Perfect complexes and completion

Beren Sanders; Paul Balmer

REVIEW 1 major objections 4 minor 19 references

Perfect complexes and completion

T0 review · 1 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that perfect complexes over an $I$-adic completion are equivalent to dualizable complexes in the derived category of $I$-complete complexes exactly when the Koszul complex of a generating sequence is unchanged by…

desk verdict A clean, correct theorem giving the right criterion for when perfect complexes over a completion match dualizable complete complexes; the noetherian case globalizes BIKP23 and the non-noetherian counterexample shows the criterion is real. read the letter →

arxiv 2411.14761 v1 pith:S7XUHPJW submitted 2024-11-22 math.AC math.ATmath.CTmath.KT

Paul Balmer , Beren Sanders This is my paper · ORCID

classification math.ACmath.ATmath.CTmath.KT MSC 18F99

keywords perfectcomplexesI-adiccompletionderivedcompletedualizableobjectsKoszultensor-triangularcategorytt-equivalencenoetherianrings

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 paper asks when the derived category of perfect complexes over the $I$-adic completion $\hat R$ can be recovered from the derived category $D(R)$ using only tensor-triangular data. The answer is a sharp criterion: recovery happens exactly when a generating sequence $s=(s_1,\dots,s_r)$ of $I$ is Koszul-complete, meaning the Koszul complex over $R$ and the one over $\hat R$ have the same homology. For noetherian rings every sequence is Koszul-complete, so the equivalence always holds there. The theorem also identifies the obstruction as an isomorphism of ring objects: the tensor-triangular completion of the unit agrees with classical ring completion precisely in the Koszul-complete case. This matters because categorical completion and classical $I$-adic completion are generally different, and the dualizable objects in the complete category are the natural replacement for perfect complexes over the completed ring.

What carries the argument

The load-bearing object is the Koszul complex $\operatorname{kos}_R(s)=\bigotimes_{i=1}^r \operatorname{cone}(s_i: R\to R)$, where each cone is a two-term complex $R\xrightarrow{s_i} R$. The categorical side is governed by the idempotent triangle $e_Y\to 1\to f_Y\to \Sigma e_Y$ in $D(R)$: $Y$-torsion is $e_Y\otimes -$ and $Y$-completion is $(-)^\wedge_Y=[e_Y,-]$. The paper compares the unit of the completed category, $\hat 1_Y=[e_Y,1]$, with the classical completed ring $\hat R_Y$. The technical heart is Theorem 4.17: if $R$ is classically $I$-adically complete, an object of $D_Y(R)$ is dualizable iff it lies in the thick subcategory generated by $e_Y$, and a derived complete complex is dualizable iff it is a perfect complex. The proof inducts on homological amplitude, using lifting of idempotent matrices from $R/I$ to $\hat R$.

What would settle it

Take the non-noetherian ring $R = \mathbb{Z}_{(p)} \oplus (\mathbb{Q}/\mathbb{Z}_{(p)})$ with $s=p$ and $Y=V(p)$. The Koszul complex $\operatorname{kos}_R(p)$ has nonzero $H_1$ (a copy of $\mathbb{Z}/p$ inside $\mathbb{Q}/\mathbb{Z}_{(p)}$), while the completion $\hat R$ is $\hat{\mathbb{Z}}_p$, for which $H_1$ of $\operatorname{kos}_{\hat R}(p)$ is zero; so $s$ is not Koszul-complete. Theorem 5.1 then predicts that $(D(R)^\wedge_Y)_d$ is not tt-equivalent to $D_{\rm perf}(\hat{\mathbb{Z}}_p)$; checking that equivalence directly would settle the theorem's prediction in this case.

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Extended reading notes

Core claim

The central assertion is Theorem 5.1. For a sequence $s=(s_1,\dots,s_r)$ with $Y=V(s_1,\dots,s_r)$, the following are equivalent: (i) $s$ is Koszul-complete, i.e. the canonical map induces a quasi-isomorphism $\operatorname{kos}_R(s)\to \operatorname{kos}_{\hat R}(s)$; (ii) there is a canonical tt-equivalence between the dualizable objects $(D(R)^\wedge_Y)_d$ and $D_{\rm perf}(\hat R_Y)$ fitting into the completion diagram; (iii) the analogous equivalence holds for the supported category $(D_Y(R))_d$; and (iv) there is an isomorphism of ring objects $\hat 1_Y\simeq \hat R_Y$ in $D(R)$. The route to the theorem combines Theorem 3.20, which characterizes when torsion and completion functors become equivalences, with Corollary 4.26, which says that when $R$ is classically complete the dualizable objects in the $Y$-complete category are exactly the perfect complexes. In the noetherian case the Koszul hypothesis is automatic, so $D_{\rm perf}(\hat R)$ is canonically tt-equivalent to the dualizable $Y$-complete complexes, and the local maximal-ideal case recovers the motivating recent result.

