Compatible Cleft Extensions and Gorenstein Modules

• Compatible cleft extensions are a class of categorical extensions characterized by acyclicity conditions and the preservation of Gorenstein projective objects.
• They unify the treatment of module categories over triangular matrix algebras, Morita context rings, and θ-extensions through natural functors and complete projective resolutions.
• The framework establishes criteria where functors l and q transfer Gorenstein projectivity, with the endofunctor F measuring obstructions via its nilpotence and exactness properties.

Compatible cleft extensions are a homological refinement of Beligiannis’s cleft extensions of abelian categories. In the formulation introduced by Qin, a compatible cleft extension is a cleft extension $(\mathcal B,\mathcal A,e,i,l)$ satisfying two acyclicity conditions on complete projective resolutions; these conditions imply that both $l$ and the left adjoint $q$ of $i$ preserve Gorenstein projective objects, and they lead to recognition criteria for Gorenstein projectivity in $\mathcal A$ (Qin, 12 Jul 2025). The same framework unifies descriptions of Gorenstein projective modules over triangular matrix rings, Morita context rings with zero homomorphisms, and $\theta$-extensions (Qin, 12 Jul 2025).

1. Cleft extensions of abelian categories and the compatibility axioms

A cleft extension of an abelian category $\mathcal B$ is the data

$$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$

together with natural isomorphisms and adjunctions making

$\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$

satisfy the following properties: $e$ is exact and faithful, $l$0 is an adjoint pair, $l$1, $l$2 is fully faithful and exact, and $l$3 is an adjoint pair, necessarily with $l$4 (Qin, 12 Jul 2025).

Associated to any cleft extension are two endofunctors

$l$5

fitting into canonically split exact sequences of functors

$l$6

One also has $l$7, and related identities of the same form (Qin, 12 Jul 2025).

A cleft extension

$l$8

is called compatible if the following two conditions hold. Condition (A) requires that for every complete projective resolution $l$9 in $q$0 and every $q$1, the complexes

$q$2

are acyclic. Condition (B) requires that for every complete projective resolution $q$3 in $q$4 and every $q$5, the complexes

$q$6

are acyclic (Qin, 12 Jul 2025).

The paper also notes that one may replace (B) by the stronger—but often easier to verify—hypotheses of Proposition 3.2 (ii), involving nilpotence of $q$7 and vanishing of higher left-derived functors of powers of $q$8 (Qin, 12 Jul 2025). A common misunderstanding is to treat compatibility as a formal consequence of the splitting $q$9; in Qin’s formulation, compatibility is an additional homological requirement expressed through complete projective resolutions.

2. Preservation of Gorenstein projective objects

A complex

$i$0

with each $i$1 is a complete $i$2-projective resolution if it is exact and $i$3 is exact for every $i$4. An object $i$5 is Gorenstein projective if it appears as the image of one of the differentials in such a resolution (Qin, 12 Jul 2025).

The main preservation theorem states that if $i$6 is a compatible cleft extension, then both

$i$7

preserve Gorenstein projectives. In particular,

$i$8

(Qin, 12 Jul 2025).

For preservation by $i$9, one starts with $\mathcal A$0 and a complete projective resolution $\mathcal A$1. Applying $\mathcal A$2 gives a complex $\mathcal A$3 of projectives in $\mathcal A$4. The exactness of $\mathcal A$5 is deduced from the split exact sequence

$\mathcal A$6

together with condition (A) and the faithfulness of $\mathcal A$7. The acyclicity of $\mathcal A$8 for $\mathcal A$9 follows from

$\theta$0

and from the fact that each $\theta$1 is a direct summand of some $\theta$2 (Qin, 12 Jul 2025).

For preservation by $\theta$3, one starts with a complete $\theta$4-projective resolution $\theta$5 and checks exactness of $\theta$6 and acyclicity of $\theta$7 for each $\theta$8, invoking condition (B) (Qin, 12 Jul 2025).

