h-Divisible Modules: Theory and Applications

• h-Divisible modules are R-modules that are realized as quotient modules of a Q-linear space or via localization, capturing a refined notion of divisibility.
• They extend classical module divisibility through relative settings using multiplicative subsets and ideal filters, linking to J-injectivity and strongly flat covers.
• In arithmetic and p-adic contexts, h-divisibility characterizes finite heights in Kisin and Wach modules, influencing Hodge–Tate weight determination and related Galois phenomena.

Searching arXiv for recent and foundational papers on h-divisible modules and related usages of the term. An h-divisible module is not a uniform notion across the literature represented here. In one standard commutative-algebra usage, for an integral domain $R$ with field of fractions $Q$, an $R$-module is h-divisible if it is a quotient module of a $Q$-linear space. In an $S$-relative version, for a multiplicative subset $S\subseteq R$, one defines

$h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$

and calls $M$ $S$-h-divisible when $h_S(M)=M$. In the general-ring theory of division in modules, the closest precise replacement is not a separate term “h-divisible” but $Q$0-injectivity relative to an ideal filter $Q$1. In a different arithmetic-geometric usage, “h-divisible” denotes finite height $Q$2 for $Q$3- or Kisin modules, expressed by annihilation of $Q$4 by a power of a distinguished element $Q$5 (Zhang, 1 Sep 2025, Tronto, 2021, Cais et al., 2014).

1. Classical module-theoretic meaning over an integral domain

Let $Q$6 be an integral domain and $Q$7 its field of fractions. In this setting, an $Q$8-module $Q$9 is divisible if $R$0 for all nonzero $R$1, and h-divisible if $R$2 is a quotient module of a $R$3-linear space. The implication

$R$4

is immediate in this framework. A central characterization is that “every divisible module is h-divisible” if and only if $R$5 is a Matlis domain, namely a domain satisfying

$R$6

meaning that the projective dimension of the $R$7-module $R$8 is at most $R$9 (Zhang, 1 Sep 2025).

This places h-divisibility strictly within localization theory: the module is not merely closed under division by nonzero elements of $Q$0, but is obtained as a quotient of an object already linear over $Q$1. The same source records the cover-theoretic consequences established earlier for domains: over any integral domain, every h-divisible module admits a strongly flat cover, while every divisible module admits a strongly flat cover if and only if $Q$2 is a Matlis domain (Zhang, 1 Sep 2025).

Within this domain setting, h-divisible modules are therefore tied simultaneously to localization, projective dimension of $Q$3, and approximation theory by strongly flat modules. A plausible implication is that the term captures a divisibility notion that is stronger than elementwise surjectivity of multiplication maps and more closely aligned with passage to the fraction field.

2. Relative h-divisibility with respect to a multiplicative subset

For a commutative ring $Q$4 with identity and a multiplicative subset $Q$5, the relative theory replaces the fraction field by the localization $Q$6. The basic notions are $Q$7-torsion-free, $Q$8-torsion, $Q$9-divisible, $S$0-reduced, and $S$1-injective. Here $S$2 is $S$3-divisible if $S$4 for every $S$5, and $S$6 is $S$7-injective if

$S$8

for every $S$9-ideal $S\subseteq R$0 of $S\subseteq R$1. The relative analogue of h-divisibility is

$S\subseteq R$2

while

$S\subseteq R$3

An equivalent characterization is

$S\subseteq R$4

Dually,

$S\subseteq R$5

Consequently, $S\subseteq R$6-h-divisible modules are closed under quotients, and $S\subseteq R$7-h-reduced modules are closed under submodules (Zhang, 1 Sep 2025).

When $S\subseteq R$8 is regular, the localization $S\subseteq R$9 governs both divisibility and injectivity. Every $h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$0-module is $h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$1-divisible, and the cited lemma states that every $h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$2-module is also $h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$3-injective. Under the same regularity hypothesis, an $h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$4-module is $h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$5-h-divisible if and only if it is a quotient of an $h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$6-injective $h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$7-module. This identifies $h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$8-h-divisibility as a quotient condition relative to the localization rather than ordinary injectivity over $h_S(M)=\sum_{f\in{}_R(R_S,M)}(f),$9.

