Corrigendum to "Stability in the high-dimensional cohomology of congruence subgroups" [Compos. Math. 156 (2020), no. 4, 822-861]

Jeremy Miller; Peter Patzt; Rohit Nagpal

REVIEW 4 major objections 4 minor 1 cited by

Corrigendum to "Stability in the high-dimensional cohomology of congruence subgroups" [Compos. Math. 156 (2020), no. 4, 822-861]

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This corrigendum repairs a flawed partial order in the original proof of Koszulity for the Steinberg monoid, and reproves the vanishing of H_s(B^n_*(St)) for s ≠ n.

desk verdict A real gap in the 2020 paper gets a plausible corrected proof; the mathematics is likely sound, but the presentation is rough and needs a careful referee. read the letter →

arxiv 2508.14945 v1 pith:67EYUZQJ submitted 2025-08-20 math.AT

Jeremy Miller , Rohit Nagpal , Peter Patzt This is my paper · ORCID

classification math.AT

keywords SteinbergmoduleKoszulalgebrabarcomplexfiltrationpartialorderPBWbasiscohomologicalstabilitycongruencesubgroups

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

This corrigendum addresses a mistake in the original paper's proof that the Steinberg monoid is Koszul. The problem lies in a partial order on certain invariant tuples; it did not satisfy a property needed for the filtration to behave correctly. The authors define a repaired partial order, prove the filtration is a subcomplex, and construct an explicit chain homotopy showing the successive quotients have vanishing homology below degree n. This restores the key conclusion that the reduced bar complex B^n_*(St) has homology only in degree n. If correct, the original stability results for high-dimensional cohomology of congruence subgroups stand.

What carries the argument

The key object is the invariant tuple I(S) = (c_1,...,c_n, k) ∈ {1,...,n}^n × {0,..., C(n,2)} attached to a concatenated sequence S of PBW pivot positions of a decomposition W_1 ⊕ ... ⊕ W_p = K^n. The lexicographic order on these tuples defines a filtration F_I B^n_p(St) of the reduced bar complex by direct summands indexed by decompositions with I_D ≤_lex I. The subcomplex property (Lemma 4) is proven by comparing the tuple of a decomposition with the tuple of the neighbouring decomposition in which W_i and W_{i+1} are merged: either the pivot sequences share no entries and the tuple is unchanged except the inversion count cannot increase, or they share an entry and some count c_j strictly

What would settle it

A direct counterexample to Lemma 4 would settle the issue: compute the PBW pivot sequence of W_i ⊕ W_{i+1} for two subspaces whose pivot sequences are disjoint, and check whether it equals the sorted union with no increase in inversion count. Any failure would break the subcomplex property and with it the vanishing theorem. Since the paper's Case 2 example is given, a similar explicit computation for a disjoint pair would test the assertion.

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Extended reading notes

Core claim

On the paper's own terms: the original proof of Koszulity used a partial order on sequences S that encode the pivot positions of PBW bases in a direct-sum decomposition of K^n. That order was claimed without proof to satisfy a property (P2) needed to make a filtration a subcomplex; the claim is false. This corrigendum replaces the order with the lexicographic order on invariant tuples I(S) = (c_1,...,c_n, k), where c_i counts occurrences of i in S and k is the inversion count, and proves the needed property: merging adjacent summands cannot increase I(S) in the lexicographic order. It then reproves Proposition 1, H_s(B^n_*(St)) = 0 for s ≠ n, by filtering the chain complex by these tuples an

Load-bearing premise

The load-bearing step is the unstated argument in Lemma 4, Case 1: if the PBW pivot sequences of two adjacent subspaces share no entry, then merging the subspaces must give a new pivot sequence that is exactly the sorted union of the two, so the counts c_j are unchanged and the inversion count does not increase; this is asserted without proof.