These results make the compatibility axioms operational: the acyclicity tests are precisely the mechanism by which Gorenstein projective objects pass through the cleft-extension functors. This suggests that the endofunctor $\theta$9 measures the obstruction to transferring complete projective resolutions across the extension.

3. Recognition of Gorenstein projective objects in $\mathcal B$0

After establishing preservation by $\mathcal B$1 and $\mathcal B$2, Qin studies when an object $\mathcal B$3 is Gorenstein projective in terms of $\mathcal B$4 and the cleft structure. Writing

$\mathcal B$5

for the natural monomorphism and epimorphism from

$\mathcal B$6

one obtains for each $\mathcal B$7 a long exact “Bar-type” diagram which, after applying $\mathcal B$8, produces a three-term complex

$\mathcal B$9

(Qin, 12 Jul 2025).

The main criterion considers the following three conditions:

1. $$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$0.
2. $$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$1 and $$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$2 is a monomorphism.
3. $$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$3 and the sequence

$$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$4

is exact in $$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$5.

The theorem proves that one always has

$$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$6

Moreover, if the composite natural transformation $$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$7 vanishes identically, then all three conditions become equivalent: $$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$8 The paper notes that this happens, for instance, in the case of trivial extensions where $$(\mathcal B,\;\mathcal A,\;e\colon\mathcal A\to\mathcal B,\;i\colon\mathcal B\to\mathcal A,\;l\colon\mathcal B\to\mathcal A)$$9 (Qin, 12 Jul 2025).

The proof of $\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$0 under $\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$1 uses the identification

$\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$2

which yields a short exact sequence

$\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$3

This exhibits $\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$4 as a trivial extension of $\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$5 by the functor $\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$6, and a generalized horseshoe argument then produces a complete $\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$7-projective resolution of $\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$8 from one in $\xymatrix@C=4em{ \mathcal B \ar@<1ex>[r]^i & \mathcal A \ar@<1ex>[l]^q \ar@<1ex>[r]^e & \mathcal B \ar@<1ex>[l]^l }$9 (Qin, 12 Jul 2025).

A persistent misconception is that the three conditions are always equivalent. Qin’s theorem is more precise: in general only

$e$0

is proved, and the converse $e$1 requires the additional hypothesis $e$2 (Qin, 12 Jul 2025).

4. Standard examples and unified module-theoretic criteria

Qin’s framework treats several familiar constructions as instances of the same abstract cleft-extension mechanism. In each case, compatibility is reduced to explicit acyclicity conditions on tensor and Hom functors, and Theorems 3.4 and 3.6 recover known descriptions of Gorenstein projective modules (Qin, 12 Jul 2025).

Example	Associated $e$3	Gorenstein projective criterion
Triangular matrix algebra $e$4	$e$5	$e$6, $e$7, $e$8 injective
Morita context ring $e$9	$l$00	$l$01, $l$02, canonical maps are isomorphisms
$l$03-extension $l$04	$l$05	$l$06 and a two-term complex is exact; if $l$07, these conditions are sufficient

For triangular matrix algebras, an object of $l$08 is a triple $l$09 with $l$10, $l$11, and $l$12. The functors are

$l$13

$l$14

$l$15

The extension is compatible precisely when $l$16 is a “compatible bimodule” in the sense of Zhang. Under the stated acyclicity hypotheses, a $l$17-module $l$18 is Gorenstein projective if and only if

$l$19

(Qin, 12 Jul 2025).

For Morita context rings with zero homomorphisms,

$l$20

a $l$21-module is a quadruple $l$22 satisfying $l$23 and $l$24. Compatibility corresponds to acyclicity of

$l$25

on complete projective resolutions. Under these hypotheses, $l$26 is Gorenstein projective if and only if

$l$27

and the canonical maps

$l$28

are isomorphisms. Equivalently, the two length-three sequences

$l$29

are exact and $l$30 are Gorenstein projective in $l$31 and $l$32 respectively (Qin, 12 Jul 2025).

