---
title: KE-Closed Subcategories Overview
url: https://www.emergentmind.com/topics/ke-closed-subcategories
type: topic
---

# KE-Closed Subcategories Overview

The expression **KE-closed subcategory** is not uniform across the literature. In tensor-triangulated work of Benson–Iyengar–Krause, it refers to a **Hom-closed colocalizing subcategory**, a usage identified with the Krause–Emmanouil terminology; in abelian and module-theoretic settings it usually denotes subcategories closed under **kernels** and **extensions**; and in categorical topology it has been used for **left Kan-extendable subcategories**. Despite the terminological divergence, the recurring theme is that a closure condition singles out subcategories controlled by an ambient spectrum, lattice, or comonadic construction [1008.3701] [2309.01044] [2511.10513].

## 1. Terminological scope and basic closure patterns

In a triangulated category \(T\), a **localizing** subcategory is a full triangulated subcategory closed under all set-indexed coproducts, while a **colocalizing** subcategory is closed under all set-indexed products. In a tensor triangulated category with internal function object \(\mathrm{Hom}(X,Y)\) and compact unit \(\mathbf 1\), a colocalizing subcategory \(S\) is **Hom-closed** if any of the following equivalent conditions hold:

\[
\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,
\]
\[
\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,
\]
\[
\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.
\]

In the stable module category \(\mathrm{StMod}(kG)\), being closed under tensor product with simples is equivalent to being Hom-closed; these are the **KE-closed** subcategories in the sense of Krause–Emmanouil [1008.3701].

In an abelian category, the phrase usually has a different meaning. A full subcategory is called **KE-closed** when it is closed under kernels and extensions. In a Grothendieck category, closure under subobjects already implies kernel-closure, because kernels are subobjects of sources; hence weakly closed or closed subcategories are automatically kernel-closed, but need not be extension-closed. This is why the abelian literature often aligns “KE-closed” more closely with **localizing** or **torsion-free** behavior than with the triangulated Hom-closed notion [2411.13706] [1405.4473].

The same initials appear again in categorical topology, where they stand for **Kan-extendable** rather than kernels-and-extensions. There a subcategory \(W\subseteq C\) is left Kan-extendable if the inclusion \(J:W\to C\) admits a pointwise left Kan extension along itself and the induced density comonad is idempotent [2511.10513]. The common label therefore hides materially different structures.

## 2. Hom-closed colocalizing subcategories and cosupport

The most systematic use of “KE-closed” in a triangulated setting is the classification of Hom-closed colocalizing subcategories via **local homology** and **cosupport**. Let \(T\) be a compactly generated \(R\)-linear triangulated category with set-indexed coproducts and products, where \(R\) is a graded-commutative noetherian ring acting via the graded center. For a specialization-closed \(\mathcal V\subseteq \mathrm{Spec}\,R\), local cohomology is the colocalization \(I_{\mathcal V}\), and local homology is its right adjoint \(A_{\mathcal V}\). For a prime \(\mathfrak p\), Benson–Iyengar–Krause define

\[
\Lambda_{\mathfrak p}:=A_{V(\mathfrak p)}A_{Z(\mathfrak p)},\qquad
Z(\mathfrak p)=\{\,\mathfrak q\in \mathrm{Spec}R\mid \mathfrak q\subseteq \mathfrak p\,\},
\]

together with the adjunction

\[
\mathrm{Hom}_T(I_{\mathfrak p}Y,X)\cong \mathrm{Hom}_T\bigl(Y,\Lambda_{\mathfrak p}X\bigr).
\]

The associated invariant is

\[
\operatorname{cosupp}_R(X)=\{\,\mathfrak p\in \operatorname{Spec}R\mid \Lambda_{\mathfrak p}X\neq 0\,\}.
\]

This cosupport detects vanishing: \(X=0\) if and only if \(\operatorname{cosupp}_R(X)=\varnothing\). It behaves well in triangles and under products, and for specialization-closed \(\mathcal V\) one has

\[
\operatorname{cosupp}_R(A_{\mathcal V}X)=\mathcal V\cap \operatorname{cosupp}_R(X),\qquad
\operatorname{cosupp}_R(V_{\mathcal V}X)=\bigl(\operatorname{Spec}R\setminus \mathcal V\bigr)\cap \operatorname{cosupp}_R(X).
\]

A key structural point is that \(\Lambda_{\mathfrak p}\) distributes over products, so for any \(U\subseteq \operatorname{Spec}R\), the class

\[
\{\,X\in T\mid \operatorname{cosupp}_R(X)\subseteq U\,\}
\]

is colocalizing. The paper also establishes the interaction with internal Hom,

\[
\operatorname{cosupp}_R(\mathrm{Hom}(X,Y))
\subseteq \operatorname{supp}_R(X)\cap \operatorname{cosupp}_R(Y),
\]

with equality under stratification [1008.3701].

