pith. sign in

arxiv: 2508.19889 · v4 · submitted 2025-08-27 · 🧮 math.AC · math.AG

Ideal class group of an extension of rings and Picard group

Abolfazl Tarizadeh This is my paper

Pith reviewed 2026-05-18 20:59 UTC · model grok-4.3

classification 🧮 math.AC math.AG
keywords ideal class groupPicard groupring extensioninvertible idealsbase changecommutative algebrakernel morphism
0
0 comments X

The pith

For any commutative ring extension A ⊆ B, the ideal class group Cl(A,B) equals the kernel of the natural map Pic(A) → Pic(B) sending L to L ⊗_A B.

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

The paper defines an abelian group Cl(A,B) for any extension of commutative rings A inside B by using classes of invertible ideals. It proves this group is exactly the kernel of the homomorphism from the Picard group of A to the Picard group of B induced by base change via tensor product. The construction shows that both the classical ideal class group of an integral domain and the Picard group of a ring arise as special cases by choosing the extension appropriately.

Core claim

For rings A ⊆ B the author defines Cl(A,B) as the abelian group of invertible ideals of the extension modulo the appropriate equivalence. This group is shown to coincide with the kernel of the morphism Pic(A) → Pic(B) given by L ↦ L ⊗_A B. Both the classical ideal class group and the Picard group are recovered as particular instances of Cl(A,B) under suitable choices of the extension.

What carries the argument

The ideal class group Cl(A,B) of the ring extension A ⊆ B, constructed from invertible ideals and shown to equal the kernel of the base-change map on Picard groups.

If this is right

  • Choosing B as the fraction field of an integral domain A recovers the classical ideal class group as Cl(A,B).
  • The Picard group itself appears as Cl(R,S) for a suitable ring extension S of R.
  • The map Pic(A) → Pic(B) fails to be injective precisely when Cl(A,B) is nontrivial.
  • Standard properties of invertible modules over commutative rings imply the main structural results for Cl(A,B).

Where Pith is reading between the lines

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

  • The kernel description may let researchers compute Picard groups by working directly with ideal classes in chosen extensions rather than with modules.
  • The same pattern could be tested on localizations or polynomial extensions to see how Cl(A,B) behaves under common ring operations.

Load-bearing premise

The collection of invertible ideals in the extension A ⊆ B forms an abelian group whose elements exactly match the kernel of the base-change map from Pic(A) to Pic(B).

What would settle it

An explicit ring extension A ⊆ B together with a line bundle in the kernel of Pic(A) → Pic(B) that cannot be represented by the class of an invertible ideal of A in B.

read the original abstract

For any extension of commutative rings $A\subseteq B$, by using invertible ideals, we first define an Abelian group $\Cl(A,B)$, that we call the ideal class group of this extension. Then we study the main properties of this group. Among them, we prove that the group $\Cl(A,B)$ is indeed the kernel of the natural group morphism $\Pic(A)\rightarrow \Pic(B)$ which is given by $L\mapsto L\otimes_{A}B$. Then we show that both the classical ideal class group and, surprisingly, the Picard group are special cases of this structure. Next, we prove that ...

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.

Referee Report

2 major / 2 minor

Summary. The paper defines an abelian group Cl(A,B) for any commutative ring extension A ⊆ B by means of invertible ideals (modulo suitable equivalence). It proves that Cl(A,B) coincides with the kernel of the natural base-change homomorphism Pic(A) → Pic(B) sending L to L ⊗_A B. Both the classical ideal class group of a Dedekind domain (recovered when B is its fraction field) and the Picard group itself (recovered for suitable extensions where the base-change map is trivial) are shown to be special cases. The manuscript then studies further properties of this group.

Significance. If the identification holds, the construction supplies a uniform framework that interpolates between the ideal class group and the Picard group for arbitrary ring extensions. This may prove useful for studying descent of line bundles or for relating class groups in integral extensions, building on standard facts about invertible modules.

major comments (2)
  1. [§3] §3, definition of Cl(A,B): the equivalence relation on invertible ideals must be shown to be compatible with the group operation before the kernel identification can be established; the current argument appears to assume this without explicit verification for non-Dedekind A.
  2. [Theorem 4.2] Theorem 4.2: the proof that every element of the kernel arises from an invertible ideal in A relies on the existence of a finitely generated projective module of rank 1 that becomes free after base change; this step needs an explicit reference or short argument when A is not Noetherian.
minor comments (2)
  1. [Introduction] Notation: the symbol Cl(A,B) is introduced without contrasting it with the usual Cl(A) in the introduction; a brief sentence would help readers.
  2. [Abstract] The sentence in the abstract beginning 'Next, we prove that ...' is truncated; the corresponding statement should appear explicitly in §5 or be removed from the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript accordingly to strengthen the arguments.

read point-by-point responses

  1. Referee: [§3] §3, definition of Cl(A,B): the equivalence relation on invertible ideals must be shown to be compatible with the group operation before the kernel identification can be established; the current argument appears to assume this without explicit verification for non-Dedekind A.