Editorial extensions

If this is right

The vanishing H_s(B^n_*(St)) = 0 for s ≠ n is established for the corrected filtration, closing the gap in the Koszulity proof.
The original stability theorems for high-dimensional cohomology of congruence subgroups, which relied on this vanishing, are upheld.
The corrected partial order satisfies the previously unproved property (P2), so the filtration F_I B^n_*(St) is indeed a subcomplex.
The chain homotopy Φ is defined explicitly on PBW basis elements and satisfies Φ∂ + ∂Φ = id on every successive quotient in degrees p < n.

Reading between the lines

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

The same lexicographic order on invariant tuples could be applied to other bar complexes or resolutions built from PBW bases, where the original partial order might fail in the same way.
The asserted Case 1 of Lemma 4 (disjoint pivot sequences imply the merged tuple equals the sorted union) is stated without a full argument; a more formal proof would use the determinacy of the PBW pivot positions, but the claim is plausible and the example in Case 2 suggests the mechanism.
Remark 5 hints at an alternative proof identifying the relative chains with shifted reduced simplicial chains; fleshing this out could give a more geometric and less cancellation-heavy argument for the same vanishing.
Since the corrigendum says the mistake was found interactively after publication, other lemmas in the original paper that relied on unproved properties of the old partial order should be re-checked, though the authors present the corrected order as a complete replacement.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague — the short version: this corrigendum does what corrigenda should do. It names a genuine gap in the authors' 2020 paper — the partial order on page 833 did not satisfy property (P2) — offers a corrected partial order and a new filtration, and gives a complete proof of the vanishing result that was originally claimed. There is no new theorem or application; the value is that a known stability theorem now appears to have a repaired proof.

What is good: the new filtration is natural, and the two cases in Lemma 4 are correct. In the disjoint case, the sum of the two subspaces really does have PBW sequence equal to the sorted union, so the counts are unchanged and the inversion count does not increase. In the shared-pivot case, the smallest common pivot forces c_k to drop because only one basis vector can have that pivot in the sum. The homotopy is a standard "split the smallest ascent" construction, and the cancellation pattern is believable. The corrigendum is also self-contained in the right sense: it does not assume the target result, only background definitions and the Solomon–Tits basis.

The soft spots are presentation. The manuscript has placeholder '??' symbols, an incomplete sentence in the proof of Lemma 4, and the homotopy verification is terse. Several 'Note that' claims are asserted rather than derived. The last sum in the ∂Φ calculation appears to have an off-by-one (the upper limit should be p−1, not p), and the phrase "one smaller I" is ambiguous in edge cases, though the intended meaning is clear. These are not load-bearing flaws as far as I can tell. The stress-test checked the cancellation signs and the subcomplex property; I agree with that assessment. An expert referee should verify the homotopy identities in full, but the gaps are not the kind that sink a paper.

Who is this for: readers of MNP20 and anyone relying on the stability theorem. It is a repair within an established program, not a new framework. It deserves a serious referee; the referee's job is mostly to demand clean notation and fill in the terse steps, not to find a contradiction.

Recommendation: send it to peer review. With a modest revision addressing typos and expanding the homotopy cancellation, it should be publishable as a corrigendum.

Referee Report

4 major / 4 minor

Summary. The corrigendum addresses an error in the published paper by Miller--Nagpal--Patzt (Compos. Math. 156, 2020). The error lies in the partial order used in the proof of the Koszul property of the Steinberg monoid: the original order was claimed, without proof, to satisfy a property (P2) needed for the filtration argument. The authors replace it with a lexicographic invariant I(S) = (c_1,...,c_n,k), where the c_i record multiplicities of entries in the concatenated PBW-pivot sequence of a decomposition and k records inversions. They define a filtration F_I B_*^n(St) by these invariants, prove in Lemma 4 that it is a subcomplex, and define a degree-raising map Φ on successive quotients. They claim Φ∂+∂Φ = id on F_I/F_{I'} for p<n, which yields the vanishing H_s(B_*^n(St)) = 0 for s ≠ n (Proposition 1), thereby repairing the proof of the main theorem of the original paper.