These constructions culminate in a local-global principle for Hom-closed colocalizing subcategories:

\[
\mathrm{Coloc}_{\mathrm{Hom}}(X)
=
\mathrm{Coloc}_{\mathrm{Hom}}\bigl(\{\Lambda_{\mathfrak p}X\mid \mathfrak p\in \operatorname{Spec}R\}\bigr).
\]

A tensor triangulated category \(T\) is **costratified** by \(R\) if each \(\Lambda_{\mathfrak p}T\) admits no proper nonzero Hom-closed colocalizing subcategories. Under costratification, Hom-closed colocalizing subcategories are classified by arbitrary subsets of \(\operatorname{supp}_R(T)\), via

\[
\mathcal C\longmapsto \bigcup_{X\in \mathcal C}\operatorname{cosupp}_R(X),
\qquad
\mathcal C_{\mathcal W}
=
\{\,X\in T\mid \operatorname{cosupp}_R(X)\subseteq \mathcal W\,\}.
\]

This is the central classification theorem for KE-closed subcategories in the triangulated sense [1008.3701].

## 3. Principal examples in triangulated and representation-theoretic contexts

The prototype application is the stable module category \(\mathrm{StMod}(kG)\) of a finite group \(G\), where \(k\) has characteristic \(p\) dividing \(|G|\). Writing \(V_G\) for the set of homogeneous primes in \(H^*(G,k)\) excluding the maximal one, \(\mathrm{StMod}(kG)\) is costratified by \(H^*(G,k)\). Consequently, subsets \(U\subseteq V_G\) correspond bijectively to colocalizing subcategories closed under tensor with simples, equivalently Hom-closed colocalizing subcategories. One description is

\[
U\longmapsto
\{\,N\in \mathrm{StMod}(kG)\mid
\mathrm{Hom}_{kG}(M,N)=0
\text{ for all } M \text{ with } V_G(M)\subseteq U\,\}.
\]

Equivalently, these are the subcategories defined by cosupport:

\[
\mathcal C_{\mathcal W}
=
\{\,X\in \mathrm{StMod}(kG)\mid
\operatorname{cosupp}_{H^*(G,k)}(X)\subseteq \mathcal W\,\},
\qquad
\mathcal W\subseteq V_G.
\]

The same paper proves a bijection between tensor-ideal localizing subcategories and Hom-closed colocalizing subcategories via orthogonals \(S\mapsto S^\perp\) and \(U\mapsto {}^\perp U\) [1008.3701].

Two further tensor-triangulated examples are treated uniformly. If \(A\) is a graded exterior algebra on generators in negative odd degrees with zero differential, then \(K(\mathrm{Inj}\,A)\) is costratified by \(\operatorname{Ext}^*_A(k,k)\cong k[x_1,\dots,x_c]\), and Hom-closed colocalizing subcategories correspond to arbitrary subsets of \(\operatorname{Spec}\operatorname{Ext}^*(k,k)\). If \(A\) is a formal dg algebra with \(H^*A\) graded-commutative and noetherian, then \(D(A)\) is costratified by \(H^*A\), independent of the chosen zig-zag of quasi-isomorphisms, and Hom-closed colocalizing subcategories of \(D(A)\) are classified by arbitrary subsets of \(\operatorname{Spec}H^*A\) [1008.3701].

The paper also records concrete small cases. For \(G=C_p\), costratification yields only two Hom-closed colocalizing subcategories in \(\mathrm{StMod}(kC_p)\): \(0\) and the whole category. For \(G=E=(C_p)^r\), one has \(H^*(E,k)\cong k[x_1,\dots,x_r]\), and for finite-dimensional modules \(M\),

\[
\operatorname{cosupp}_G(M)=\operatorname{supp}_G(M).
\]

This identifies KE-closed subcategories with those determined by the classical support varieties of modules [1008.3701].

