---
title: 'h-Divisible Modules: Theory and Applications'
url: https://www.emergentmind.com/topics/h-divisible-module
type: topic
---

# h-Divisible Modules: Theory and Applications

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 $J$-injectivity relative to an ideal filter $J$. In a different arithmetic-geometric usage, “h-divisible” denotes finite height $\le h$ for $(\varphi,\Gamma)$- or Kisin modules, expressed by annihilation of $\operatorname{coker}(1\otimes\varphi)$ by a power of a distinguished element $E(u)$ [2509.01045] [2111.14363] [1412.3174].

## 1. Classical module-theoretic meaning over an integral domain

Let $R$ be an integral domain and $Q$ its field of fractions. In this setting, an $R$-module $M$ is divisible if $sM=M$ for all nonzero $s\in R$, and h-divisible if $M$ is a quotient module of a $Q$-linear space. The implication
$$
\text{h-divisible} \Longrightarrow \text{divisible}
$$
is immediate in this framework. A central characterization is that “every divisible module is h-divisible” if and only if $R$ is a Matlis domain, namely a domain satisfying
$$
{}_R Q \leq 1,
$$
meaning that the projective dimension of the $R$-module $Q$ is at most $1$ [2509.01045].

This places h-divisibility strictly within localization theory: the module is not merely closed under division by nonzero elements of $R$, but is obtained as a quotient of an object already linear over $Q$. 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 $R$ is a Matlis domain [2509.01045].

Within this domain setting, h-divisible modules are therefore tied simultaneously to localization, projective dimension of $Q$, 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 $R$ with identity and a multiplicative subset $S\subseteq R$, the relative theory replaces the fraction field by the localization $R_S$. The basic notions are $S$-torsion-free, $S$-torsion, $S$-divisible, $S$-reduced, and $S$-injective. Here $M$ is $S$-divisible if $sM=M$ for every $s\in S$, and $M$ is $S$-injective if
$$
{}_R^1(R/I,M)=0
$$
for every $S$-ideal $I$ of $R$. The relative analogue of h-divisibility is
$$
\text{S-}h\text{-divisible} \iff h_S(M)=M,
$$
while
$$
\text{S-}h\text{-reduced} \iff h_S(M)=0.
$$
An equivalent characterization is
$$
M\ \text{is S-}h\text{-divisible}\ \Longleftrightarrow\ \exists\ \text{an epimorphism}\ R_S^{(\kappa)}\twoheadrightarrow M.
$$
Dually,
$$
M\ \text{is S-}h\text{-reduced}\ \Longleftrightarrow\ {}_R(R_S,M)=0.
$$
Consequently, $S$-h-divisible modules are closed under quotients, and $S$-h-reduced modules are closed under submodules [2509.01045].

When $S$ is regular, the localization $R_S$ governs both divisibility and injectivity. Every $R_S$-module is $S$-divisible, and the cited lemma states that every $R_S$-module is also $S$-injective. Under the same regularity hypothesis, an $R$-module is $S$-h-divisible if and only if it is a quotient of an $S$-injective $R$-module. This identifies $S$-h-divisibility as a quotient condition relative to the localization rather than ordinary injectivity over $R$.

A standard caution is that $S$-h-divisible need not coincide with “quotient of an injective module.” The example
$$
R:=\mathbb{Z}(+)\mathbb{Q}/\mathbb{Z}
$$
with $S$ the set of all non-zero-divisors shows that $R$ is $S$-h-divisible because it is a total ring of quotients, but $R$ is not a quotient of an injective $R$-module. This separates the $S$-relative theory sharply from the absolute one.

## 3. General-ring divisibility via ideal filters and $J$-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 $J$. An ideal filter $J$ of a ring $R$ is a non-empty set of right ideals such that $I,I'\in J$ implies $I\cap I'\in J$, and $I\in J$ together with $I\subseteq I'$ implies $I'\in J$. Typical examples are
$$
p^\infty := \{ I \text{ right ideal } \mid I \supseteq p^nR \text{ for some } n\ge 1\},
$$
and
$$
\infty := \{ I \mid I \supseteq nR \text{ for some } n\ge 1\}.
$$
For left $R$-modules $M\subseteq N$ and a right ideal $I\subseteq R$, the $I$-division module is
$$
IMN := \{x\in N \mid Ix\subseteq M\},
$$
and the $J$-division module is
$$
JMN := \bigcup_{I\in J} IMN.
$$
The associated $J$-torsion submodule is $N[J]:=J0N$ [2111.14363].

