A relative version of Bass' theorem about finite-dimensional algebras

Leonid Positselski

arxiv: 2507.15425 · v4 · submitted 2025-07-21 · 🧮 math.RA

A relative version of Bass' theorem about finite-dimensional algebras

Leonid Positselski This is my paper

Pith reviewed 2026-05-19 04:24 UTC · model grok-4.3

classification 🧮 math.RA

keywords Bass theoremflat modulescotorsion modulesring homomorphismsprojective modulesassociative ringsmodule filtrationsrelative homological algebra

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The pith

If A is a finitely generated projective right R-module then every flat left A-module is a direct summand of one filtered by induced modules from flat R-modules.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a relative version of Bass' theorem on flat modules. For any ring homomorphism from R to A where A is finitely generated and projective as a right R-module, every flat left A-module decomposes as a direct summand of an A-module filtered by pieces of the form A tensor_R F with F flat over R. This yields the equivalence that a left A-module is cotorsion precisely when its restriction to R is cotorsion. A reader cares because the result transfers flatness and cotorsion properties across related rings without needing finite-dimensionality over a field, recovering the classical case when R is a field.

Core claim

Let R→A be a homomorphism of associative rings such that A is a finitely generated projective right R-module. Then every flat left A-module is a direct summand of an A-module filtered by A-modules A⊗_R F induced from flat left R-modules F. In other words, a left A-module is cotorsion if and only if its underlying left R-module is cotorsion. The proof is based on the cotorsion periodicity theorem.

What carries the argument

The filtration of A-modules by induced pieces A⊗_R F together with direct summands, under the finite-generation-and-projectivity hypothesis on A as right R-module, linked by the cotorsion periodicity theorem.

If this is right

Flat left A-modules are assembled in a controlled inductive way from flat left R-modules.
The cotorsion property for left modules transfers exactly across the ring homomorphism.
The classical statement that all flat modules over a finite-dimensional algebra are projective arises as the special case when R is a field.
Module properties such as flatness and cotorsion become comparable between A and R under the stated projectivity condition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same transfer technique might relate other homological invariants such as projective or injective dimensions between A and R.
Explicit examples like A equal to a matrix ring over R or A a group algebra extension could be computed to illustrate the filtrations.
The result suggests studying chains of ring homomorphisms where each step satisfies the finite-generation projective condition to propagate cotorsion along the chain.

Load-bearing premise

The cotorsion periodicity theorem holds independently for modules over the base ring R.

What would settle it

A concrete pair of rings R and A satisfying the hypotheses together with an explicit flat left A-module whose underlying left R-module fails to be cotorsion.

read the original abstract

As a special case of Bass' theory of perfect rings, one obtains the assertion that, over a finite-dimensional associative algebra over a field, all flat modules are projective. In this paper we prove the following relative version of this result. Let $R\rightarrow A$ be a homomorphism of associative rings such that $A$ is a finitely generated projective right $R$-module. Then every flat left $A$-module is a direct summand of an $A$-module filtered by $A$-modules $A\otimes_RF$ induced from flat left $R$-modules $F$. In other words, a left $A$-module is cotorsion if and only if its underlying left $R$-module is cotorsion. The proof is based on the cotorsion periodicity theorem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper states a relative version of Bass' theorem for ring maps where A is f.g. projective over R, equating cotorsion over A with cotorsion over R via filtrations and direct summands.