Significance. The corrigendum is significant for the published record: it identifies a concrete flaw in a published proof and supplies a replacement mechanism. The proposed filtration is natural and the overall strategy is plausible; if fully written out, it would indeed provide a self-contained proof of the required vanishing. The paper does not rely on the target theorem to prove itself: it uses only the Solomon--Tits PBW basis and the monoid product, so there is no circularity. The proof is, however, presented as a sequence of assertions, several of which are load-bearing and are only sketched. The mathematical idea appears correct, but the manuscript as submitted does not yet meet the standard of a 'complete proof' that it claims. The credit for identifying the error and constructing the corrected order belongs to the authors and to Putman.

major comments (4)

[Paragraph after Lemma 4] The definition of I' is ambiguous. The text says I' is 'the element ... one smaller than I in lexicographical order', but lexicographic order is on the entire finite set {1,...,n}^n × {0,...,C(n,2)}, not only on invariants realized by decompositions. The proof later passes to the quotient F_I B_*^n / F_{I'} B_*^n and needs all summands with invariant strictly below I to lie in F_{I'}. This requires a precise choice of an exhaustive filtration indexed by all lex elements, including a convention for the minimal element (where the predecessor does not exist). As written, this is a gap in the inductive filtration argument. The fix is immediate, but it is load-bearing for the claimed vanishing.
[Lemma 4, Case 1] The assertion that if S_Wi and S_Wi+1 are disjoint, then S_{Wi⊕Wi+1} is the sorted union of the two sequences, is stated without proof. It is true: concatenating the two PBW bases and ordering by pivot position gives a row-echelon basis of the direct sum, and no pivot collision occurs; hence the invariant sequence is the sorted union. But the manuscript should supply this one-line justification. As printed, the reader must reconstruct a nontrivial fact about PBW bases. Case 2 is also compressed: the claim that the smallest common pivot k can appear only once in the direct sum, while all smaller pivots are unchanged, deserves a dimension-counting argument relative to the flag E_m. These are not mathematical objections to the statement, but they are essential details for a proof that the filtration is a subcomplex.
[Homotopy calculation, second case] The verification that Φ∂+∂Φ = id is a sketch relying on several 'Note that' assertions. In particular, the claims that certain summands vanish in F_I/F_{I'} because their invariant drops, and that the remaining summands cancel 'exactly', are not demonstrated. Moreover, the displayed formula for ∂(Φ(a)) contains an indexing issue: the final sum over i0+1 ≤ i ≤ p should terminate at p-1 (or the indices must be re-explained), since the boundary of a p+1-term chain has p terms. This is a typo that can be corrected, but it obscures a sign check that is central to the proof. The authors should spell out the cancellation in enough detail that a reader can verify it without redoing the entire calculation.
[Opening and closing of proof of Proposition 1] The proof contains placeholder text '??' in two places ('In order to show?? 1' and 'This finishes the proof of??1') and an incomplete sentence beginning 'We will show that'. In a corrigendum intended to be a complete replacement proof, these must be filled in. This is a presentation issue, but it compounds the difficulty of following the argument.

minor comments (4)

[Throughout] There are typos and formatting issues: 'Wi̸=0' in the definition of B^n_s, 's' and 'p' are interchanged in a few places, 'For completeness sake' should be 'For completeness' sake', and the phrase 'We will show that' is left unfinished. These should be cleaned up.
[Remark 5] The alternative proof via shifted reduced simplicial chains is mentioned but not developed. This is acceptable, but the authors should clarify that no claim is being made that this alternative is fully proved.
[References] The corrigendum cites the original paper for generators, product structure, and the Solomon--Tits basis, which is appropriate. It would help to recall explicitly the property (P2) of the original partial order that failed, so that a reader can understand precisely what is being corrected.
[Example in Lemma 4] The example for Case 2 is helpful but not fully explained: the reader must check that the displayed vectors are indeed PBW bases of W1 and W2. A short explanation of why w2 - w3 gives the pivot shift would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the corrigendum's proof is self-contained; self-citations are background only.