A homomorphism $\varphi:M\to N$ is a $J$-map if $J(\varphi(M))N=N$, equivalently if $N/\varphi(M)$ is $J$-torsion. The corresponding generalization of injectivity is the notion of a $J$-injective module: a left $R$-module $Q$ is $J$-injective if for every $J$-injective monomorphism $i:M\to N$ and every homomorphism $f:M\to Q$, there exists $g:N\to Q$ with $g\circ i=f$. The Baer-type criterion states that $Q$ is $J$-injective if and only if for every two-sided ideal $I\in J$ and every $f:I\to Q$, there exists $g:R\to Q$ extending $f$.

In this framework, the paper explicitly notes that it does not define “h-divisible module.” The closest and precise general notion is $J$-injectivity. For $R=\mathbb{Z}$ and $J=p^\infty$, one has
$$
p^\infty\text{-injective} \iff p\text{-divisible},
$$
and for $J=\infty$, $J$-injectivity coincides with divisibility by all integers, hence with injectivity for $\mathbb{Z}$-modules. The theory also furnishes generalized injective hulls: every left $R$-module admits a $J$-hull, and $J$-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 $J$, and the natural exact notion is $J$-injectivity rather than an undifferentiated adjective “h-divisible.”

## 4. Homological and tensor formulations: the $w$-h-divisible variant

A separate development appears in the $w$-theoretic study of Prüfer $v$-multiplication domains. There the ambient ring is an integral domain $R$ with quotient field $K$, 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 $A$ over a domain, the classical notion of h-divisible can be understood by the condition
$$
Tor^R_1(X,A)=0 \quad \text{for every } R\text{-module } X,
$$
equivalently by flatness of $A$, and equivalently by injectivity of the Davis map
$$
\mu_{A,B}:A\otimes_R B\to \mathcal{T}\otimes_K\mathcal{S}
$$
for every torsion-free $B$, where $\mathcal{T}=K\otimes_R A$ and $\mathcal{S}=K\otimes_R B$ [2509.13617].

The cited paper then introduces the $w$-analogue. A torsion-free module $A$ is $w$-h-divisible if
$$
Tor^R_1(X,A)\ \text{is GV-torsion for all }X,
$$
equivalently if $A$ is $w$-flat. The main equivalence states that $R$ is a Prüfer $v$-multiplication domain if and only if the following hold: $Tor^R_2(M,N)$ is GV-torsion for all $M,N$; equivalently $w$-w.gl.dim$(R)\leq 1$; equivalently $Tor^R_1(X,A)$ is GV-torsion for all $X$ and torsion-free $A$; equivalently the Davis map has GV-torsion kernel. In this setting, every torsion-free module is $w$-h-divisible.

Several structural consequences mirror classical Prüfer theory only after $w$-localization. Torsion submodules are $w$-pure, and for finitely generated or $w$-finitely generated $M$, the sequence
$$
0\to T(M)\to M\to M/T(M)\to 0
$$
$w$-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 $M\otimes_R N$ for arbitrary torsion-free $M,N$. 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: $R$ is a PVMD if and only if every pure $w$-injective divisible $R$-module is injective.

## 5. Divisibility formalism and Kummer theory

The ideal-filter theory of division has direct arithmetic applications. Let $K$ be a field with separable closure $K_s$, let $G$ be a commutative algebraic group over $K$, let $R\subseteq End_K(G)$, let $M\subseteq G(K)$ be an $R$-submodule, let $J$ be a complete ideal filter of $R$, and set
$$
T:=G(K)[J], \qquad \Gamma:=JMG(K).
$$
If $T$ is $J$-injective, then $\Gamma$ is a saturated and normal $(J,T)$-extension of $M$. The Galois sequence
$$
1 \to Gal(K(\Gamma)\mid K(T)) \to Gal(K(\Gamma)\mid K) \to Gal(K(T)\mid K) \to 1
$$
is aligned with the module-theoretic sequence
$$
1 \to Hom(\Gamma/sat(M),T) \to Aut_M(\Gamma) \to Aut_{tor(M)}(T) \to 1
$$
through the embeddings
$$
\kappa: Gal(K(\Gamma)\mid K(T)) \hookrightarrow Hom(\Gamma/sat(M),T),
$$
$$
\tau: Gal(K(T)\mid K) \hookrightarrow Aut_{tor(M)}(T),
$$
and
$$
\rho: Gal(K(\Gamma)\mid K) \hookrightarrow Aut_M(\Gamma)
$$
[2111.14363].

