, since its kernel is $Q_0B_0=0$ and it is epic. The denominator identifications $Q_0Y'=Q_0Y$, $Q_0(B/B_0)=Q_0B$ thus identify the original subobject with $Q_0P\hookrightarrow Q_0B$, with its specified inclusion.

For an ascending sequence of quotient subobjects, choose lifts $P_n\subset B$ by the first paragraph. Replace them by $P_1+\cdots+P_n$. These are now an ascending sequence upstairs, and their quotient images are exactly the original sequence: an exact additive quotient preserves finite sums of subobjects as images of their sum maps, and the largest of the first $n$ quotient subobjects is the $n$-th one. Noetherianness of $B$ stabilizes the lifted sequence and hence the original one. Every quotient object is represented by some $B$, proving the second assertion. $\square$

Theorem 5.2. Let $\mathcal T=\mathcal T_{\mathcal S}$. With their actual quotient functors there is an exact equivalence $\mathcal A/\mathcal T\simeq\operatorname{Ind}(\mathcal B/\mathcal S). \tag{5.1}$ The quotient is locally noetherian. Its full noetherian subcategory is $\mathcal B/\mathcal S$, via its constant embedding. In particular the quotient of any ambient noetherian object is noetherian, and every noetherian quotient object has such a representative.

Proof. Transport the previously constructed exact Ind functor $L=\operatorname{Ind}(Q_0)$ through the exact realization $\operatorname{Ind}(\mathcal B)\simeq\mathcal A$. Its kernel is the actual embedded $\operatorname{Ind}(\mathcal S)$, by the complete kernel theorem in the Ind quotient lesson. Theorem 3.1 identifies that kernel with $\mathcal T$. That lesson's exact quotient equivalence, full section adjunction and invertible counit give (5.1), identifying the induced functor with the specified quotient. Thus its Hom maps are the ones of that constructed quotient, rather than just a category with the same object classes.

Lemma 5.1 makes the small abelian category $\mathcal B/\mathcal S$ noetherian. The complete noetherian Ind criterion in the preceding lesson makes its completion locally noetherian. The exact constant-finite-piece theorem identifies its finitely presented objects with $\mathcal B/\mathcal S$; in a locally noetherian category these are exactly its noetherian objects. Restricting $L$ to a constant $B$ gives the constant $Q_0B$. Since every object of the small quotient is represented by some $B$, all the asserted finite-part statements follow. $\square$

6. The section functor and injectives

Denote the quotient by $Q\colon\mathcal A\to\mathcal A/\mathcal T$ and its right adjoint by $R$. The earlier Ind construction gives, for $B\in\mathcal B$, $\operatorname{Hom}_{\mathcal A}(B,RD) \simeq\operatorname{Hom}_{\mathcal A/\mathcal T}(QB,D). \tag{6.1}$ Its presheaf on $\mathcal B$ is left exact and hence the specified Ind object. The adjunction extends (6.1) to all ambient objects by their filtered presentations. Its counit $QRD\to D$ is invertible, so the section is fully faithful. These constructions and the natural units and counits are retained from the complete Ind quotient proof.

Proposition 6.1. The section preserves filtered colimits and injective objects. The torsion coreflection takes an ambient injective to an injective of $\mathcal T$. It need not give an injective of $\mathcal A$. The section need not preserve noetherian objects.

Proof. Form the canonical comparison $\operatorname{colim}_iRD_i\to R(\operatorname{colim}_iD_i)$ from the section's images of the colimit arrows. Test it with each noetherian $B$. Finite presentation of $B$ and of $QB$, together with (6.1), identifies the induced map with the natural isomorphisms $\begin{aligned} \operatorname{Hom}(B,\operatorname{colim}_iRD_i) &=\operatorname{colim}_i\operatorname{Hom}(QB,D_i)\\ &=\operatorname{Hom}(QB,\operatorname{colim}_iD_i)\\ &=\operatorname{Hom}(B,R\operatorname{colim}_iD_i). \end{aligned} \tag{6.2}$ The noetherian objects include a detecting generating family, so the comparison is invertible. This verifies its actual maps.

