Quasi-Faithfully Flat Extensions

• Quasi-faithfully flat extensions are finite ring maps defined via an auxiliary bimodule, relaxing classical flatness to preserve k-torsionfreeness.
• They enable a change-of-rings framework that compares classical and Gorenstein transposes, ensuring Ext-vanishing properties persist across rings.
• They are pivotal in transferring extension-closedness, quasi-k-Gorenstein properties, and finite representation type, with applications to Frobenius extensions and skew group rings.

Searching arXiv for the specified paper and closely related work. Search 1: exact paper by title keywords. Searching arXiv for: "Base change of (Gorenstein) transpose, k-torsionfree modules, and quasi-faithfully flat extensions" Quasi-faithfully flat extensions are finite ring homomorphisms $\varphi\colon R\to A$ designed to retain much of the change-of-rings control usually associated with faithfully flat extensions, while permitting substantially weaker hypotheses. In the setting where $R$ is a two-sided Noetherian ring and modules are finitely generated, the notion is used to compare transpose constructions, $k$-torsionfree modules, extension-closedness, quasi-$k$-Gorensteiness, and finite representation type across $R$ and $A$. The framework developed in "Base change of (Gorenstein) transpose, $k$-torsionfree modules, and quasi-faithfully flat extensions" establishes a close relationship between the classical transpose of an $A$-module and the Gorenstein transpose of a suitable syzygy over $R$, extends a result of Zhao, and gives applications to Frobenius extensions and skew group rings (Liu, 16 Jul 2025).

1. Ambient setting and $k$-torsionfree modules

Throughout, $R$0 is a Noetherian ring on both sides, and

$R$1

For $R$2, choose a projective presentation

$R$3

and apply $R$4. This yields an exact sequence of right modules

$R$5

where $R$6 is the transpose of $R$7 (Liu, 16 Jul 2025).

For each integer $R$8, the module $R$9 is called $k$0-torsionfree when

$k$1

The full subcategory of $k$2-torsionfree modules is denoted by $k$3. This definition packages Ext-vanishing of the transpose into a categorical condition that is stable enough to support descent and ascent arguments.

In the paper’s broader program, $k$4-torsionfree modules are the main invariants transported across finite ring homomorphisms. A central feature is that their behavior can be controlled not merely by flatness of $k$5 over $k$6, but by the existence of an auxiliary bimodule implementing an appropriate base-change mechanism.

2. Definition of quasi-faithfully flat extensions

A finite ring homomorphism $k$7 is called a quasi-faithfully flat extension if there exists an $k$8-$k$9-bimodule $k$0 such that:

1. $k$1 is finitely generated and projective as a left $k$2-module,
2. $k$3 is faithfully flat as a right $k$4-module, and
3. $k$5 is flat over $k$6 (Liu, 16 Jul 2025).

This definition weakens the classical requirement of faithful flatness on both sides for $k$7 itself. When $k$8 is flat and faithfully flat over both $k$9 and $R$0, one recovers the usual notion of faithfully flat extension. The key distinction is that quasi-faithful flatness is witnessed by a possibly different bimodule $R$1, rather than by $R$2 alone.

Several examples clarify the scope of the notion. Any faithfully flat extension is automatically quasi-faithfully flat. If $R$3 is a field and $R$4 is a Frobenius $R$5-algebra in the sense that $R$6 as $R$7-$R$8-bimodules, then for any $R$9-algebra map $A$0 one may take

$A$1

In this case $A$2 is free, hence projective, over $A$3, faithfully flat over $A$4, and $A$5 is free over $A$6. Consequently, every $A$7-algebra homomorphism out of a Frobenius $A$8-algebra is quasi-faithfully flat. There are also non-projective examples: the canonical surjection

$A$9

is quasi-faithfully flat although $k$0 is not projective over $k$1 (Liu, 16 Jul 2025).

A common misconception is that a useful “faithful flatness” substitute must force $k$2 itself to be projective or flat over $k$3. The example above shows that quasi-faithful flatness is genuinely weaker: it is formulated so that the transfer of homological properties can still proceed through $k$4.

3. Base change for transpose and detection of $k$5-torsionfreeness

The mechanism behind the theory is a functorial comparison between transpose over $k$6 and transpose after tensoring to $k$7. If $k$8 and $k$9 is an $A$0-$A$1-bimodule that is projective and finitely generated over $A$2, then there is a canonical isomorphism of right $A$3-modules

$A$4

This transfer-of-transpose statement is the basic change-of-rings computation from which the later results follow (Liu, 16 Jul 2025).

Under additional flatness hypotheses, transpose comparison yields preservation and detection of $A$5-torsionfreeness. If $A$6 is flat over $A$7 and $A$8 is flat over $A$9, then for each $R$0,

$R$1

If, in addition, $R$2 is faithfully flat on the right, then the converse also holds. Thus under a quasi-faithfully flat extension one obtains the equivalence

$R$3

The paper’s conceptual summary identifies the heart of the argument as the transpose isomorphism together with flatness assumptions that allow one to compare Ext-groups and detect their vanishing across the two rings. This is the precise sense in which quasi-faithfully flat extensions serve as a substitute for classical faithfully flat base change: they preserve the homological criterion defining $R$4-torsionfree modules (Liu, 16 Jul 2025).

