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Factoring maps to big Cohen-Macaulay algebras through blowups

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Pith's one-line read Any sufficiently functorial big Cohen-Macaulay algebra factors through the derived global sections of every proper birational map.

desk verdict Clean formal extraction: any weakly functorial BCM assignment factors through derived global sections of every proper birational model. read the letter →

arxiv 2607.08965 v1 pith:CDX3S5QN submitted 2026-07-09 math.AC math.AG

Karl Schwede This is my paper · ORCID
classification math.ACmath.AG MSC 13D2213H1014B0513A30
keywords bigCohen-MacaulayalgebrasderivedsplintersReesCousincomplexesbirationalmorphismsweakfunctorialitylocalcohomology
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that maps from a reduced equidimensional Noetherian local ring into a big Cohen-Macaulay algebra always factor, in the derived category, through the derived global sections of any proper birational modification of the spectrum. Big Cohen-Macaulay algebras have long been a tool for proving vanishing and purity statements in commutative algebra; the new result says that once such an algebra is assigned in a weakly functorial way, the factorization through blow-ups (and more generally through proper birational maps) is automatic. The argument first reduces to the Rees algebra of an ideal of positive height, then uses a Sancho-de-Salas comparison to kill the fiber of the structure map after local cohomology, and finally employs a Cousin filtration argument to promote that vanishing to an actual factorization. The payoff is that several known purity and splinter properties of rings that admit big Cohen-Macaulay algebras become formal consequences of the assignment itself rather than of special constructions such as absolute integral closures.

What carries the argument

The combination of a Sancho-de-Salas triangle for the Rees algebra of a positive-height ideal with Sharp’s characterization of weakly balanced big Cohen-Macaulay modules by exactness of the height-filtration Cousin complex; together they convert local-cohomology vanishing into a global factorization.

What would settle it

Exhibit a reduced equidimensional local ring R together with a weakly functorial balanced big Cohen-Macaulay algebra B_R and a blow-up Y o Spec R for which the composition of the structure map with the natural map R o RΓ(Y, O_Y) fails to vanish on the fiber in the derived category.

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Extended reading notes

Core claim

For a reduced universally catenary equidimensional Noetherian local ring R, any weakly functorial balanced big Cohen-Macaulay algebra assignment R ↦ B_R, and any proper birational map Y → Spec R, the structure map R o B_R factors in the derived category as R o RΓ(Y, O_Y) o B_R.

Load-bearing premise

The existence of a balanced big Cohen-Macaulay algebra assignment that is weakly functorial at least for surjections coming from localizations of Rees algebras; without that commutative diagram the comparison has nothing to map into.

Editorial extensions

If this is right

  • Weakly BCM-regular rings are automatically birational derived splinters.
  • The main purity statements previously proved for absolute integral closures or their p-adic completions become formal consequences of any sufficiently functorial BCM assignment.
  • The Briançon–Skoda theorem and related integral-closure results for pseudo-rational and Du Bois singularities follow from the same formal factorization once a weakly functorial BCM algebra is known to exist.
  • The same factorization holds after localization, so the result globalizes to any reduced universally catenary locally equidimensional Noetherian ring that admits a weak CM-assignment.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the assignment can be upgraded so that the maps are of derived commutative algebras, the factorization would also control higher multiplicative structure on the derived global sections.
  • The same formal argument may apply to other classes of algebras characterized by vanishing of local cohomology (for instance certain perfectoid or almost Cohen-Macaulay algebras) once weak functoriality for Rees surjections is verified.
  • Question 3.9 in the paper suggests a natural next test: whether the factorization can be realized by maps of derived rings rather than merely of complexes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper proves that if (R, m) is a reduced universally catenary equidimensional Noetherian local ring and R ↦ B_R is a sufficiently weakly functorial balanced big Cohen-Macaulay algebra assignment (compatible at least with surjections from localizations of Rees algebras), then for every proper birational map Y o Spec R the structure map R o B_R factors in D(R) through RΓ(Y, O_Y). The argument reduces via Chow’s lemma to the blow-up of an ideal of positive height, compares the fiber of R o RΓ(Y, O_Y) with the degree-zero part of a Sancho-de-Salas sequence (Theorem 3.1), and then shows that any map from that fiber into a weakly cohomologically Cohen-Macaulay module that vanishes after local cohomology at every prime is already zero in D(R) (Lemma 3.2). Two independent proofs of the vanishing lemma are given (Cousin filtration and middle-perverse t-structure). The result recovers that weakly BCM-regular rings are birational derived splinters and, combined with earlier work, the Briançon–Skoda theorem of Rodríguez-Villalobos–Schwede.

