Relatively Cotorsion Modules

• Relatively cotorsion modules are modules defined by the vanishing of Ext¹ with respect to a specific test class, generalizing the notion of classical cotorsion modules.
• They leverage cotorsion pairs, filtration techniques, and derived-categorical approaches to build and classify various homological approximation systems.
• These methods facilitate distinguishing between classical and relative cotorsion theories, elucidating their behavior under localization, finiteness conditions, and relative homomorphisms.

Relatively cotorsion modules are modules defined by Ext-orthogonality with respect to a specified class of test modules rather than with respect to all flat modules. For a class $\mathcal A$ of left $R$-modules, the corresponding relative cotorsion class is

$$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$

This construction places classical cotorsion theory into a broader framework: if $\mathcal A=\mathcal F$ is the class of flat modules, then $\mathcal A^\perp=\mathcal F^\perp$ is the usual class of cotorsion modules; if $\mathcal A$ is replaced by flat Mittag–Leffler modules, finitely $n$-presented modules, Gorenstein projectives, $S$-flat modules, or modules induced along a ring homomorphism, one obtains distinct relative cotorsion theories with their own approximation, filtration, and derived-categorical behavior (Cortés-Izurdiaga, 2016). The modern literature treats these classes not as isolated variants, but as instances of cotorsion pairs, Tor-pairs, balanced pairs, and higher or weak cotorsion structures (Bravo et al., 2015).

1. General formalism and the role of cotorsion pairs

A cotorsion pair in a module category is a pair $(\mathcal A,\mathcal B)$ such that

$$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$

where orthogonality is computed with $R$0. In this setting, modules in $R$1 are cotorsion relative to $R$2, while modules in $R$3 are the corresponding left-hand test objects. The cotorsion pair generated by a class $R$4 is

$R$5

and this is the basic mechanism by which relative cotorsion classes arise (Cortés-Izurdiaga, 2016).

Several papers emphasize that relative cotorsion theory is primarily a theory of orthogonality classes rather than a single distinguished notion. In the $R$6 context, one studies

$R$7

obtaining complete or perfect cotorsion pairs for every ring and every $R$8 (Bravo et al., 2015). In the Gorenstein-projective context, the pair

$R$9

is hereditary over every ring, so $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$0 is the relative cotorsion class with respect to Gorenstein projectives (Cortés-Izurdiaga et al., 2021). In higher-dimensional variants, left and right $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$1-cotorsion pairs replace degree-$$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$2 vanishing by vanishing up to degree $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$3, together with bounded relative resolution conditions (Huerta et al., 2019).

Completeness, heredity, and approximation are central structural properties. Completeness means the existence of special precovers and preenvelopes; hereditary means $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$4 for all $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$5 and all $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$6, $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$7. In Grothendieck categories, cotorsion pairs generated by a set are complete by the Eklof–Trlifaj theorem, and this mechanism is repeatedly used in the relative setting (Positselski, 9 Sep 2025). This suggests that relative cotorsion modules are best viewed as the right-hand side of a homological approximation system, not merely as isolated Ext-vanishing objects.

2. Principal relative cotorsion classes in the literature

Different choices of the testing class $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$8 produce distinct relative cotorsion theories. The following families recur in the literature.

Test class $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$9	Relative cotorsion class	Typical source
Flat modules $\mathcal A=\mathcal F$0	$\mathcal A=\mathcal F$1	Classical cotorsion
Flat Mittag–Leffler modules $\mathcal A=\mathcal F$2	$\mathcal A=\mathcal F$3	(Cortés-Izurdiaga, 2016)
Finitely $\mathcal A=\mathcal F$4-presented modules $\mathcal A=\mathcal F$5	$\mathcal A=\mathcal F$6-inj	(Bravo et al., 2015)
Gorenstein projectives $\mathcal A=\mathcal F$7	$\mathcal A=\mathcal F$8	(Cortés-Izurdiaga et al., 2021)
$\mathcal A=\mathcal F$9-flat modules $\mathcal A^\perp=\mathcal F^\perp$0	$\mathcal A^\perp=\mathcal F^\perp$1	(Bennis et al., 2024)
$\mathcal A^\perp=\mathcal F^\perp$2-flat and $\mathcal A^\perp=\mathcal F^\perp$3-projective modules along $\mathcal A^\perp=\mathcal F^\perp$4	$\mathcal A^\perp=\mathcal F^\perp$5 or $\mathcal A^\perp=\mathcal F^\perp$6	(Positselski, 9 Sep 2025)
Classes from semidualizing bimodules	$\mathcal A^\perp=\mathcal F^\perp$7 or $\mathcal A^\perp=\mathcal F^\perp$8-pd$\mathcal A^\perp=\mathcal F^\perp$9	(Tang et al., 2019)