## 4. KE-closed subcategories in abelian and Grothendieck categories

In Grothendieck categories, the vocabulary shifts. A **weakly closed** subcategory is closed under subobjects, quotients, and arbitrary direct sums. A **closed** subcategory is a weakly closed subcategory also closed under arbitrary products. A **localizing** subcategory is weakly closed and extension-closed. Since kernels are subobjects, every weakly closed, and therefore every closed, subcategory is kernel-closed; but closed subcategories need not be extension-closed. For this reason, “KE-closed” aligns with **localizing** only if one intends closure under kernels and extensions together with the usual abelian-subcategory axioms [2411.13706].

The quotient theorem for closed subcategories is formulated in terms of a localizing subcategory \(Y\subseteq X\) and the Gabriel quotient \(T:X\to X/Y\). Assuming \(X\) satisfies \((AB4^*)\), closed subcategories of \(X/Y\) correspond not to arbitrary closed subcategories of \(X\), but to **\(Y\)-closed** ones: closed, \(Y\)-essentially stable, and \(Y\)-torsionfree generated. The assignments
\[
Z\mapsto T(Z),\qquad
Z'\mapsto (T^{-1}(Z'))'
\]
give inverse bijections between \(Y\)-closed subcategories of \(X\) and closed subcategories of \(X/Y\) [2411.13706].

For quasi-coherent sheaves on a locally noetherian scheme \(X\), Kanda–Matsui–Mizuno classify several closure notions by **local filters** of subobjects of \(\mathcal O_X\). In particular, **closed subcategories** of \(\mathrm{QCoh}(X)\) correspond to **principal local filters**, equivalently to quasi-coherent ideal subsheaves \(\mathcal I\subseteq \mathcal O_X\); the corresponding closed subcategory is
\[
\{\,\mathcal M\in \mathrm{QCoh}(X)\mid \mathcal M\mathcal I=0\,\}.
\]
These closed subcategories are in bijection with closed subschemes of \(X\). **Localizing** subcategories correspond to local filters closed under products, equivalently to specialization-closed subsets of \(X\), and **bilocalizing** subcategories correspond to idempotent ideal sheaves \(\mathcal I\) with \(\mathcal I^2=\mathcal I\), equivalently to open-and-closed subsets [1405.4473].

These results show that in abelian geometry the decisive distinction is usually not between localizing and colocalizing, but between **product closure**, **extension closure**, and **torsion-theoretic quotient behavior**.

## 5. Commutative noetherian rings: kernels-and-extensions as a depth-theoretic notion

For a commutative noetherian ring \(R\), both Kobayashi–Saito and the later Bass-function classification take **KE-closed** to mean an additive subcategory of \(\mathsf{mod}\,R\) closed under kernels and extensions. The structural theorem is that a KE-closed subcategory is exactly a **torsion-free class in a torsion-free class**. If \(F(\mathcal X)\) denotes the torsion-free closure of \(\mathcal X\), then
\[
\mathcal X \text{ is KE-closed }
\Longleftrightarrow
\mathcal X \text{ is a torsion-free class of } F(\mathcal X).
\]
This yields the dimension-sensitive consequence that if \(\dim R\le 1\), then KE-closed subcategories coincide with torsion-free classes [2309.01044].

For two-dimensional normal domains, the picture changes. If \(R\) is a two-dimensional noetherian normal domain, then
\[
\mathrm{ke}(\mathsf{mod}\,R)=\mathrm{torf}(\mathsf{mod}\,R)\cup \mathrm{dom.resol}(\mathsf{mod}\,R),
\]
and if \(R\) is local, then
\[
\mathrm{ke}(\mathsf{mod}\,R)=\mathrm{torf}(\mathsf{mod}\,R)\cup \{\mathrm{cm}\,R\}.
\]
Thus the maximal Cohen–Macaulay subcategory \(\mathrm{cm}\,R\) is the canonical non-torsion-free example in the two-dimensional normal local case [2309.01044].