4. Extension-closedness and change of rings

A full subcategory $R$5 is extension closed if every short exact sequence

$R$6

with $R$7 also has $R$8. For the subcategories $R$9, quasi-faithfully flat extensions support both descent and ascent results (Liu, 16 Jul 2025).

The descent statement is direct: if $k$0 is quasi-faithfully flat and $k$1 is extension closed in $k$2, then $k$3 is extension closed in $k$4. The ascent statement requires stronger homological input. Suppose $k$5 is finite and satisfies

$k$6

for some projective $k$7 and $k$8. If either $k$9 or $R$00 is commutative satisfying the usual Gorenstein-depth condition $R$01, then extension-closedness of $R$02 implies extension-closedness of $R$03.

In the especially important case $R$04, these two directions combine into a clean equivalence. If $R$05 is a finite quasi-faithfully flat extension, with $R$06 Noetherian, and

$R$07

then for every $R$08 the category $R$09 is extension closed in $R$10 if and only if $R$11 is extension closed in $R$12. The paper further notes that in this special case one has the stronger identification

$R$13

so the theory collapses to a particularly transparent form of ascent and descent.

5. Consequences for quasi-$R$14-Gorenstein algebras

For a Noetherian algebra $R$15, being left quasi-$R$16-Gorenstein means that in the minimal injective resolution

$R$17

one has

$R$18

Huang’s theorem identifies this condition with extension-closedness of each $R$19 for $R$20 (Liu, 16 Jul 2025).

Because quasi-faithfully flat extensions preserve extension-closedness under the hypotheses above, they also preserve quasi-$R$21-Gorensteiness. If $R$22 satisfies the hypotheses of the equivalence theorem for extension-closedness and both $R$23 and $R$24 are Noetherian algebras, then for every $R$25,

$R$26

This yields an affirmative answer to a question posed by Zhao in the case where both rings are Noetherian algebras.

A notable special case is that of Frobenius extensions. If $R$27 is a Frobenius extension of Noetherian algebras and $R$28 is also faithfully flat over $R$29, then $R$30 is left quasi-$R$31-Gorenstein if and only if $R$32 is. In the commutative case, the paper observes that this forces left-right symmetry of quasi-$R$33-Gorensteinness for $R$34 (Liu, 16 Jul 2025).

6. Frobenius extensions, finite representation type, and skew group rings

A full subcategory $R$35 has finite representation type if, up to isomorphism, it contains only finitely many indecomposable objects. Under the Krull-Remak-Schmidt hypothesis, such as when $R$36 is artinian or Henselian local, Frobenius extensions also transfer finiteness properties of $R$37 (Liu, 16 Jul 2025).

If $R$38 is a Frobenius extension and Krull-Remak-Schmidt holds over both $R$39 and $R$40, then two complementary statements hold. First, if $R$41 is separable and $R$42 has finite representation type, then $R$43 has finite representation type. Second, if $R$44 is split and $R$45 has finite representation type, then $R$46 has finite representation type. The separable case gives ascent, while the split case gives descent.

The standard application is to skew group rings. If $R$47 is a finite group acting on an artinian ring $R$48 and $R$49 is invertible in $R$50, then the skew group ring $R$51 is a separable, split Frobenius extension of $R$52. Hence, for each $R$53,

$R$54

The same argument, applied to Gorenstein-projective modules, gives an equivalence of Cohen-Macaulay finiteness: $R$55 These consequences place quasi-faithfully flat and Frobenius-type change-of-rings phenomena in direct contact with representation-theoretic finiteness questions.

7. Conceptual role within transpose and syzygy theory

The topic is situated within a broader change-of-rings analysis of transpose and Gorenstein transpose. Under suitable homological conditions on $R$56 over $R$57, the paper establishes a close relationship between the classical transpose of a finitely generated left $R$58-module and the Gorenstein transpose of a certain syzygy module of that module over $R$59 (Liu, 16 Jul 2025). The quasi-faithfully flat framework then isolates the hypotheses needed for this comparison to control the more concrete invariant of $R$60-torsionfreeness.

The resulting picture is structurally coherent. Transpose comparison yields equivalence of Ext-vanishing conditions; Ext-vanishing controls membership in $R$61; preservation of $R$62 feeds into extension-closedness; extension-closedness characterizes quasi-$R$63-Gorensteiness in the Noetherian algebra setting; and in the Frobenius, separable, and split contexts, the same formalism reaches finite representation type and skew group rings. A plausible implication is that quasi-faithfully flatness is best understood not as a variant of flatness in isolation, but as a change-of-rings device tailored to transpose-based homological invariants.

Within this framework, the notion’s significance lies in its precision. It is weak enough to include examples where $R$64 is not projective over $R$65, yet strong enough to make $R$66-torsionfreeness, extension-closedness, and several Gorenstein-flavored finiteness properties transport reliably across finite extensions.