The $\mathcal A$0 hierarchy interpolates between several standard theories. For $\mathcal A$1, $\mathcal A$2-injectives are injective modules and $\mathcal A$3-flat modules are flat modules; for $\mathcal A$4, $\mathcal A$5-injectives are absolutely pure modules; for $\mathcal A$6, $\mathcal A$7-injectives are absolutely clean modules and $\mathcal A$8-flat modules are level modules (Bravo et al., 2015). This gives a graded family of relative cotorsion theories controlled by finiteness conditions on presentations.

A different kind of relativity arises from multiplicative subsets. If $\mathcal A$9 is multiplicative, an $n$0-module is $n$1-weakly cotorsion when $n$2, and an $n$3-module is $n$4-cotorsion when $n$5 for every $n$6-flat module $n$7 (Positselski et al., 2017). The latter yields a hereditary perfect cotorsion pair $n$8 over any commutative ring (Bennis et al., 2024).

Relative cotorsion may also be encoded by local depth data. Over a commutative noetherian ring, hereditary cotorsion pairs cogenerated by pure-injective modules of finite injective dimension are classified by integer-valued functions $n$9 bounded by $S$0, and the left class is exactly the class of modules $S$1 satisfying

$S$2

(Herbera et al., 2024). Here the “relative cotorsion” condition is converted into a system of local depth inequalities.

3. When relative cotorsion coincides with classical cotorsion

One of the most studied questions is whether a relative cotorsion class actually recovers the classical cotorsion modules. A basic example is the class $S$3 of flat Mittag–Leffler modules. The main result of “The cotorsion pair generated by the class of flat Mittag–Leffler modules” states that if every flat left $S$4-module is filtered by totally ordered direct limits of projective modules, then

$S$5

so the cotorsion pair generated by $S$6 coincides with the Enochs flat–cotorsion pair (Cortés-Izurdiaga, 2016). This covers countable rings, left perfect rings, and discrete valuation domains.

The mechanism behind this identification is a closure theorem: if a class $S$7 is closed under direct sums and $S$8-free modules, then $S$9 contains all totally ordered direct limits of modules from $(\mathcal A,\mathcal B)$0 (Cortés-Izurdiaga, 2016). Applied to $(\mathcal A,\mathcal B)$1 and combined with Eklof’s filtration theorem, this moves from projectives to totally ordered direct limits of projectives, and then from these limits to all flat modules under the stated ring-theoretic filtration hypothesis.

A similar recovery phenomenon occurs in relative Bass theory for a ring homomorphism $(\mathcal A,\mathcal B)$2 with $(\mathcal A,\mathcal B)$3 finitely generated projective as a right $(\mathcal A,\mathcal B)$4-module. In that situation, a left $(\mathcal A,\mathcal B)$5-module is cotorsion over $(\mathcal A,\mathcal B)$6 if and only if its underlying $(\mathcal A,\mathcal B)$7-module is cotorsion: $(\mathcal A,\mathcal B)$8 Equivalently, every flat left $(\mathcal A,\mathcal B)$9-module is a direct summand of an $$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$0-module filtered by induced modules $$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$1 with $$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$2 flat over $$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$3 (Positselski, 21 Jul 2025). This yields a precise relative version of the familiar fact that over a finite-dimensional algebra over a field, every flat module is projective.

The literature also records explicit failures of the hypotheses used in such identifications. For flat Mittag–Leffler modules, there exists a von Neumann regular ring with a flat module that is not a totally ordered direct limit of projectives, so the filtration hypothesis is not automatic (Cortés-Izurdiaga, 2016). Likewise, for a ring map $$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$4 without the projectivity hypothesis, cotorsion over $$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$5 need not be detected by restriction to $$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$6; the example $$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$7, $$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$8 exhibits flat non-projective $$\mathcal A={}^{\perp_1}\mathcal B \qquad\text{and}\qquad \mathcal B=\mathcal A^{\perp_1},$$9-modules while every $R$00-vector space is cotorsion over $R$01 (Positselski, 21 Jul 2025). These examples delimit the range of equivalence theorems without asserting a general failure of relative identification beyond the stated hypotheses.