For injective preservation, transpose a map from a subobject of $\mathcal A$ into $RD$ across $Q\dashv R$. Exactness of $Q$ preserves the subobject inclusion, so injectivity of $D$ supplies the extension. Transpose back. The same proof applies to the exact inclusion $\mathcal T\hookrightarrow\mathcal A$ and its right adjoint $\tau$, establishing injectivity inside $\mathcal T$. An extension along a general ambient monomorphism is a different assertion, and Exercise 8.2 shows its failure. Exercise 8.1 gives a noetherian quotient object whose section is not ambient noetherian. $\square$

7. Transport through an equivalence

Proposition 7.1. An exact equivalence $F\colon\mathcal A\to\mathcal A'$ transports the correspondence (3.1), the torsion coreflection, and the quotient and its section, with their natural structural maps.

Proof. A quasi-inverse transports subobjects, their inclusion order, quotient objects and exact rows. It therefore preserves and reflects noetherianness and Serre closure. An equivalence also preserves every colimit that exists: applying its quasi-inverse to a proposed cocone converts the universal Hom bijection into the original one. Thus $F\mathcal T$ is localizing, and its intersection with the finite part is $F\mathcal S$. Theorem 3.1 gives the same correspondence on both sides.

The subobject $F\tau M\subset FM$ is the largest $F\mathcal T$-subobject: every such subobject transports back to a $\mathcal T$-subobject of $M$. The unique comparison to $\tau'FM$ respects their inclusions and is invertible. It is natural by monic cancellation. Equivalently it is the right-adjoint comparison of the transported inclusion, so preserves the adjunction maps.

The exact functor from $\mathcal A$ to $\mathcal A'/F\mathcal T$ induced by $F$ kills $\mathcal T$. The full exact quotient universal property factors it through $\mathcal A/\mathcal T$. The quasi-inverse factors in the other direction, and their unit and counit transformations extend through the quotient fractions. These factors are inverse exact equivalences with those specified natural transformations. Their right-adjoint comparison transports the section and its counit. On finite objects the same construction is $\mathcal B/\mathcal S\to\mathcal B'/F\mathcal S$; its Ind extension agrees with the preceding quotient comparison, since both preserve filtered colimits and agree on the constants and all their arrows. The complete finite-part extension theorem proves that uniqueness. This proves the simultaneous compatibility claimed. $\square$

8. Four graded exercises with complete solutions

Exercise 8.1 — Classify all localizing subcategories of abelian groups (advanced)

Classify the Serre subcategories of finitely generated abelian groups and hence all localizing subcategories of $\mathsf{Ab}$. Give the quotient for each proper class. For each nonzero proper class, exhibit a noetherian quotient object whose section is not noetherian, and explain the zero class separately.

Solution. Integer ideals are principal by division by a least positive element; ascending chains of nonzero ideals stabilize because their generators are divisors of the first generator. Thus $\mathbb Z$ is a noetherian generator, and the noetherian objects of $\mathsf{Ab}$ are precisely the finitely generated groups, by the preceding lesson.

A proper Serre class cannot contain a group with an element of infinite order: its generated subgroup is $\mathbb Z$, and every finitely generated group is a quotient of a finite sum of copies of $\mathbb Z$. A finitely generated torsion group is finite: choose generators killed by positive integers $n_i$; their direct sum of cyclic groups maps onto it and is finite. A finite abelian group has a finite filtration with factors $\mathbb Z/p$: a nonzero element generates a finite cyclic subgroup containing a subgroup of prime order, and induction applies to the smaller quotient. A simple nonzero finite group is consequently exactly $\mathbb Z/p$.

It follows that a proper Serre class is determined by a set $\Pi$ of primes, and consists of the finite groups all of whose composition-factor primes belong to $\Pi$. Both directions hold: subquotients of such finite groups have the same prime restriction, extensions concatenate filtrations, and a Serre class containing those simple groups contains every group with such a filtration. The extra class is the whole finitely generated category.

Theorem 3.1 gives the corresponding proper localizing class $\mathcal T_\Pi$: its groups have each element killed by a positive integer whose prime factors are in $\Pi$. Indeed the cyclic subgroup of each element must be finite with that prime restriction; conversely every finitely generated subgroup of such a group is finite with that restriction. The full finitely generated category corresponds to all of $\mathsf{Ab}$; the class of all finite groups corresponds to ordinary torsion groups, with $\Pi$ equal to all primes.