The defect of Kummer surjectivity is controlled by the exact sequence
$$
0 \to J\,sat(M)\,sat(G(K))/sat(M) \to \ker(im(\kappa)) \to H^1(im(\tau),T).
$$
The main abstract Kummer theorem states that if the $End(T)$-submodule of $Hom(\Gamma/sat(M),T)$ generated by $im(\kappa)$ is finitely generated, and if there exist positive integers $d,n,m$ such that
$$
d\cdot J\,sat(M)\,sat(G(K)) \subseteq sat(M),
$$
$$
n\cdot H^1(im(\tau),T)=0,
$$
and the subring of $End(T)$ generated by $im(\tau)$ contains $m\cdot End(T)$, then
$$
im(\kappa)\supseteq dnm\cdot Hom(\Gamma/sat(M),T).
$$

For elliptic curves this becomes effective. If $K$ is a number field, $E/K$ an elliptic curve, $R=End_K(E)$, and $J=\infty$, then
$$
T=E(K)_{tors},
$$
and
$$
\Gamma=\{x\in E(K)\mid nx\in M \text{ for some } n\in \mathbb{Z}_{>0}\}.
$$
The torsion module $E(K)[J]$ is $J$-injective. Under effective computability of the abelian group structures of $E(K)$ and $M$, there exists an effectively computable $c>0$ such that the index of $im(\kappa)$ in $Hom(\Gamma/sat(M),T)$ divides $c$. The resulting degree bound is
$$
\frac{n^{2\rk_R(M)}[K(n^{-1}M):K(E[n])]}{} \qquad \text{divides} \qquad c
$$
for every $n\ge 1$, with $c$ depending only on the $R$-module structure of $M$ and on the image of the $\ell$-adic Galois representations attached to $E$. 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 $p$-adic Hodge theory

In the theory of Wach and Kisin modules attached to $p$-divisible groups, the adjective “h-divisible” is used in a different sense. Let $\mathfrak{M}$ be a finite free module over
$$
S:=W(k)[[u]]
$$
equipped with semilinear Frobenius $\varphi$ and a compatible $\Gamma$-action. Writing
$$
\varphi^*\mathfrak{M}:=S\otimes_{\varphi,S}\mathfrak{M},
$$
the module is said to be of finite height $\le h$, or “h-divisible,” if
$$
E(u)^h\cdot \mathrm{coker}(1\otimes\varphi)=0,
$$
where
$$
1\otimes\varphi:\varphi^*\mathfrak{M}\to \mathfrak{M}.
$$
Equivalently, $1\otimes\varphi$ becomes an isomorphism after inverting $E(u)$ [1412.3174].

In the cyclotomic realization used there, the analogous Barsotti–Tate condition for a BT$(G)$-module $(M,\varphi)$ requires $\operatorname{coker}(1\otimes\varphi)$ to be annihilated by $E$, and finite height $\le h$ is the analogous condition with $E^h$. The parameter $h$ controls Hodge–Tate weights: for crystalline representations with weights in $[0,h]$, the associated Kisin modules have height $\le h$. For $p$-divisible groups over $\mathscr{O}_K$, the Hodge–Tate weights lie in $\{0,1\}$, so $h=1$ is the relevant case.

This height language underlies the classification theorem
$$
\mathrm{pdiv}(\mathscr{O}_K)\ \overset{\sim}{\longleftrightarrow}\ \mathrm{BT}(G)^{I_K},
$$
compatible with duality and change of $K$ and $r$. It is also compatible with the Kisin–Ren theory of finite $E$-height $(\varphi,\Gamma)$-modules and with Wach modules. For a crystalline representation $V$, the Wach module $\mathcal{N}(V)$ recovers the filtered Frobenius module through
$$
D_{\mathrm{cris}}(V)\cong \big(\mathcal{N}(V)\otimes_S B_{\mathrm{cris}}^+\big)^{\Gamma=1},
$$
with filtration
$$
\mathrm{Fil}^i\,D_{\mathrm{cris}}(V)=\mathrm{image}\left(t^i\,\mathcal{N}(V)\otimes_S B_{\mathrm{cris}}^+\longrightarrow \mathcal{N}(V)\otimes_S B_{\mathrm{cris}}^+\right).
$$
The examples $G=\mathbb{Q}_p/\mathbb{Z}_p$ and $G=\mu_{p^\infty}$ exhibit height $0$ and height $1$, 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 $Q$ or $R_S$; and in the general-ring/Kummer setting, the exact analogue is best expressed through $J$-injectivity rather than through a stand-alone term.

Source: https://www.emergentmind.com/topics/h-divisible-module