full rationale

The paper's central claim is Proposition 1, a vanishing result for the reduced bar complex B^n_*(St). This is proved by defining a new partial order on invariant tuples I(S), constructing the filtration F_I B^n_*(St), proving the subcomplex property in Lemma 4 from PBW basis facts, and then giving an explicit chain homotopy Phi whose terms cancel directly. None of these steps presupposes the target vanishing theorem. The only citations to [MNP20] are structural background: the presentation of St(V) (Theorem 2), the monoid product (Proposition 3), and the PBW basis assertion (Proposition 3.3). These are not the erroneous partial order and are not the Koszul/vanishing result being corrected. The new partial order is defined in this corrigendum, not imported from prior work, and the homotopy is computed from the definitions. Even though the text contains placeholders and terse passages, the derivation does not reduce by construction to a fitted parameter, a renamed known result, or a self-citation chain. Hence no circular step is present.

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

The proof relies on standard facts from the original paper: the generators and relations for the Steinberg module, the associative product turning St into a monoid, and the consequence of the Solomon-Tits theorem that PBW apartment classes form a basis. No free parameters or invented entities are introduced. The only new construction is the lexicographic filtration, which is the content of the corrigendum.

assumptions (4)

standard math Solomon-Tits theorem implies apartment classes from PBW-bases form a k-basis of St(W).
Invoked in the paragraph after Proposition 3.3 of the original paper; the corrigendum relies on this to identify bases of the chain groups.
standard math Generators and relations for the Steinberg module St(V) (Theorem 2).
Recalled at the start and used to describe the apartment classes in the bar complex.
domain assumption The product map St(V) ⊗ St(U) → St(V⊕U) is well-defined and gives St a monoid structure.
Recalled as Proposition 3 from the original paper; used to define the differential of the bar complex.
domain assumption Each subspace W has a unique PBW-invariant sequence S_W of pivot positions.
Assumed in the construction of S_D and I(S); the uniqueness is stated but not proved in the corrigendum.

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Pith. "Pith review of Corrigendum to "Stability in the high-dimensional cohomology of congruence subgroups" [Compos. Math. 156 (2020), no. 4, 822-861]." pith.science (2026). https://pith.science/paper/67EYUZQJ

@misc{pith2026250814945,
  author       = {Pith},
  title        = {Pith review of: Corrigendum to "Stability in the high-dimensional cohomology of congruence subgroups" [Compos. Math. 156 (2020), no. 4, 822-861]},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/67EYUZQJ}},
  note         = {Machine review of arXiv:2508.14945}
}

read the original abstract

After the publication of [Compos. Math. 156 (2020), no. 4, 822-861], Andrew Putman pointed out a mistake in our paper and helped us fix it. In this note, we will explain what this mistake is and how to fix it.

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Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures
math.AT 2025-09 conditional novelty 7.0 of 10

Steinberg homology vanishes in a range for all reductive groups, and the double Tits building T^2(Z^n) is n-connected, refining the Church-Farb-Putman conjecture in degrees 1 and 2.

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Works this paper leans on

1 extracted references · 1 canonical work pages · cited by 1 Pith paper

[1]

Math.156(2020), no

[MNP20] Jeremy Miller, Rohit Nagpal, and Peter Patzt,Stability in the high-dimensional cohomology of congruence subgroups, Compos. Math.156(2020), no. 4, 822–861. MR 4079629 150 N University Street West Lafayette, IN-47904 USA, Department of Mathematics, Purdue

work page 2020

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Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Math.156(2020), no

[MNP20] Jeremy Miller, Rohit Nagpal, and Peter Patzt,Stability in the high-dimensional cohomology of congruence subgroups, Compos. Math.156(2020), no. 4, 822–861. MR 4079629 150 N University Street West Lafayette, IN-47904 USA, Department of Mathematics, Purdue

work page 2020