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arxiv: 2210.02951 · v8 · submitted 2022-10-06 · 🧮 math.AC · math.AG· math.KT

On the Grothendieck ring and the relation of its group of units with the Picard group

Abolfazl Tarizadeh This is my paper

Pith reviewed 2026-05-24 10:53 UTC · model grok-4.3

classification 🧮 math.AC math.AGmath.KT
keywords Grothendieck ringPicard groupidempotentsexact sequenceK0commutative ringsprojective modulestrace ideal
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The pith

For any commutative ring A there exists an exact sequence 0 → Pic(A) → K0(A)* → B(A) → 0.

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

The paper proves a canonical isomorphism for modules generated by idempotents that is key to establishing relations in the Grothendieck ring. It shows that for any commutative ring A, there is an exact sequence of abelian groups 0 → Pic(A) → K0(A)* → B(A) → 0. This sequence is split exact for Dedekind domains and Noetherian one-dimensional rings. Further, it gives isomorphisms B(A) ≅ B(K0(A)) ≅ H0(A)* and applications to lifting idempotents via K0 maps.

Core claim

If e and e' are idempotents of a commutative ring A, then there is a canonical isomorphism Ae ⊕ Ae' ≅ Ae/Ae(1-e') ⊕ Ae'/Ae'(1-e) ⊕ A(e+e'-2ee'). This is used to prove that for any ring A the sequence 0 → Pic(A) → K0(A)* → B(A) → 0 is exact at the beginning and end. The paper also shows that B(A) ≅ B(K0(A)) ≅ H0(A)* canonically, that a ring map lifts idempotents iff the induced K0 map does, and that the support of a finitely generated projective module is the whole spectrum iff its trace ideal is the unit ideal.

What carries the argument

The canonical isomorphism Ae ⊕ Ae' ≅ Ae/Ae(1-e') ⊕ Ae'/Ae'(1-e) ⊕ A(e+e'-2ee') for idempotents e and e'.

If this is right

  • The sequence is split exact for Dedekind domains and Noetherian one-dimensional rings.
  • A ring morphism A to B lifts idempotents if and only if K0(A) to K0(B) does.
  • If B has finitely many maximal ideals then K0(A) to K0(B) is surjective.
  • The support of a finitely generated projective module is the whole prime spectrum if and only if its trace ideal is the unit ideal.

Where Pith is reading between the lines

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

  • The exact sequence may help compute the units of K0(A) explicitly for additional classes of rings beyond those already identified.
  • The connection between B(A) and H0(A)* could link this construction to cohomology-based invariants in algebraic geometry.
  • Applications to whether certain K-theory maps are surjective might follow from the idempotent lifting criterion when the target has few maximal ideals.

Load-bearing premise

The canonical isomorphism Ae ⊕ Ae' ≅ Ae/Ae(1-e') ⊕ Ae'/Ae'(1-e) ⊕ A(e+e'-2ee') holds for all idempotents e, e' and is sufficient to establish the exactness of the sequence at K0(A)*.

What would settle it

A commutative ring A for which the natural map from Pic(A) to the units of K0(A) fails to be injective, or for which the cokernel is not B(A).

read the original abstract

As the first main result of this article, we prove that if $e$ and $e'$ are idempotents of a commutative ring $A$, then there is a canonical isomorphism of $A$-modules: $$Ae\oplus Ae'\simeq Ae/Ae(1-e')\oplus Ae'/Ae'(1-e)\oplus A(e+e'-2ee').$$ This result plays an important role in proving several results on the Grothendieck ring $K_{0}(A)$. Especially, we first show that for any ring $A$ there is a complex of Abelian groups which is exact at the beginning and end: $$\xymatrix{0\ar[r]&\Pic(A)\ar[r]&K_{0}(A)^{\ast} \ar[r]&\mathscr{B}(A)\ar[r]&0.}$$ Then we show that the above sequence is split exact for some certain rings $A$ (including Dedekind domains or more generally Noetherian one dimensional rings). The next main result asserts that for any ring $A$ we have the canonical isomorphisms of Abelian groups $\mathscr{B}(A)\simeq\mathscr{B}\big(K_{0}(A)\big)\simeq H_{0}(A)^{\ast}$. As an application, we show that a morphism of rings $A\rightarrow B$ lifts idempotents if and only if the induced ring map $K_{0}(A)\rightarrow K_{0}(B)$ lifts idempotents. If moreover, $B$ has finitely many maximal ideals then the map $K_{0}(A)\rightarrow K_{0}(B)$ is surjective. Finally, we show that the support of a finitely generated projective module is the whole prime spectrum if and only if its trace ideal is the whole unit ideal.