The 2025 classification refines this by attaching to each KE-closed subcategory \(\mathcal C\subseteq \mathsf{mod}\,R\) the function
\[
f_{\mathcal C}(\mathfrak p)=\inf_{M\in \mathcal C}\operatorname{depth}(R_{\mathfrak p},M_{\mathfrak p}),
\]
whose finiteness locus is \(\operatorname{Supp}(\mathcal C)\). It satisfies Bass-type constraints: \(\operatorname{dom}(f_{\mathcal C})\) is specialization-closed; if \(\mathfrak p\) is minimal in the domain then \(f_{\mathcal C}(\mathfrak p)=0\); and for saturated inclusions \(\mathfrak p\subsetneq \mathfrak q\), one has \(f_{\mathcal C}(\mathfrak q)\le f_{\mathcal C}(\mathfrak p)+1\). These are the axioms of a **Bass function**. The associated subcategory is
\[
X_f=\{\,M\in \mathsf{mod}\,R\mid \operatorname{depth}(R_{\mathfrak p},M_{\mathfrak p})\ge f(\mathfrak p)\text{ for all }\mathfrak p\,\}.
\]

The paper proves that every KE-closed subcategory is reconstructed from its function:
\[
\mathcal X=X_{f_{\mathcal X}}.
\]
Under the hypothesis that \(R\) is \((S_2)\)-excellent in the sense of Česnavičius, KE-closed subcategories are classified by **\(2\)-Bass functions**, producing a bijection
\[
\mathrm{ke}(\mathsf{mod}\,R)\;\longleftrightarrow\; \mathrm{Bass}_2(\operatorname{Spec}R).
\]
This places KE-closed subcategories as the “\(n=2\)” layer above Serre subcategories (\(n=0\)) and torsion-free classes (\(n=1\)) [2509.05767].

## 6. Finite-dimensional algebras, classifying spaces, and further extensions

For a finite-dimensional basic algebra \(\Lambda\), the paper on \( \tau^{-1}\)-rigid modules uses a slightly different convention: a full additive subcategory of \(\mathrm{mod}\text{-}\Lambda\) is KE-closed if it is closed under **kernels of epimorphisms** and extensions. In \(\mathrm{mod}\text{-}\Lambda\), this is equivalent to being closed under **subobjects and extensions**, hence to being a **torsion-free class**. This identification is central to the paper’s description of decreasing sequences of maximal join intervals in the lattice \(\mathrm{torf}(\mathrm{mod}\text{-}\Lambda)\), and to the bijection between cogen-preordered \( \tau^{-1}\)-rigid modules and contravariantly finite ICE-sequences [2410.01963].

The closely related theory of **ICE-closed** subcategories, meaning closure under images, cokernels, and extensions, is developed for hereditary artin algebras. Over a Dynkin quiver \(Q\), ICE-closed subcategories of \(\mathrm{mod}\,kQ\) are in bijection with isomorphism classes of basic rigid \(kQ\)-modules via
\[
U\longmapsto \operatorname{cok}U,
\]
and every ICE-closed subcategory is a torsion class inside some wide subcategory. In type \(A_n\), the total number of ICE-closed subcategories equals the \(n\)-th large Schröder number [2005.05536]. This does not directly classify KE-closed subcategories in the kernels-and-extensions sense, but it clarifies the neighboring closure notions used in representation theory.

At a more abstract level, a complete lattice \(\Phi\) of subcategories of a fixed type admits a **classifying space** \(K(\Phi)\), and the subspace \(K_{gp}(\Phi)\) of generally prime points classifies the **g-primely generated** subcategories of that type. For a lattice of KE-closed subcategories \(\Phi_{KE}\), this yields a classification by closed subsets of \(K_{gp}(\Phi_{KE})\) whenever KE-closed subcategories are g-primely generated. The framework also shows a limitation: non-distributive lattices, or lattices with non-g-primely-generated elements, need not be classifiable by a single topological space in this sense [1709.02982].

A different categorical extension appears in locale theory. There, “KE-closed” is used as shorthand for **left Kan-extendable**. If \(W\subseteq C\) is left Kan-extendable and closeable in a bicomplete category, then the category \(W_l[C]\) of \(W\)-generated objects is coreflective and cartesian closed. Applied to compact strongly Hausdorff locales \(KHLoc\subseteq HLoc\), this produces the cartesian closed category
\[
kHLoc=(KHLoc)_l[HLoc]
\]
of compactly generated strongly Hausdorff locales [2511.10513].

Taken together, these literatures show that “KE-closed subcategory” is a family of context-dependent closure notions rather than a single invariant definition. In tensor-triangulated categories it is governed by cosupport and local homology; in abelian and module categories by kernel–extension behavior, torsion theory, and depth; in representation theory by torsion-free lattices and rigid objects; and in categorical topology by idempotent density comonads.

Source: https://www.emergentmind.com/topics/ke-closed-subcategories