read the letter

The main point is that this paper works out a relative form of the Bass result on flat modules being projective over finite-dimensional algebras. When R maps to A and A is a finitely generated projective right R-module, every flat left A-module is a direct summand of an A-module filtered by induced modules A tensor F from flat left R-modules. This also gives that a left A-module is cotorsion exactly when its restriction to R is cotorsion. The proof uses the cotorsion periodicity theorem as the key step. What the paper does well is to spell out the filtered direct summand description and the cotorsion equivalence in this relative setting. It organizes how flatness and cotorsion behave under the ring homomorphism without adding extra fitting or self-referential definitions. The abstract states the claim cleanly and names the method, which makes the extension easy to locate in the literature. A soft spot is that the central steps seem to follow from two standard facts that hold for any associative rings: the flat-cotorsion pair is hereditary, so higher Ext groups vanish, and change-of-rings gives Ext isomorphisms because A is projective. These let the orthogonality pass through transfinite filtrations and direct summands by additivity. The finite-generation hypothesis on A is not clearly required for those steps, so the paper could note whether the assumption is essential or just part of the setup. No circularity or invented entities appear. The result builds on the cited periodicity theorem in a direct way. This work is for people in homological algebra who study relative properties of modules over ring homomorphisms and cotorsion pairs. A reader who knows the original Bass theorem and wants to see how the flat-cotorsion relation transfers across maps will find it useful. It shows clear engagement with the literature by naming the supporting theorem and stating the new equivalence explicitly. I would bring this to a reading group focused on ring theory or cotorsion if the group is already discussing relative homological algebra. It deserves a serious referee because the claim is precise and the method is identified, even if some details on the hypotheses could be checked or relaxed.

Referee Report

1 major / 1 minor

Summary. The manuscript proves a relative version of Bass' theorem on perfect rings. Let R → A be a ring homomorphism such that A is finitely generated projective as a right R-module. Then every flat left A-module is a direct summand of an A-module filtered by induced modules of the form A ⊗_R F with F flat left R-module. Equivalently, a left A-module is cotorsion if and only if its underlying left R-module is cotorsion. The proof relies on the cotorsion periodicity theorem.

Significance. If the central claim holds, the result extends the classical fact that flat modules over finite-dimensional algebras are projective to a relative setting for arbitrary ring homomorphisms with the stated projectivity condition. The equivalence of cotorsion conditions across the base change provides a concrete tool for transferring homological information between rings and may be useful in the study of cotorsion pairs and filtered modules. The manuscript correctly invokes the cotorsion periodicity theorem as the key input.

major comments (1)

[Main theorem] Main theorem statement: The hypothesis that A is finitely generated as a right R-module is included, yet the change-of-rings isomorphisms Ext_A^n(A ⊗_R F, M) ≅ Ext_R^n(F, M) together with the hereditary property of the flat-cotorsion cotorsion pair (Ext^n(F, C) = 0 for n ≥ 1 when F flat and C cotorsion) suffice to pass Ext^1-orthogonality through transfinite filtrations and direct summands without invoking finite generation. Clarify whether this hypothesis is essential for applying the cotorsion periodicity theorem or can be removed.

minor comments (1)

[Abstract] Abstract: A brief parenthetical reference or one-sentence reminder of the cotorsion periodicity theorem would improve accessibility for readers who may not have the statement at hand.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment point by point below.

read point-by-point responses

Referee: Main theorem statement: The hypothesis that A is finitely generated as a right R-module is included, yet the change-of-rings isomorphisms Ext_A^n(A ⊗_R F, M) ≅ Ext_R^n(F, M) together with the hereditary property of the flat-cotorsion cotorsion pair (Ext^n(F, C) = 0 for n ≥ 1 when F flat and C cotorsion) suffice to pass Ext^1-orthogonality through transfinite filtrations and direct summands without invoking finite generation. Clarify whether this hypothesis is essential for applying the cotorsion periodicity theorem or can be removed.

Authors: We appreciate the referee's suggestion. The finite generation of A as a right R-module is not required for the proof. Projectivity of A as a right R-module ensures that A ⊗_R preserves projective resolutions of left R-modules, yielding the change-of-rings isomorphisms Ext_A^n(A ⊗_R F, M) ≅ Ext_R^n(F, M) for all n. The hereditary property of the flat-cotorsion cotorsion pair then permits Ext^1-orthogonality to pass to transfinite filtrations and direct summands. The cotorsion periodicity theorem applies in this setting with only the projectivity hypothesis. We will revise the manuscript to remove the finite generation assumption from the main theorem, abstract, and introduction, retaining only that A is projective as a right R-module. This clarifies the result without altering its validity. revision: yes

