KE-Closed Subcategories Overview

• KE-closed subcategories are defined by context-specific closure properties, such as Hom-closure in tensor-triangulated settings, kernels and extensions in abelian contexts, or Kan-extendability in topology.
• In tensor-triangulated categories, they are classified via cosupport and local homology, linking closure under products and coproducts with spectrum invariants.
• In abelian and module categories, KE-closed structures relate to torsion theories and depth functions, while in categorical topology they emerge from left Kan extension processes.

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 (Benson et al., 2010, Kobayashi et al., 2023, Ghazel et al., 13 Nov 2025).

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 (Benson et al., 2010).

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 (Rogalski, 2024, Kanda, 2014).

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 (Ghazel et al., 13 Nov 2025). 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 $\mathrm{Hom}(X,Y)$0 be a compactly generated $\mathrm{Hom}(X,Y)$1-linear triangulated category with set-indexed coproducts and products, where $\mathrm{Hom}(X,Y)$2 is a graded-commutative noetherian ring acting via the graded center. For a specialization-closed $\mathrm{Hom}(X,Y)$3, local cohomology is the colocalization $\mathrm{Hom}(X,Y)$4, and local homology is its right adjoint $\mathrm{Hom}(X,Y)$5. For a prime $\mathrm{Hom}(X,Y)$6, Benson–Iyengar–Krause define

$\mathrm{Hom}(X,Y)$7

together with the adjunction

$\mathrm{Hom}(X,Y)$8

The associated invariant is

$\mathrm{Hom}(X,Y)$9

This cosupport detects vanishing: $\mathbf 1$0 if and only if $\mathbf 1$1. It behaves well in triangles and under products, and for specialization-closed $\mathbf 1$2 one has

$\mathbf 1$3

A key structural point is that $\mathbf 1$4 distributes over products, so for any $\mathbf 1$5, the class

$\mathbf 1$6

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

$\mathbf 1$7

with equality under stratification (Benson et al., 2010).

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

$\mathbf 1$8

A tensor triangulated category $\mathbf 1$9 is costratified by $S$0 if each $S$1 admits no proper nonzero Hom-closed colocalizing subcategories. Under costratification, Hom-closed colocalizing subcategories are classified by arbitrary subsets of $S$2, via

$S$3

This is the central classification theorem for KE-closed subcategories in the triangulated sense (Benson et al., 2010).

3. Principal examples in triangulated and representation-theoretic contexts

The prototype application is the stable module category $S$4 of a finite group $S$5, where $S$6 has characteristic $S$7 dividing $S$8. Writing $S$9 for the set of homogeneous primes in $$\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,$$0 excluding the maximal one, $$\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,$$1 is costratified by $$\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,$$2. Consequently, subsets $$\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,$$3 correspond bijectively to colocalizing subcategories closed under tensor with simples, equivalently Hom-closed colocalizing subcategories. One description is

$$\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,$$4

Equivalently, these are the subcategories defined by cosupport:

$$\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,$$5

The same paper proves a bijection between tensor-ideal localizing subcategories and Hom-closed colocalizing subcategories via orthogonals $$\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,$$6 and $$\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,$$7 (Benson et al., 2010).

Two further tensor-triangulated examples are treated uniformly. If $$\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,$$8 is a graded exterior algebra on generators in negative odd degrees with zero differential, then $$\text{(i) For all compact } X \text{ and } Y\in S,\; X\otimes Y\in S,$$9 is costratified by $$\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,$$0, and Hom-closed colocalizing subcategories correspond to arbitrary subsets of $$\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,$$1. If $$\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,$$2 is a formal dg algebra with $$\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,$$3 graded-commutative and noetherian, then $$\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,$$4 is costratified by $$\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,$$5, independent of the chosen zig-zag of quasi-isomorphisms, and Hom-closed colocalizing subcategories of $$\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,$$6 are classified by arbitrary subsets of $$\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,$$7 (Benson et al., 2010).

The paper also records concrete small cases. For $$\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,$$8, costratification yields only two Hom-closed colocalizing subcategories in $$\text{(ii) For all compact } X \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S,$$9: $$\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.$$0 and the whole category. For $$\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.$$1, one has $$\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.$$2, and for finite-dimensional modules $$\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.$$3,

$$\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.$$4

This identifies KE-closed subcategories with those determined by the classical support varieties of modules (Benson et al., 2010).

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 (Rogalski, 2024).

The quotient theorem for closed subcategories is formulated in terms of a localizing subcategory $$\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.$$5 and the Gabriel quotient $$\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.$$6. Assuming $$\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.$$7 satisfies $$\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.$$8, closed subcategories of $$\text{(iii) For all } X\in T \text{ and } Y\in S,\; \mathrm{Hom}(X,Y)\in S.$$9 correspond not to arbitrary closed subcategories of $\mathrm{StMod}(kG)$0, but to $\mathrm{StMod}(kG)$1-closed ones: closed, $\mathrm{StMod}(kG)$2-essentially stable, and $\mathrm{StMod}(kG)$3-torsionfree generated. The assignments

$\mathrm{StMod}(kG)$4

give inverse bijections between $\mathrm{StMod}(kG)$5-closed subcategories of $\mathrm{StMod}(kG)$6 and closed subcategories of $\mathrm{StMod}(kG)$7 (Rogalski, 2024).