A standard caution is that $M$0-h-divisible need not coincide with “quotient of an injective module.” The example

$M$1

with $M$2 the set of all non-zero-divisors shows that $M$3 is $M$4-h-divisible because it is a total ring of quotients, but $M$5 is not a quotient of an injective $M$6-module. This separates the $M$7-relative theory sharply from the absolute one.

3. General-ring divisibility via ideal filters and $M$8-injective modules

For rings that are unitary and not necessarily commutative, the theory of division in modules replaces h-divisibility by a divisibility formalism indexed by an ideal filter $M$9. An ideal filter $S$0 of a ring $S$1 is a non-empty set of right ideals such that $S$2 implies $S$3, and $S$4 together with $S$5 implies $S$6. Typical examples are

$S$7

and

$S$8

For left $S$9-modules $h_S(M)=M$0 and a right ideal $h_S(M)=M$1, the $h_S(M)=M$2-division module is

$h_S(M)=M$3

and the $h_S(M)=M$4-division module is

$h_S(M)=M$5

The associated $h_S(M)=M$6-torsion submodule is $h_S(M)=M$7 (Tronto, 2021).

A homomorphism $h_S(M)=M$8 is a $h_S(M)=M$9-map if $Q$00, equivalently if $Q$01 is $Q$02-torsion. The corresponding generalization of injectivity is the notion of a $Q$03-injective module: a left $Q$04-module $Q$05 is $Q$06-injective if for every $Q$07-injective monomorphism $Q$08 and every homomorphism $Q$09, there exists $Q$10 with $Q$11. The Baer-type criterion states that $Q$12 is $Q$13-injective if and only if for every two-sided ideal $Q$14 and every $Q$15, there exists $Q$16 extending $Q$17.

In this framework, the paper explicitly notes that it does not define “h-divisible module.” The closest and precise general notion is $Q$18-injectivity. For $Q$19 and $Q$20, one has

$Q$21

and for $Q$22, $Q$23-injectivity coincides with divisibility by all integers, hence with injectivity for $Q$24-modules. The theory also furnishes generalized injective hulls: every left $Q$25-module admits a $Q$26-hull, and $Q$27-hulls are unique up to isomorphism commuting with the embedding. Over left-Noetherian rings, product-closed ideal filters are complete, which supplies the idempotence and closure properties needed for this construction.

This suggests that, over a general ring, any attempt to speak of “h-divisibility” should first specify the class of ideals or elements encoding division. In the cited formalism, that data is carried by $Q$28, and the natural exact notion is $Q$29-injectivity rather than an undifferentiated adjective “h-divisible.”

4. Homological and tensor formulations: the $Q$30-h-divisible variant

A separate development appears in the $Q$31-theoretic study of Prüfer $Q$32-multiplication domains. There the ambient ring is an integral domain $Q$33 with quotient field $Q$34, and torsion is replaced by GV-torsion, defined using finitely generated GV-ideals. Within this framework, the classical Hattori–Davis theory is restated homologically: for a torsion-free module $Q$35 over a domain, the classical notion of h-divisible can be understood by the condition

$Q$36

equivalently by flatness of $Q$37, and equivalently by injectivity of the Davis map

$Q$38

for every torsion-free $Q$39, where $Q$40 and $Q$41 (Zhang et al., 17 Sep 2025).

The cited paper then introduces the $Q$42-analogue. A torsion-free module $Q$43 is $Q$44-h-divisible if

$Q$45

equivalently if $Q$46 is $Q$47-flat. The main equivalence states that $Q$48 is a Prüfer $Q$49-multiplication domain if and only if the following hold: $Q$50 is GV-torsion for all $Q$51; equivalently $Q$52-w.gl.dim$Q$53; equivalently $Q$54 is GV-torsion for all $Q$55 and torsion-free $Q$56; equivalently the Davis map has GV-torsion kernel. In this setting, every torsion-free module is $Q$57-h-divisible.