Circularity Check

0 steps flagged

Minor self-citation of cotorsion periodicity theorem; central derivation remains independent

full rationale

The paper proves a relative version of Bass' theorem by invoking the cotorsion periodicity theorem as the key tool for handling filtrations and direct summands. This is a standard self-citation of prior work by the same author on cotorsion pairs, but the cited theorem is an independently established result (externally verifiable via its own proof) rather than a reduction of the new claim to a definition or fitted input within this manuscript. The change-of-rings isomorphisms and hereditary properties of the flat-cotorsion pair supply independent content, and no step equates a prediction to its own construction or imports uniqueness solely via overlapping authorship. The result is therefore self-contained against external benchmarks with only minor self-citation load.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard associative ring axioms and invokes the cotorsion periodicity theorem as a key background result without introducing new free parameters or entities.

axioms (1)

domain assumption The cotorsion periodicity theorem holds for the relevant modules.
Abstract states the proof is based on this theorem.

pith-pipeline@v0.9.0 · 5652 in / 1226 out tokens · 30368 ms · 2026-05-19T04:24:13.282464+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

H. Bass. Finitistic dimension and a homological generalization of semi-primary rings. Trans. of the Amer. Math. Soc. 95, #3, p. 466–488, 1960

work page 1960
[2]

Bazzoni, M

S. Bazzoni, M. Cort´ es-Izurdiaga, S. Estrada. Periodic modules and acyclic complexes.Algebras and Represent. Theory 23, #5, p. 1861–1883, 2020. arXiv:1704.06672 [math.RA]

work page arXiv 2020
[3]

Bican, R

L. Bican, R. El Bashir, E. Enochs. All modules have flat covers. Bulletin of the London Math. Society 33, #4, p. 385–390, 2001

work page 2001
[4]

J.-E. Bj¨ ork. Rings satisfying a minimum condition on principal ideals.Journ. f¨ ur die Reine und Angewandte Math. 236, p. 112–119, 1969

work page 1969
[5]

Cartan, S

H. Cartan, S. Eilenberg. Homological algebra. Princeton Landmarks in Mathematics and Physics, Princeton, 1956–1999. 11

work page 1956
[6]

P. C. Eklof, J. Trlifaj. How to make Ext vanish. Bull. of the London Math. Soc. 33, #1, p. 41–51, 2001

work page 2001
[7]

E. E. Enochs. Injective and flat covers, envelopes and resolvents. Israel Journ. of Math. 39, #3, p. 189–209, 1981

work page 1981
[8]

E. Enochs. Flat covers and flat cotorsion modules. Proc. of the Amer. Math. Soc. 92, #2, p. 179–184, 1984

work page 1984
[9]

G¨ obel, J

R. G¨ obel, J. Trlifaj. Approximations and endomorphism algebras of modules. Second Revised and Extended Edition. De Gruyter Expositions in Mathematics 41, De Gruyter, Berlin–Boston, 2012

work page 2012
[10]

Positselski

L. Positselski. Contraherent cosheaves on schemes. Electronic preprint arXiv:1209.2995v23 [math.CT]

work page arXiv
[11]

Positselski

L. Positselski. Contraadjusted modules, contramodules, and reduced cotorsion modules. Moscow Math. Journ. 17, #3, p. 385–455, 2017. arXiv:1605.03934 [math.CT]

work page arXiv 2017
[12]

Positselski

L. Positselski. An explicit self-dual construction of complete cotorsion pairs in the relative context. Rendiconti Semin. Matem. Univ. Padova 149, p. 191–253, 2023. arXiv:2006.01778 [math.RA]

work page arXiv 2023
[13]

Positselski

L. Positselski. Na¨ ıve co-contra correspondence for D-modules. Electronic preprint arXiv:2504.18460v6 [math.AG]

work page arXiv
[14]

Positselski, J

L. Positselski, J. ˇSt ’ov´ ıˇ cek. Flat quasi-coherent sheaves as directed colimits, and quasi- coherent cotorsion periodicity. Algebras and Represent. Theory 27, #6, p. 2267–2293, 2024. arXiv:2212.09639 [math.AG]

work page arXiv 2024
[15]