Significance. The factorization property was previously known only for the concrete constructions R^{+} (char p) and ĉR^{+} (mixed characteristic). Establishing it for any weakly functorial BCM assignment unifies those results and supplies a formal reason why the property holds whenever such algebras exist. The argument is short, classical (Cousin complexes, Sancho-de-Salas, perverse t-structures), and supplies two independent proofs of the key vanishing statement. The applications to derived splinters and Briançon–Skoda are immediate and clean. The manuscript is therefore a useful structural contribution to the theory of big Cohen-Macaulay algebras.

minor comments (5)
  1. Definition 2.5 introduces a “weak CM-assignment” while the abstract and introduction speak of “sufficiently functorial” or “weakly functorial” balanced big Cohen-Macaulay algebra assignments. A single sentence equating the two terminologies would avoid any momentary confusion.
  2. In the proof of Theorem 3.1 the identification [RΓ_{S>0} S]_0 ≅ K• is asserted by reference to Lipman; a one-line reminder that the degree-zero part of the Sancho-de-Salas sequence is precisely the fiber of R o RΓ(Y, O_Y) would make the comparison self-contained.
  3. Lemma 2.2 sketches the weakly balanced case; the balanced case is cited to the literature. Since the main theorems only need the weakly balanced/cohomological version, the sketch is sufficient, but a parenthetical remark that the balanced localization statement is classical would be helpful.
  4. The AI-acknowledgement section is unusually detailed. It is honest and does not affect the mathematics, but the journal may wish to decide whether such a section should appear in the published version or be moved to a supplementary note.
  5. Typographical: “Sancho de Salas” is sometimes hyphenated and sometimes not; “cR+” versus “ĉR+” appears inconsistently in the introduction. Standardize throughout.

Circularity Check

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No significant circularity: the factorization is proved from classical Cousin/Sancho-de-Salas vanishing under an explicit external functoriality hypothesis on BCM algebras.

full rationale

The load-bearing chain (support bound (3.0.1), Sancho-de-Salas comparison in Theorem 3.1, Cousin filtration induction in Lemma 3.2 via Sharp’s external characterization, and localization+Chow reduction in Corollary 3.7) does not define any quantity in terms of the target factorization, fit parameters, or import a uniqueness theorem from the author’s prior work as a black-box force. Self-citations (MS21, MMGS26, RS24) appear only for recovery of known corollaries and background on BCM-regular rings; they are not used to justify the vanishing maps or the Cousin exactness. The sole non-classical input is the weakly functorial BCM assignment of Definition 2.5 (compatible with surjections from localizations of Rees algebras), which is stated explicitly as a hypothesis and known independently from R+/cR+/André constructions. The two proofs of Lemma 3.2 are self-contained against that hypothesis. Hence the derivation is independent of its conclusion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper works entirely inside classical commutative algebra. No free parameters are fitted. The only non-standard inputs are the existence of weakly functorial BCM algebras (known by prior deep work of Hochster-Huneke, André, Bhatt) and the classical theory of Cousin complexes (Sharp). No new entities are postulated.

assumptions (4)
  • domain assumption Existence of balanced big Cohen-Macaulay algebras for excellent local domains in all characteristics (Hochster-Huneke, André, Bhatt).
    Invoked throughout as the source of the algebras B_R and B_S; without them the statement is vacuous.
  • standard math Sharp's characterization: an R-module is weakly balanced BCM iff its augmented Cousin complex is exact (Theorem 2.7).
    Used to replace B_R by its Cousin complex in the proof of Lemma 3.2.
  • standard math Localization preserves the weakly balanced BCM property for universally catenary equidimensional rings (Lemma 2.2).
    Needed to reduce the global statement to local statements at every prime.
  • domain assumption Weak functoriality of the BCM assignment with respect to surjections from localizations of Rees algebras (Definition 2.5 and the diagram in Theorem 3.5).
    The precise hypothesis that makes the Sancho-de-Salas comparison produce a zero map into B_R.

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Pith. "Pith review of Factoring maps to big Cohen-Macaulay algebras through blowups." pith.science (2026). https://pith.science/paper/CDX3S5QN

@misc{pith2026260708965,
  author       = {Pith},
  title        = {Pith review of: Factoring maps to big Cohen-Macaulay algebras through blowups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CDX3S5QN}},
  note         = {Machine review of arXiv:2607.08965}
}
abstract

We show that for a reduced universally catenary equidimensional Noetherian local ring $R$, any sufficiently functorial big Cohen-Macaulay algebra assignment $R \mapsto B_R$, and any proper birational map $Y \to \mathrm{Spec} R$, there exists a factorization $R \to {\bf R}\Gamma(Y, \mathcal{O}_Y) \to B_R$ in the derived category of $R$-modules.

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