4. Relative cotorsion from finiteness, localization, and semidualizing data

The $R$02 framework organizes relative cotorsion via finiteness of presentations. A module is finitely $R$03-presented if it admits an exact sequence

$R$04

with each $R$05 finitely generated projective, and the corresponding right orthogonal $R$06 consists of the $R$07-injective modules (Bravo et al., 2015). For every ring and every $R$08, the pair

$R$09

is complete, while

$R$10

is perfect (Bravo et al., 2015). When $R$11 is $R$12-coherent, these classes stabilize: $R$13 and the associated cotorsion pairs become hereditary.

Localization produces another relative theory. For a multiplicative subset $R$14, $R$15-strongly flat modules are defined as those orthogonal to $R$16-weakly cotorsion modules, and under hypotheses such as countability of $R$17 or $R$18, flat $R$19 is $R$20-strongly flat exactly when $R$21 is projective for all $R$22 and $R$23 is projective (Positselski et al., 2017). The later notion of $R$24-cotorsion instead uses all $R$25-flat modules as test objects; the pair $R$26 is hereditary and perfect, and $R$27 is $R$28-perfect if and only if every module is $R$29-cotorsion (Bennis et al., 2024).

Semidualizing bimodules furnish a more elaborate relative homological algebra. Given a semidualizing $R$30-bimodule $R$31, one defines $R$32-projective, $R$33-flat, and $R$34-injective classes, together with the derived vanishing classes

$R$35

and the corresponding classes on the $R$36-side defined by Tor-vanishing (Tang et al., 2019). Under the cograde hypotheses of Theorem 4.19 in that paper, the pairs

$R$37

are complete cotorsion pairs (Tang et al., 2019). Here relative cotorsion is encoded by bounded $R$38-injective or $R$39-projective dimension rather than by flatness.

These constructions show that “relative cotorsion module” is not tied to a single geometric or algebraic source. It may reflect finiteness of relations, localization at multiplicative subsets, or semidualizing adjunction data, depending on which testing class governs the orthogonality.

5. Gorenstein, higher, and weak variants

Relative cotorsion theory interacts strongly with Gorenstein homological algebra. For any ring, $R$40 is a hereditary cotorsion pair, so modules in $R$41 are cotorsion relative to Gorenstein projectives (Cortés-Izurdiaga et al., 2021). Under the hypothesis that projective modules are $R$42-pure-injective for some infinite regular $R$43, the class of Gorenstein projectives is deconstructible, and the cotorsion pair is complete (Cortés-Izurdiaga et al., 2021). The same paper also proves that, assuming $R$44 does not exist, $R$45-pure-injective and pure-injective coincide.

A related but distinct development concerns projectively coresolved Gorenstein flat modules. The class $R$46 forms the left-hand side of a complete hereditary cotorsion pair

$R$47

and Gorenstein flat modules satisfy a short-exact-sequence characterization involving $R$48 and flat modules (Šaroch et al., 2018). In particular, there is a hereditary cotorsion pair

$R$49

generated by a set, and $R$50 is a covering class over every ring (Šaroch et al., 2018). Dually, the pair $R$51 is hereditary and perfect, so Gorenstein injectives also sit as the right class of a robust relative cotorsion theory (Šaroch et al., 2018).

Higher-dimensional generalizations are formalized by $R$52-cotorsion pairs. A left $R$53-cotorsion pair $R$54 requires

$R$55

together with the existence of short exact sequences whose kernels have $R$56-resolution dimension at most $R$57 (Huerta et al., 2019). This recovers ordinary cotorsion pairs when $R$58 and provides an appropriate language for higher Gorenstein and cluster-tilting situations.

There is also a weaker theory motivated by $R$59-tilting. A left weak cotorsion pair $R$60 in $R$61 requires $R$62 and approximation sequences on both sides, but does not require $R$63 (Buan et al., 2021). In such a pair,

$R$64

in the paper’s notation $R$65, so the left-hand class is still a relative cotorsion class, though the right-hand side is weaker than in classical cotorsion theory (Buan et al., 2021). For a support $R$66-tilting module $R$67, the pair

$R$68

is a left weak cotorsion pair (Buan et al., 2021).

6. Relative constructions along ring homomorphisms and derived-categorical frameworks

A particularly explicit relative theory starts from a ring homomorphism $R$69 and a hereditary complete cotorsion pair $R$70 in $R$71-Mod. Under suitable hypotheses, the induced cotorsion pair $R$72 in $R$73-Mod can be described by cofiltrations or filtrations built from coinduced or induced modules (Positselski, 2020). If every $R$74-module has finite $R$75-resolution dimension bounded by $R$76 and $R$77 is preserved by $R$78, then

$R$79

meaning that $R$80 consists exactly of direct summands of modules with finite cofiltrations by coinduced modules $R$81, $R$82 (Positselski, 2020). Under countable product hypotheses, one obtains $R$83-indexed or $R$84-indexed cofiltrations.