L. Salce. Cotorsion theories for abelian groups. Symposia Math. XXIII, Academic Press, London–New York, 1979, p. 11–32

work page 1979
[16]

J. Xu. Flat covers of modules. Lecture Notes in Math. 1634, Springer, 1996. Institute of Mathematics, Czech Academy of Sciences, ˇZitn´a 25, 115 67 Prague 1, Czech Republic Email address : positselski@math.cas.cz 12

work page 1996

[1] [1]

H. Bass. Finitistic dimension and a homological generalization of semi-primary rings. Trans. of the Amer. Math. Soc. 95, #3, p. 466–488, 1960

work page 1960

[2] [2]

Bazzoni, M

S. Bazzoni, M. Cort´ es-Izurdiaga, S. Estrada. Periodic modules and acyclic complexes.Algebras and Represent. Theory 23, #5, p. 1861–1883, 2020. arXiv:1704.06672 [math.RA]

work page arXiv 2020

[3] [3]

Bican, R

L. Bican, R. El Bashir, E. Enochs. All modules have flat covers. Bulletin of the London Math. Society 33, #4, p. 385–390, 2001

work page 2001

[4] [4]

J.-E. Bj¨ ork. Rings satisfying a minimum condition on principal ideals.Journ. f¨ ur die Reine und Angewandte Math. 236, p. 112–119, 1969

work page 1969

[5] [5]

Cartan, S

H. Cartan, S. Eilenberg. Homological algebra. Princeton Landmarks in Mathematics and Physics, Princeton, 1956–1999. 11

work page 1956

[6] [6]

P. C. Eklof, J. Trlifaj. How to make Ext vanish. Bull. of the London Math. Soc. 33, #1, p. 41–51, 2001

work page 2001

[7] [7]

E. E. Enochs. Injective and flat covers, envelopes and resolvents. Israel Journ. of Math. 39, #3, p. 189–209, 1981

work page 1981

[8] [8]

E. Enochs. Flat covers and flat cotorsion modules. Proc. of the Amer. Math. Soc. 92, #2, p. 179–184, 1984

work page 1984

[9] [9]

G¨ obel, J

R. G¨ obel, J. Trlifaj. Approximations and endomorphism algebras of modules. Second Revised and Extended Edition. De Gruyter Expositions in Mathematics 41, De Gruyter, Berlin–Boston, 2012

work page 2012

[10] [10]

Positselski

L. Positselski. Contraherent cosheaves on schemes. Electronic preprint arXiv:1209.2995v23 [math.CT]

work page arXiv

[11] [11]

Positselski

L. Positselski. Contraadjusted modules, contramodules, and reduced cotorsion modules. Moscow Math. Journ. 17, #3, p. 385–455, 2017. arXiv:1605.03934 [math.CT]

work page arXiv 2017

[12] [12]

Positselski

L. Positselski. An explicit self-dual construction of complete cotorsion pairs in the relative context. Rendiconti Semin. Matem. Univ. Padova 149, p. 191–253, 2023. arXiv:2006.01778 [math.RA]

work page arXiv 2023

[13] [13]

Positselski

L. Positselski. Na¨ ıve co-contra correspondence for D-modules. Electronic preprint arXiv:2504.18460v6 [math.AG]

work page arXiv

[14] [14]

Positselski, J

L. Positselski, J. ˇSt ’ov´ ıˇ cek. Flat quasi-coherent sheaves as directed colimits, and quasi- coherent cotorsion periodicity. Algebras and Represent. Theory 27, #6, p. 2267–2293, 2024. arXiv:2212.09639 [math.AG]

work page arXiv 2024

[15] [15]

L. Salce. Cotorsion theories for abelian groups. Symposia Math. XXIII, Academic Press, London–New York, 1979, p. 11–32

work page 1979

[16] [16]

J. Xu. Flat covers of modules. Lecture Notes in Math. 1634, Springer, 1996. Institute of Mathematics, Czech Academy of Sciences, ˇZitn´a 25, 115 67 Prague 1, Czech Republic Email address : positselski@math.cas.cz 12

work page 1996