Let $W$ be the multiplicative set generated by $\Pi$, and $\Lambda=W^{-1}\mathbb Z$. Localization $M\mapsto W^{-1}M$ is exact: represent a class by $m/s$; it is zero exactly when some $w\in W$ kills $m$. This test proves injectivity after localization of a monomorphism, and lifts and the same vanishing test prove exactness at the middle and surjectivity. Its kernel is precisely $\mathcal T_\Pi$. Restriction of $\Lambda$-modules is its right adjoint by the rule extending a linear map uniquely to $m/s\mapsto s^{-1}f(m)$. Its counit is invertible because all denominators already act invertibly. The complete exact quotient/section comparison therefore gives $\mathsf{Ab}/\mathcal T_\Pi\simeq\Lambda\text{-}\operatorname{Mod}, \tag{8.1}$ with that actual localization functor. For $\Pi=\varnothing$ this is identity; for all primes it is rationalization. The quotient by the whole category is zero.

If $p\in\Pi$, the quotient object $\Lambda$ is noetherian by Theorem 5.2 as the image of $\mathbb Z$. Its section is the underlying group $\Lambda$, which is not finitely generated: finitely many rational generators have a common denominator $d$, while $1/p^n$ is not an integer multiple of $1/d$ once $n$ exceeds the exponent of $p$ in $d$. Hence the section does not preserve noetherian objects. $\square$

Exercise 8.2 — A quiver diamond and injective torsion (intermediate)

Let $\mathcal A$ be the category of all arrows $f\colon M\to N$ of vector spaces over a field $k$. Classify its localizing subcategories, compute their torsion coreflections and quotient sections, and exhibit an ambient injective whose torsion part is not ambient injective.

Solution. Kernels, cokernels and colimits are componentwise, so this is Grothendieck. The two finite-dimensional generators $(k\xrightarrow1 k)$, $(0\to k)$ have Hom functors equal to evaluation at $M,N$; these jointly detect isomorphisms. Finite-dimensional arrows are noetherian by dimension. Every arrow is a directed union of finite-dimensional subarrows: choose finite spans in $M$, and a finite span in $N$ containing their images. Finite generation then makes every noetherian object finite-dimensional, and conversely. This proves local noetherianness and its finite part.

The finite category has two simple objects $S_1=(k\to0)$, $S_2=(0\to k)$. In fact every finite arrow has the exact row $0\to(0\to N)\to(M\xrightarrow fN)\to(M\to0)\to0, \tag{8.2}$ and filtrations of the finite spaces give factors of those two types. A nonzero simple arrow must therefore be one of them. Its Serre classes, and consequently its localizing classes, form the four-member inclusion diamond: zero, $\mathcal T_1=\{M\to0\}$, $\mathcal T_2=\{0\to N\}$, and all arrows. The two middle classes are incomparable and their generated join is all arrows by (8.2).

Their torsion parts are respectively $\tau_1(M\xrightarrow fN)=(\ker f\to0), \qquad\tau_2(M\xrightarrow fN)=(0\to N), \tag{8.3}$ with their evident inclusion maps. Quotient by $\mathcal T_1$ is evaluation at $N$; its right section takes $V$ to $(V\xrightarrow1V)$. A map from $M\to N$ to that section is determined by its $N$-component, with $M$-component its composite with $f$. Quotient by $\mathcal T_2$ is evaluation at $M$; its section takes $V$ to $(V\to0)$, and maps into it are exactly maps $M\to V$. Thus both displayed adjunctions have identity counits and exact quotient functors with the required kernels.

The arrow $I=(k\xrightarrow1k)$ is injective: $\operatorname{Hom}(-,I)=\operatorname{Hom}_k(N,k)$, whose extension property is basis extension, and the $N$-component of a monomorphism is monic. But $\tau_2I=S_2$ is not ambient injective. Its identity map cannot extend along $S_2\hookrightarrow I$: every map $I\to S_2$ has zero first component, and its commuting square forces its second component to be zero too. Internally $\mathcal T_2\simeq k\text{-}\operatorname{Mod}$, so $S_2$ is injective there. This verifies exactly the distinction in Proposition 6.1. $\square$

Exercise 8.3 — A nonzero localizing class invisible to finite presentation (advanced)

Construct a commutative valuation domain $R$ whose nonzero maximal ideal $\mathfrak m$ is idempotent and not finitely generated. Prove that $\mathcal T=\{M\mid\mathfrak mM=0\} \tag{8.4}$ is a nonzero localizing subcategory of $R\text{-}\operatorname{Mod}$, but contains no nonzero finitely presented module. Explain why a finitely presented regular generator does not suffice for Theorem 3.1 with its noetherian finite part replaced by the finitely presented objects.