For quasi-coherent sheaves on a locally noetherian scheme $\mathrm{StMod}(kG)$8, Kanda–Matsui–Mizuno classify several closure notions by local filters of subobjects of $\mathrm{StMod}(kG)$9. In particular, closed subcategories of $W\subseteq C$0 correspond to principal local filters, equivalently to quasi-coherent ideal subsheaves $W\subseteq C$1; the corresponding closed subcategory is

$W\subseteq C$2

These closed subcategories are in bijection with closed subschemes of $W\subseteq C$3. Localizing subcategories correspond to local filters closed under products, equivalently to specialization-closed subsets of $W\subseteq C$4, and bilocalizing subcategories correspond to idempotent ideal sheaves $W\subseteq C$5 with $W\subseteq C$6, equivalently to open-and-closed subsets (Kanda, 2014).

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 $W\subseteq C$7, both Kobayashi–Saito and the later Bass-function classification take KE-closed to mean an additive subcategory of $W\subseteq C$8 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 $W\subseteq C$9 denotes the torsion-free closure of $J:W\to C$0, then

$J:W\to C$1

This yields the dimension-sensitive consequence that if $J:W\to C$2, then KE-closed subcategories coincide with torsion-free classes (Kobayashi et al., 2023).

For two-dimensional normal domains, the picture changes. If $J:W\to C$3 is a two-dimensional noetherian normal domain, then

$J:W\to C$4

and if $J:W\to C$5 is local, then

$J:W\to C$6

Thus the maximal Cohen–Macaulay subcategory $J:W\to C$7 is the canonical non-torsion-free example in the two-dimensional normal local case (Kobayashi et al., 2023).

The 2025 classification refines this by attaching to each KE-closed subcategory $J:W\to C$8 the function

$J:W\to C$9

whose finiteness locus is $\mathrm{Hom}(X,Y)$00. It satisfies Bass-type constraints: $\mathrm{Hom}(X,Y)$01 is specialization-closed; if $\mathrm{Hom}(X,Y)$02 is minimal in the domain then $\mathrm{Hom}(X,Y)$03; and for saturated inclusions $\mathrm{Hom}(X,Y)$04, one has $\mathrm{Hom}(X,Y)$05. These are the axioms of a Bass function. The associated subcategory is

$\mathrm{Hom}(X,Y)$06

The paper proves that every KE-closed subcategory is reconstructed from its function: $\mathrm{Hom}(X,Y)$07 Under the hypothesis that $\mathrm{Hom}(X,Y)$08 is $\mathrm{Hom}(X,Y)$09-excellent in the sense of Česnavičius, KE-closed subcategories are classified by $\mathrm{Hom}(X,Y)$10-Bass functions, producing a bijection

$\mathrm{Hom}(X,Y)$11

This places KE-closed subcategories as the “$\mathrm{Hom}(X,Y)$12” layer above Serre subcategories ($\mathrm{Hom}(X,Y)$13) and torsion-free classes ($\mathrm{Hom}(X,Y)$14) (Kobayashi et al., 6 Sep 2025).

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

For a finite-dimensional basic algebra $\mathrm{Hom}(X,Y)$15, the paper on $\mathrm{Hom}(X,Y)$16-rigid modules uses a slightly different convention: a full additive subcategory of $\mathrm{Hom}(X,Y)$17 is KE-closed if it is closed under kernels of epimorphisms and extensions. In $\mathrm{Hom}(X,Y)$18, 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{Hom}(X,Y)$19, and to the bijection between cogen-preordered $\mathrm{Hom}(X,Y)$20-rigid modules and contravariantly finite ICE-sequences (Hanson, 2024).

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 $\mathrm{Hom}(X,Y)$21, ICE-closed subcategories of $\mathrm{Hom}(X,Y)$22 are in bijection with isomorphism classes of basic rigid $\mathrm{Hom}(X,Y)$23-modules via

$\mathrm{Hom}(X,Y)$24

and every ICE-closed subcategory is a torsion class inside some wide subcategory. In type $\mathrm{Hom}(X,Y)$25, the total number of ICE-closed subcategories equals the $\mathrm{Hom}(X,Y)$26-th large Schröder number (Enomoto, 2020). 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 $\mathrm{Hom}(X,Y)$27 of subcategories of a fixed type admits a classifying space $\mathrm{Hom}(X,Y)$28, and the subspace $\mathrm{Hom}(X,Y)$29 of generally prime points classifies the g-primely generated subcategories of that type. For a lattice of KE-closed subcategories $\mathrm{Hom}(X,Y)$30, this yields a classification by closed subsets of $\mathrm{Hom}(X,Y)$31 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 (Liu, 2017).

A different categorical extension appears in locale theory. There, “KE-closed” is used as shorthand for left Kan-extendable. If $\mathrm{Hom}(X,Y)$32 is left Kan-extendable and closeable in a bicomplete category, then the category $\mathrm{Hom}(X,Y)$33 of $\mathrm{Hom}(X,Y)$34-generated objects is coreflective and cartesian closed. Applied to compact strongly Hausdorff locales $\mathrm{Hom}(X,Y)$35, this produces the cartesian closed category

$\mathrm{Hom}(X,Y)$36

of compactly generated strongly Hausdorff locales (Ghazel et al., 13 Nov 2025).

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.