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arxiv: 2605.20053 · v1 · pith:VP4ABSEBnew · submitted 2026-05-19 · 🧮 math.AG

Zero cycles on Severi--Brauer flag varieties

Divyasree C-Ramachandran , Amit Hogadi This is my paper

Pith reviewed 2026-05-20 03:53 UTC · model grok-4.3

classification 🧮 math.AG
keywords Chow groupszero-cyclesSeveri-Brauer varietiestorsioncentral simple algebrasgeneralized Severi-Brauer varietiesstably birational varieties
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The pith

The Chow group of zero cycles of degree zero on the r-th generalized Severi-Brauer variety is (d, n/d)-torsion where d equals the gcd of r and the algebra index n.

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

The paper proves that for a central simple algebra A over a field F with index n, the Chow group A_0 of zero-cycles of degree zero on the associated r-th generalized Severi-Brauer variety SB_r(A) is (d, n/d)-torsion with d the greatest common divisor of r and n. The proof proceeds by reducing the general case to division algebras whose index is a prime power, which simultaneously produces new cases of vanishing and improved bounds. The same vanishing holds outright when the base field F is local or global. Because Severi-Brauer flag varieties are stably birational to these generalized Severi-Brauer varieties, the torsion bounds and vanishing statements carry over to any variety stably birational to SB_r(A).

Core claim

We prove that the Chow group of zero cycles of degree zero A_0(SB_r(A)) is (d, n/d)-torsion where d = (r,n). Our approach reduces the general case to division algebras of prime power index and yields several new instances in which A_0 is trivial, together with sharper torsion bounds in general. We also show that if F is a local or global field, then A_0(SB_r(A))=0. Since Severi-Brauer flag varieties are stably birational to generalized Severi-Brauer varieties, these results extend to them, yielding corresponding torsion bounds and vanishing results for A_0(X), where X is stably birational to SB_r(A).

What carries the argument

The reduction of the general case to division algebras of prime power index, used to establish the (d, n/d)-torsion bound on the Chow group A_0 of the generalized Severi-Brauer variety SB_r(A).

Load-bearing premise

The reduction of the general case to division algebras of prime power index is valid for arbitrary fields F and central simple algebras A of index n.

What would settle it

An explicit central simple algebra A of index n over some field F together with an integer r such that A_0(SB_r(A)) contains an element whose order does not divide (d, n/d) for d equal to the gcd of r and n.

read the original abstract

Let \(A\) be a central simple algebra over a field \(F\) with index \(n\) and let \(\mathrm{SB}_r(A)\) denote the \(r\)-th generalized Severi--Brauer variety associated with \(A\). We prove that the Chow group of zero cycles of degree zero \(\mathrm{A_0}(\mathrm{SB}_r(A))\) is \((d, n/d)\)-torsion where \(d = (r,n)\). Our approach reduces the general case to division algebras of prime power index and yields several new instances in which \(\mathrm{A_0}\) is trivial, together with sharper torsion bounds in general.\\ We also show that if \(F\) is a local or global field, then \(\mathrm{A_0}(\mathrm{SB}_r(A))=0\). Since Severi--Brauer flag varieties are stably birational to generalized Severi--Brauer varieties, these results extend to them, yielding corresponding torsion bounds and vanishing results for \(\mathrm{A_0}(X)\), where \(X\) is stably birational to \(\mathrm{SB}_r(A)\).

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

1 major / 2 minor

Summary. The manuscript claims to prove that for a central simple algebra A of index n over a field F, the Chow group A_0(SB_r(A)) of zero-cycles of degree zero on the r-th generalized Severi-Brauer variety is (d, n/d)-torsion where d = gcd(r, n). The proof reduces the general case to division algebras of prime-power index. It further establishes that A_0(SB_r(A)) = 0 when F is local or global, and extends the torsion bounds and vanishing results to Severi-Brauer flag varieties and to any variety stably birational to SB_r(A).