The 2025 paper on relative Bass theory specializes this philosophy to the classical flat–cotorsion pair and proves that, when $R$85 is finitely generated projective as a right $R$86-module, cotorsion over $R$87 is exactly cotorsion over $R$88 under restriction of scalars (Positselski, 21 Jul 2025). The proof uses derived change-of-rings adjunctions, the relative bar resolution, and the cotorsion periodicity theorem asserting that in an acyclic complex of cotorsion modules, all cocycles are cotorsion (Positselski, 21 Jul 2025).

Derived categories of the second kind provide a broader ambient framework. Given a hereditary complete cotorsion pair $R$89 generated by a set in a Grothendieck category, the Becker coderived category of $R$90 and the Becker contraderived category of $R$91 are naturally equivalent: $R$92 For cotorsion pairs associated with a ring homomorphism, such as the flaprojective pair $R$93 and the relatively cotorsion pair $R$94, these equivalences are controlled by periodicity properties (Positselski, 9 Sep 2025). In particular, for $R$95 with $R$96 projective over $R$97, the “Relatively Cotorsion Conjecture” asserts that the associated coderived and contraderived categories coincide with the extremal projective and flat cases precisely when relative cotorsion periodicity holds (Positselski, 9 Sep 2025).

This derived-categorical perspective does not replace module-theoretic relative cotorsion; rather, it reframes it. Relative cotorsion modules become the fibrant or contraadjusted objects underlying coderived–contraderived equivalences, Quillen adjunctions, and recollement patterns.

7. Structural themes, examples, and open directions

Several structural themes recur across the literature. First, relative cotorsion classes are often deconstructible or generated by a set, which yields completeness by Eklof–Trlifaj (Bravo et al., 2015). Second, many left-hand classes are resolving while the right-hand classes are coresolving; this is explicit for $R$98, for $R$99, and for cotorsion pairs arising from semidualizing bimodules (Bennis et al., 2024). Third, filtration and cofiltration methods are pervasive: flat modules filtered by totally ordered direct limits of projectives, $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$00-modules cofiltered by $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$01, and weakly cotorsion modules generated from $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$02 and $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$03 are representative examples (Cortés-Izurdiaga, 2016).

A common misconception is that “relative cotorsion” always means classical cotorsion with a different name. The literature shows otherwise. In some contexts, such as flat Mittag–Leffler modules over suitable rings or restriction along certain finite projective ring extensions, the relative class coincides with the classical cotorsion class (Cortés-Izurdiaga, 2016). In others, such as $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$04-injective, $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$05, or $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$06-cotorsion theories, it does not collapse to the flat–cotorsion theory and instead tracks different homological constraints (Bravo et al., 2015).

Another point of potential confusion is terminological. The phrase “cotorsion” may refer either to right Ext-orthogonals in the Enochs sense or, in the stabilization-theoretic sense, to the cotorsion quotient functor

$$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$07

where $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$08 is the trace of injectives in $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$09 (Martsinkovsky et al., 2016). That functorial notion is formally dual to a torsion functor and is distinct from cotorsion pairs, even though the same word appears in both theories.

Open directions are explicitly recorded in several places. For the flat Mittag–Leffler theory, the failure of the filtration hypothesis does not settle whether $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$10 in general (Cortés-Izurdiaga, 2016). For the relative Bass theorem, it remains open whether finite presentation of $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$11 as a right $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$12-module, without projectivity, suffices for the cotorsion equivalence (Positselski, 21 Jul 2025). In the coderived–contraderived framework, periodicity conjectures are formulated both for flaprojective and relatively cotorsion pairs associated with a ring homomorphism (Positselski, 9 Sep 2025).

Taken together, these developments present relatively cotorsion modules as a flexible and technically rich family of Ext-orthogonality classes. Their significance lies not only in the individual examples—flat Mittag–Leffler, $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$13, Gorenstein, $$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$14-flat, semidualizing, or relative-to-$$\mathcal A^\perp=\{\,M\mid \operatorname{Ext}^1_R(A,M)=0\text{ for all }A\in\mathcal A\,\}.$$15—but in the recurrent homological pattern: a chosen testing class determines a cotorsion pair, the cotorsion pair controls approximations and filtrations, and those approximations in turn interact with local algebra, model structures, and derived categories.