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

0 major / 3 minor

Summary. The paper proves a canonical isomorphism of A-modules Ae ⊕ Ae' ≅ Ae/Ae(1-e') ⊕ Ae'/Ae'(1-e) ⊕ A(e+e'-2ee') for idempotents e, e' in a commutative ring A. This is used to construct maps yielding an exact sequence 0 → Pic(A) → K0(A)* → B(A) → 0 for any commutative ring A; the sequence is shown to be split exact when A is a Dedekind domain or more generally a Noetherian one-dimensional ring. Further results include canonical isomorphisms B(A) ≅ B(K0(A)) ≅ H0(A)*, a characterization of when a ring homomorphism A → B lifts idempotents in terms of the induced map on K0, surjectivity of K0(A) → K0(B) when B has finitely many maximal ideals, and a criterion for a finitely generated projective module to have full support in terms of its trace ideal.

Significance. If the derivations hold, the work supplies an explicit relation linking the Picard group to the units of the Grothendieck ring K0(A) via the auxiliary group B(A), together with concrete isomorphisms and applications to idempotent lifting and projective-module support. The canonical module isomorphism for pairs of idempotents is a self-contained algebraic identity that may facilitate explicit computations; the exact sequence and its splitting in low-dimensional cases add structural information to the study of K0(A) for commutative rings.

minor comments (3)
  1. [Section introducing B(A)] §2 (or wherever B(A) is introduced): the definition of the group B(A) should be stated explicitly before the exact sequence is constructed, including its generators and relations, to make the surjectivity argument self-contained.
  2. [Statement of the exact sequence] The notation K0(A)* for the group of units is used without prior clarification whether it denotes multiplicative units or an additive group; a sentence clarifying the group operation would prevent ambiguity.
  3. [Split-exactness paragraph] The proof that the sequence splits for Noetherian one-dimensional rings relies on the structure theorem for projective modules over such rings; a brief reference to the relevant theorem (e.g., the fact that projectives are direct sums of ideals) would strengthen readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. No specific major comments appear in the report, so we have nothing further to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central derivation begins with a direct, explicit verification of the canonical A-module isomorphism for any pair of idempotents e, e' (via explicit maps and module structures). This isomorphism is then used, together with the universal property of K0(A) and the standard idempotent-direct-summand correspondence, to define the maps Pic(A) → K0(A)* and K0(A)* → B(A) and to verify exactness at the two ends. All steps rely on these standard, externally established facts rather than any fitted parameter, self-referential definition, or load-bearing self-citation chain. The subsequent results on splitting, canonical isomorphisms B(A) ≅ B(K0(A)) ≅ H0(A)*, and applications to lifting idempotents likewise flow from the same non-circular base. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions and properties of commutative rings, projective modules, the Grothendieck construction K0(A), and the Picard group; no numerical fitting or new postulated entities appear in the abstract.

axioms (1)
  • domain assumption Standard definitions and exactness properties of the Grothendieck ring K0(A) and Picard group Pic(A) in commutative algebra.
    These are invoked to define the groups appearing in the exact sequence.

pith-pipeline@v0.9.0 · 5872 in / 1324 out tokens · 63568 ms · 2026-05-24T10:53:12.786653+00:00 · methodology

discussion (0)

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Reference graph

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