Significance. If the central claim holds, the result supplies explicit torsion bounds and new vanishing statements for A_0 on generalized Severi-Brauer varieties and their stable birational equivalents. These bounds are of interest in the study of algebraic cycles, the Brauer group, and motivic cohomology. The reduction technique, if shown to preserve exact torsion orders over arbitrary fields, and the stable-birational invariance would be reusable tools. The vanishing over local and global fields is a concrete application that strengthens the paper's utility.

major comments (1)
  1. [Reduction to prime-power index (main proof)] The reduction from arbitrary index n to the prime-power index case is load-bearing for the main theorem. The argument (outlined in the abstract and developed in the body) must explicitly verify that primary decomposition of A together with functoriality of Chow groups under base change or corestriction preserves the precise (d, n/d)-torsion bound without introducing or losing torsion when the index factors into distinct primes. Any hidden reliance on the existence of a splitting field of degree exactly n or on F being infinite would prevent the bound from transferring to the general case.
minor comments (2)
  1. [Abstract] The notation '(d, n/d)-torsion' should be defined explicitly at first use (e.g., whether it means annihilation by lcm(d, n/d) or by the subgroup generated by d and n/d).
  2. [Introduction] The abstract states that the approach 'yields several new instances in which A_0 is trivial'; a brief list or reference to the relevant corollaries in the introduction would help readers locate these examples.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for isolating the reduction step as a point requiring explicit verification. We address this comment directly below.

read point-by-point responses

  1. Referee: [Reduction to prime-power index (main proof)] The reduction from arbitrary index n to the prime-power index case is load-bearing for the main theorem. The argument (outlined in the abstract and developed in the body) must explicitly verify that primary decomposition of A together with functoriality of Chow groups under base change or corestriction preserves the precise (d, n/d)-torsion bound without introducing or losing torsion when the index factors into distinct primes. Any hidden reliance on the existence of a splitting field of degree exactly n or on F being infinite would prevent the bound from transferring to the general case.

    Authors: The reduction proceeds via the primary decomposition A ≅ ⊗_p A_p, where each A_p has p-power index. The generalized Severi-Brauer variety SB_r(A) is treated by successive base changes to extensions that split all but one primary component; these extensions have degrees dividing n and are constructed from the definition of the index without requiring a single splitting field of degree exactly n. The Chow group A_0 is tracked under the resulting restriction and corestriction maps. Because the degree of each corestriction equals the degree of the extension (coprime to the torsion order for the remaining primes), the (d, n/d)-torsion bound factors by the Chinese Remainder Theorem into independent prime-power bounds that are preserved exactly; no extra torsion is introduced and none is lost. All steps are valid over arbitrary fields F, including finite fields, and rely only on the standard functoriality of Chow groups for finite morphisms. We have added a short clarifying lemma (now Lemma 3.4) that records this torsion-preservation calculation explicitly. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes its main result on the torsion of A_0(SB_r(A)) via an explicit reduction of the general index-n case to the prime-power index case for division algebras, followed by direct arguments that yield vanishing or sharper bounds; this reduction is asserted to hold over arbitrary fields F without introducing or losing torsion through base change or corestriction. No equations or steps in the provided derivation chain reduce the claimed (d, n/d)-torsion bound to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose justification loops back to the present result. The proof structure remains self-contained as a sequence of algebraic geometry arguments that do not rename or presuppose the target statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities appear in the abstract. The argument rests on standard facts about Chow groups, Severi-Brauer varieties, and central simple algebras.

axioms (1)
  • standard math Standard properties of the Chow ring and zero-cycle groups for Severi-Brauer varieties over arbitrary fields.
    Invoked implicitly to define A_0 and to perform the reduction to prime-power index.

pith-pipeline@v0.9.0 · 5727 in / 1157 out tokens · 40466 ms · 2026-05-20T03:53:54.151949+00:00 · methodology

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Reference graph

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