Solution. Take compatible indeterminates $x_n$, $n\ge0$, with $x_n=x_{n+1}^2$, and set $R_n=k[x_n]_{(x_n)},\qquad K=\bigcup_{n\ge0}k(x_n),\qquad R=\bigcup_{n\ge0}R_n\subset K. \tag{8.5}$ The embeddings send $x_n$ to $x_{n+1}^2$. Nonzero denominators with nonzero constant term remain invertible in the next local ring, so these are compatible embeddings of domains. Every nonzero rational function in $k(x_n)$ is $x_n^a u$ for an integer $a$ and a unit $u\in R_n$: extract the powers of $x_n$ from numerator and denominator. Define its value to be $a/2^n$. Under the embedding the exponent doubles, so this is well-defined on $K^\times$; multiplication adds values. Factoring out the smaller power in a sum at a common stage gives $v(f+g)\ge\min(v(f),v(g))$ whenever that sum is nonzero. An element is in $R$ exactly when its value is nonnegative. Thus every nonzero element of $K$ or its inverse lies in $R$, proving the valuation-domain property. Elements of value zero are units, while those of positive value, together with zero, form its maximal ideal $\mathfrak m$; evaluation of rational functions regular at zero is a compatible ring homomorphism identifying $R/\mathfrak m$ with $k$.

The ideal is nonzero. It is not finitely generated: any finite family of its nonzero elements has a least positive value $q$, and every combination of them has value at least $q$ when nonzero. Choose $n$ with $2^{-n}<q$. Then $x_n\in\mathfrak m$ is outside the generated ideal; an empty nonzero family generates zero and also misses it. Moreover every nonzero $a\in\mathfrak m$ is a product of two elements of $\mathfrak m$. Choose $n$ with $0<2^{-n}<v(a)$; then $a=x_n(a/x_n)$, and both factors have positive value. Zero is a product as well. Hence $\mathfrak m^2=\mathfrak m$. In particular $R$ is not noetherian: the principal ideals $(x_n)$ strictly increase as $n$ increases.

The class (8.4) is closed under subobjects, quotients and arbitrary coproducts. For an exact row $0\to L\to M\to N\to0$ whose ends are annihilated by $\mathfrak m$, we have $\mathfrak mM\subset L$, and hence $\mathfrak m^2M=0$. Idempotency gives $\mathfrak mM=0$. This proves extension closure and thus the full localizing property. It contains the nonzero module $k=R/\mathfrak m$.

This $k$ is not finitely presented. Here is a direct filtered-colimit obstruction, retaining the comparison map. Let $F$ run through the finitely generated subideals of $\mathfrak m$, directed by finite sum. The quotient diagram has colimit $R/\mathfrak m=k$: its relations kill precisely their union $\mathfrak m$. If $k$ were finitely presented, its identity would factor as $k\xrightarrow{s}R/F\to k$ for some $F$. Lift $s(1)$ to $t\in R$. Its residue is $1$, so $t$ is a unit of the local ring. Linearity and $\mathfrak m k=0$ give $\mathfrak m t\subset F$, forcing $\mathfrak m\subset F$, a contradiction. Thus the categorical Hom-colimit definition already proves non-finite-presentation; equivalently a module with finitely many generators and relations has that Hom property by moving its finitely many generator images and relation equations to a common filtered stage.

Any finitely presented $M$ in (8.4) is finitely generated. It is therefore a finite-dimensional $k$-space, since its finite generators have only their residue-field scalars acting. If $M\ne0$, a basis decomposes it as $k^d$ with $d\ge1$, and $k$ is an $R$-linear retract of $M$. Retracts of finitely presented objects are finitely presented: their Hom functors are the images of the natural splitting idempotents, and filtered Hom comparisons commute with those splittings. This is also the complete retract argument in the finite-piece prerequisite. It contradicts the preceding calculation. Consequently $\mathcal T$ has zero intersection with the finitely presented subcategory, just as the zero localizing class does.