For $l$33-extensions, let $l$34 be a ring, $l$35 an $l$36-$l$37-bimodule, and $l$38 an associative bimodule map. The $l$39-extension

$l$40

has module category $l$41 as a cleft extension of $l$42 with

$l$43

If the requisite acyclicity conditions hold, then $l$44 implies $l$45 and

$l$46

is exact. In the special case $l$47, these conditions are also sufficient (Qin, 12 Jul 2025).

These examples explain why compatible cleft extensions are useful: they provide a single formalism for several ring-theoretic constructions whose Gorenstein projective objects were previously described case by case.

5. Relation to adjacent cleft-extension theories

Compatible cleft extensions belong to a broader categorical program around cleft extensions of abelian categories. Earlier work by Ma and Zheng investigated the behavior of Igusa–Todorov distances, extension dimension, and Rouquier dimension under cleft extensions, under assumptions such as exactness of $l$48, projective preservation by $l$49, nilpotence $l$50, and, in the Rouquier-dimension theorem, left-perfectness of $l$51 (Ma et al., 2024). Their examples include Morita context rings, trivial extension rings, tensor rings, and arrow removals (Ma et al., 2024). Qin’s notion of compatibility is narrower and more homological: it is designed to preserve complete projective resolutions and to characterize Gorenstein projective objects (Qin, 12 Jul 2025).

Subsequent work by Karakikes studies $l$52-equivariant cleft extensions. If a finite group $l$53 acts on both categories and the cleft data are $l$54-equivariant, then one obtains a lifted cleft extension

$l$55

and the restriction functor associated to a cleft extension induces a singular equivalence if and only if its equivariant counterpart does, under hypotheses involving $l$56, vanishing of $l$57 for all sufficiently large $l$58, and reflection of finite projective dimension by $l$59 (Karakikes, 24 Jun 2026). In the same paper, the skew group ring of a $l$60-equivariant $l$61-extension is shown to be isomorphic to a $l$62-extension of the base skew group ring (Karakikes, 24 Jun 2026).

The phrase “cleft extension” also has a long independent history in Hopf-algebraic settings. For weak Hopf algebras, the categories of $l$63-cleft extensions of an algebra $l$64 and of unitary crossed products of $l$65 by $l$66 are equivalent (Guccione et al., 2018). For Hopf algebroids, Han and Schauenburg show the equivalence of left-cleft extensions, $l$67-twisted crossed products, and Hopf Galois extensions with normal basis properties (Han et al., 2024). These theories are structurally related through crossed-product and Galois ideas, but they are formulated in different ambient categories and serve different purposes.

6. Conceptual role and significance

The central accomplishment of the compatible theory is twofold. First, it proves that both $l$68 and $l$69 preserve Gorenstein projective objects. Second, it gives necessary conditions for an object of $l$70 to be Gorenstein projective, and shows that these necessary conditions are also sufficient in some special case (Qin, 12 Jul 2025). As applications, it unifies some known results on the description of Gorenstein projective modules over triangular matrix rings, Morita context rings with zero homomorphisms, and $l$71-extensions (Qin, 12 Jul 2025).

Conceptually, the framework isolates the endofunctor $l$72 and the short exact sequence

$l$73

as the mechanism controlling how far the larger category is from the base category. The recognition theorem shows that the exactness of

$l$74

can replace a direct construction of a complete projective resolution in $l$75 when $l$76 (Qin, 12 Jul 2025). This suggests that compatible cleft extensions function as a transfer principle for Gorenstein homological algebra.

A second point of significance is methodological unification. The same cleft-extension formalism encompasses triangular matrix algebras, Morita context rings with zero homomorphisms, and $l$77-extensions, and later cleft-extension work connects the framework to dimension theory and singularity categories (Ma et al., 2024, Karakikes, 24 Jun 2026). A plausible implication is that further interaction between compatibility, equivariance, and singularity categories may extend the reach of the theory beyond the concrete examples already treated.

Compatible cleft extensions should therefore be understood not as a synonym for cleft extensions in general, but as a specific homological structure on cleft extensions of abelian categories, tailored to the preservation and detection of Gorenstein projective objects (Qin, 12 Jul 2025).