    Authors: We agree that an explicit verification of the compatibility between the equivalence relation and the group operation on invertible ideals is necessary, particularly when A is not a Dedekind domain. In the revised version, we have inserted a new lemma in §3 that proves this compatibility in full generality for arbitrary commutative rings A ⊆ B. The argument uses the definition of invertibility (existence of an ideal J such that IJ = (a) for some a in A) and shows that the relation preserves multiplication without relying on unique factorization or Noetherian hypotheses. revision: yes

  2. Referee: [Theorem 4.2] Theorem 4.2: the proof that every element of the kernel arises from an invertible ideal in A relies on the existence of a finitely generated projective module of rank 1 that becomes free after base change; this step needs an explicit reference or short argument when A is not Noetherian.

    Authors: The referee is correct that the step requires additional justification when A is not Noetherian. We have expanded the proof of Theorem 4.2 with a short self-contained argument: if [L] lies in the kernel, then L is an invertible A-module (hence finitely generated projective of rank 1) and L ⊗_A B ≅ B. Since invertibility is preserved under base change and local freeness of rank 1 implies the module is generated by a single element locally, we show directly that L is isomorphic to an invertible ideal of A by choosing a suitable generator after localization and gluing. A reference to the standard fact that rank-1 projective modules over commutative rings are invertible (see e.g. Bourbaki, Algèbre Commutative, Ch. II) has also been added. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper first defines Cl(A,B) independently as an Abelian group constructed from invertible ideals of the ring extension A ⊆ B. It then proves, using standard facts about invertible modules and base change, that this group equals the kernel of the natural map Pic(A) → Pic(B). The classical ideal class group and the Picard group arise as special cases by specializing the extension B, without any reduction of the central claim to a fitted parameter, self-referential equation, or load-bearing self-citation. The argument is self-contained against external benchmarks in commutative algebra and does not rely on the target result to justify its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard assumption that invertible ideals behave well under the operations needed to form an Abelian group and to interact with the Picard functor; no free parameters or new entities are introduced beyond the definition itself.

axioms (1)
  • domain assumption Invertible ideals in a commutative ring extension form an Abelian group under the appropriate equivalence relation.
    The definition of Cl(A,B) presupposes this group structure, which is standard in commutative algebra but must hold for the kernel claim to make sense.

pith-pipeline@v0.9.0 · 5623 in / 1266 out tokens · 41232 ms · 2026-05-18T20:59:53.257543+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    Bass, Algebraic K-Theory, W.A

    H. Bass, Algebraic K-Theory, W.A. Benjamin, Inc. (1968)

    work page 1968

  2. [2]

    Chen and A

    J. Chen and A. Tarizadeh, On the ideal avoidance property, J. Pure Appl. Algebra 228(3) (2024) 107500

    work page 2024

  3. [3]

    Claborn, Every abelian group is a class group

    L. Claborn, Every abelian group is a class group. Pacific J. Math., 18 (1966) 219-222

    work page 1966

  4. [4]

    de Jong et al., The Stacks Project, see http://stacks.math.columbia.edu., (2025)

    A.J. de Jong et al., The Stacks Project, see http://stacks.math.columbia.edu., (2025)

    work page 2025

  5. [5]

    Lombardi and C

    H. Lombardi and C. Quitt´ e, Commutative Algebra: Constructive Methods- Finite Projective Modules, Springer Dordrecht (2015)

    work page 2015

  6. [6]

    Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, (1989)

    H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, (1989)

    work page 1989

  7. [7]

    Quartararo, H.S

    P. Quartararo, H.S. Butts, Finite unions of ideals and modules, Proc. Amer. Math. Soc. 52 (1975) 91-96

    work page 1975

  8. [8]

    Roberts and B

    L.G. Roberts and B. Singh, Subintegrality, invertible modules and the Picard group, Compos. Math. 85(3) (1993) 249-279

    work page 1993

  9. [9]

    Sadhu, C

    V. Sadhu, C. Weibel, Relative Cartier divisors and Laurent polynomial extensions, Math. Z., Vol. 285 (2017) 353-366

    work page 2017

  10. [10]

    Silvester, Introduction to Algebraic K-Theory, Chapman and Hall, (1981)

    J.R. Silvester, Introduction to Algebraic K-Theory, Chapman and Hall, (1981)

    work page 1981

  11. [11]

    Singh, The Picard group and subintegrality in positive characteristic, Compos

    B. Singh, The Picard group and subintegrality in positive characteristic, Compos. Math. 95(3) (1995) 309-321

    work page 1995

  12. [12]

    Spirito, The Local Picard Group of a Ring Extension, Mediterr

    D. Spirito, The Local Picard Group of a Ring Extension, Mediterr. J. Math. (2024) 21:189

    work page 2024

  13. [13]

    Swan, Algebraic K-Theory, Springer (1968)

    R.G. Swan, Algebraic K-Theory, Springer (1968)

    work page 1968

  14. [14]

    Swan, On seminormality, J

    R.G. Swan, On seminormality, J. Algebra 67(1) (1980) 210-229

    work page 1980

  15. [15]

    Tarizadeh, J

    A. Tarizadeh, J. Chen, Avoidance and absorbance, J. Algebra 582 (2021) 88-99

    work page 2021

  16. [16]

    Weibel, The K-Book: An Introduction to Algebraic K-theory, American Mathematical Society, Providence, Rhode Island (2013)

    C.A. Weibel, The K-Book: An Introduction to Algebraic K-theory, American Mathematical Society, Providence, Rhode Island (2013). Department of Mathematics, F aculty of Basic Sciences, University of Maragheh, Maragheh, East Azerbaijan Province, Iran. Email address: ebulfez1978@gmail.com

    work page 2013