The regular module $R$ is nevertheless a finitely presented generator: evaluation at $1$ identifies its Hom with underlying elements and commutes with every filtered module colimit. The two different localizing classes are therefore not determined by their finitely presented objects under this weaker hypothesis. The noetherian-subobject exhaustion used in Sections 1–3 is the step that fails. $\square$

Exercise 8.4 — Matrix Morita transport of a prime-subset quotient (intermediate)

Let $A=\operatorname{Mat}_d(\mathbb Z)$, $d\ge1$, and $e=E_{11}$. For a set of primes $\Pi$, transport Exercise 8.1 to left $A$-modules. Give its localizing class, torsion counit, quotient and section in actual matrix coordinates. Compute the torsion subobject of $M=Ae\otimes_{\mathbb Z} (\mathbb Z\oplus\mathbb Z/p^2\oplus\mathbb Z/q), \tag{8.6}$ where $p\in\Pi$ and $q\notin\Pi$ are distinct primes.

Solution. The equivalences are $M\mapsto eM$ and $N\mapsto Ae\otimes_{\mathbb Z}N$. Their actual reconstruction map and its inverse are $ae\otimes u\longmapsto au, \qquad m\longmapsto\sum_{a=1}^dE_{a1}\otimes E_{1a}m. \tag{8.7}$ The composite on $M$ is identity because $\sum_aE_{a1}E_{1a}=1$. For the other composite write the column $ae$ as $\sum_a a_{a1}E_{a1}$; multiplication by $E_{1b}$ selects $a_{b1}$, so the reconstruction returns the same column tensor. Also $e(Ae\otimes N)\simeq N$ by its first coordinate. These maps are natural and left $A$-linear, the two functors are exact, and their displayed composites give the adjunction identifications.

The desired class is $\mathcal T^A_\Pi=\{M\mid eM\in\mathcal T_\Pi\}$. Proposition 7.1 proves its localizing and finite-piece assertions. Its torsion subobject is $Ae\otimes\tau_\Pi(eM)\longrightarrow M, \qquad ae\otimes u\longmapsto au, \tag{8.8}$ which is monic because it is the exact reconstruction of the torsion inclusion. Maximality is preserved by the equivalence, so (8.8) is precisely the torsion counit, not just an isomorphic object with an unspecified map.

With $\Lambda=W^{-1}\mathbb Z$ as in Exercise 8.1, an actual quotient functor is $M\longmapsto\Lambda\otimes_{\mathbb Z}eM \quad\text{into }\Lambda\text{-}\operatorname{Mod}. \tag{8.9}$ It is exact and has kernel $\mathcal T^A_\Pi$. Its right section sends a $\Lambda$-module $D$ to $Ae\otimes_{\mathbb Z}D$, with $D$ restricted to $\mathbb Z$. Indeed the natural Hom bijection (8.7), followed by the localization adjunction, identifies maps into that section with $\Lambda$-linear maps from (8.9) to $D$. The counit is the identity localization of the first coordinate of this reconstructed column. It is invertible, giving $A\text{-}\operatorname{Mod}/\mathcal T^A_\Pi\simeq\Lambda\text{-}\operatorname{Mod}$ with its specified quotient and section. Composing with matrix Morita equivalence over $\Lambda$ also identifies the quotient with $\operatorname{Mat}_d(\Lambda)\text{-}\operatorname{Mod}$.

For (8.6), its $e$-coordinate is the displayed direct sum of three abelian groups. Its prime-subset torsion is exactly the middle summand $\mathbb Z/p^2$: no nonzero integer in the first summand is torsion, and any integer built from primes in $\Pi$ acts invertibly on $\mathbb Z/q$, since it is relatively prime to $q$ and the integer Bezout relation gives its inverse. Thus (8.8) is the inclusion of the middle column summand $Ae\otimes\mathbb Z/p^2$. Its quotient value (8.9) is $\Lambda\oplus\mathbb Z/q$, since localization kills the middle summand and is identity on the last one. All these comparisons use the actual matrix-unit maps above. $\square$

9. Freely accessible further reading

Ryo Kanda's freely accessible Atom spectra of Grothendieck categories, Proposition 7.2.1 and Corollary 7.2.7, states the noetherian Serre/localizing correspondence. The proof above uses noetherian subobject exhaustion and the complete Ind quotient interface; it uses no atom-spectrum classification theorem as an unproved input.

The Stacks Project, Serre quotients, treats the ordinary exact quotient. Its Valuation rings, especially Lemma 10.50.15, explains the principal finite-ideal property. The explicit dyadic valuation ring, invisible finite-part class, quiver calculations and matrix quotient maps are constructed and proved in this lesson.