Several structural consequences mirror classical Prüfer theory only after $Q$58-localization. Torsion submodules are $Q$59-pure, and for finitely generated or $Q$60-finitely generated $Q$61, the sequence

$Q$62

$Q$63-splits. The same paper emphasizes an important correction to naive tensor intuition: even over a PVMD, one cannot replace the Davis-map condition by literal GV-torsion-freeness of $Q$64 for arbitrary torsion-free $Q$65. The correct statement is that the kernel of the Davis map is GV-torsion. It also gives a module-theoretic characterization of PVMDs by injectivity: $Q$66 is a PVMD if and only if every pure $Q$67-injective divisible $Q$68-module is injective.

5. Divisibility formalism and Kummer theory

The ideal-filter theory of division has direct arithmetic applications. Let $Q$69 be a field with separable closure $Q$70, let $Q$71 be a commutative algebraic group over $Q$72, let $Q$73, let $Q$74 be an $Q$75-submodule, let $Q$76 be a complete ideal filter of $Q$77, and set

$Q$78

If $Q$79 is $Q$80-injective, then $Q$81 is a saturated and normal $Q$82-extension of $Q$83. The Galois sequence

$Q$84

is aligned with the module-theoretic sequence

$Q$85

through the embeddings

$Q$86

$Q$87

and

$Q$88

(Tronto, 2021).

The defect of Kummer surjectivity is controlled by the exact sequence

$Q$89

The main abstract Kummer theorem states that if the $Q$90-submodule of $Q$91 generated by $Q$92 is finitely generated, and if there exist positive integers $Q$93 such that

$Q$94

$Q$95

and the subring of $Q$96 generated by $Q$97 contains $Q$98, then

$Q$99

For elliptic curves this becomes effective. If $R$00 is a number field, $R$01 an elliptic curve, $R$02, and $R$03, then

$R$04

and

$R$05

The torsion module $R$06 is $R$07-injective. Under effective computability of the abelian group structures of $R$08 and $R$09, there exists an effectively computable $R$10 such that the index of $R$11 in $R$12 divides $R$13. The resulting degree bound is

$R$14

for every $R$15, with $R$16 depending only on the $R$17-module structure of $R$18 and on the image of the $R$19-adic Galois representations attached to $R$20. The paper states that this extends explicit Kummer bounds known for CM curves, due to Javan Peykar, and for the non-CM case, due to Lombardo and the author, in a unified divisibility-based framework.

6. “h-divisible” as finite height in $R$21-adic Hodge theory

In the theory of Wach and Kisin modules attached to $R$22-divisible groups, the adjective “h-divisible” is used in a different sense. Let $R$23 be a finite free module over

$R$24

equipped with semilinear Frobenius $R$25 and a compatible $R$26-action. Writing

$R$27

the module is said to be of finite height $R$28, or “h-divisible,” if

$R$29

where

$R$30

Equivalently, $R$31 becomes an isomorphism after inverting $R$32 (Cais et al., 2014).

In the cyclotomic realization used there, the analogous Barsotti–Tate condition for a BT$R$33-module $R$34 requires $R$35 to be annihilated by $R$36, and finite height $R$37 is the analogous condition with $R$38. The parameter $R$39 controls Hodge–Tate weights: for crystalline representations with weights in $R$40, the associated Kisin modules have height $R$41. For $R$42-divisible groups over $R$43, the Hodge–Tate weights lie in $R$44, so $R$45 is the relevant case.

This height language underlies the classification theorem

$R$46

compatible with duality and change of $R$47 and $R$48. It is also compatible with the Kisin–Ren theory of finite $R$49-height $R$50-modules and with Wach modules. For a crystalline representation $R$51, the Wach module $R$52 recovers the filtered Frobenius module through

$R$53

with filtration

$R$54

The examples $R$55 and $R$56 exhibit height $R$57 and height $R$58, respectively.

The coexistence of this finite-height usage with the algebraic notions above is a recurrent source of ambiguity. In the arithmetic-geometric setting, “h-divisible” measures the Frobenius height of a semilinear module; in commutative algebra, it measures quotient-liftability from a localization such as $R$59 or $R$60; and in the general-ring/Kummer setting, the exact analogue is best expressed through $R$61-injectivity rather than through a stand-alone term.