Load-bearing premise

The proof relies on a compact-generation fact: inside the subcategory of complexes supported on $Y$, the small objects are exactly the $Y$-supported perfect complexes; if that fact failed for some non-noetherian ring, the main equivalence would not be proven by this route.

Editorial extensions

If this is right

For a noetherian ring $R$, the perfect complexes over $\hat R$ are canonically tt-equivalent to the dualizable objects in both the derived complete category $D(R)^\wedge_Y$ and the supported category $D_Y(R)$, with the completion diagram commuting.
For a general ring, the categorical equivalence holds for an ideal iff every (equivalently, any one) generating sequence is Koszul-complete, so a finite homology computation decides a categorical question.
When the criterion holds, the classical completion functor $D_{\rm perf}(R)\to D_{\rm perf}(\hat R)$ factors canonically through categorical $Y$-completion on dualizable objects, so the two notions of completion agree at the level of perfect complexes.
The local noetherian theorem for a maximal ideal, previously known only in that special setting, is recovered as a corollary of the global criterion.

Reading between the lines

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

Editorial inference: outside the noetherian world, the natural way to build a completed derived category is to take the dualizable objects $(D(R)^\wedge_Y)_d$ and pass to its Ind-completion; the paper's Corollary 4.26 shows this construction matches classical completion for rings that are already complete.
Editorial inference: Koszul-completeness gives a practical test that can be applied to non-noetherian examples by comparing only the homology of one Koszul complex, which may be easier than constructing full tt-equivalences.
Editorial inference: whether the image of $D_{\rm perf}(R)$ in $(D(R)^\wedge_Y)_d$ agrees with the full dualizable part remains open in general tensor-triangular categories; in $D(R)$ the two coincide by Corollary 4.26, but the paper notes the general question is unresolved.

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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

Rough take: this is a solid paper. The main result is a necessary and sufficient condition on a sequence s for Dperf(Rhat) to be canonically tt-equivalent to the dualizable Y-complete complexes, and the condition is that the Koszul complexes over R and Rhat have the same homology. That's a genuine new criterion, and it's the right one: when R is noetherian it always holds, so you get a global version of the BIKP23 local theorem, and Example 3.16 shows the condition is not vacuous in general. Corollary 4.26, which characterizes dualizable objects in the Y-complete category when R is classically complete, is also independently useful.

The paper does what it sets out to do. The proofs are detailed and the strategy is coherent: Section 2 reduces the abstract completion question to fully-faithfulness and equivalence in torsion categories, Section 3 connects that to Koszul complexes, and Section 4 does the real work by induction on homological amplitude, avoiding residue fields and noetherianity. I looked for post-hoc exclusions or circular reasoning. BIKP23 is recovered as a special case, not assumed. The non-noetherian counterexample in Remark 5.3 is a genuine failure outside the hypotheses, not a patched exception.

Soft spots, in proportion. The paper is dense and leans on substantial cited machinery -- Balmer-Favi idempotents, Neeman's compact-generation theorem for localizing ideals, Thomason classification. That's normal for the area, but it means the target audience is specialists. The one step I'd flag is the citation of Neeman [Nee92] just after (4.21), used to conclude that kos(s) ⊗ d is compact in TY when d is dualizable. The reader worried about it; I think the worry is misplaced. That theorem is standard and does apply to D(R) with Thomason subsets, so it is not a vulnerability. Lemma 4.15(a) is terse, but the needed boundedness conclusion also follows from the long exact homology sequence of the triangle d → d → K(s)⊗d, so the terseness is not a gap. The main criterion, Koszul-completeness, is concrete but not always easy to check; Proposition 3.17 shows it is automatic noetherianly, and Example 3.16 shows it fails otherwise. That is about all.

Who it's for: anyone working on tt-geometry, derived completion, or the interface between commutative algebra and tensor-triangulated categories. The noetherian global statement and the counterexample will both be cited.

I'd send it to a serious referee. The main theorems are proven in detail, the claims are honest, and the result settles a natural question. My confidence is not machine-level -- I didn't verify every diagram -- but the internal consistency is high and the weakest assumptions are standard theorems. Accept.

Referee Report

1 major / 4 minor

Summary. The paper studies when the category of perfect complexes over the I-adic completion of a commutative ring is canonically equivalent to the dualizable objects in the derived category of Y-complete complexes. The authors introduce a concrete criterion, Koszul-completeness of a generating sequence, and prove (Theorem 5.1) that this criterion is equivalent to the existence of the desired tt-equivalence, to an analogous equivalence for the torsion-side category, and to an isomorphism between the tt-completion of the unit and the classical completion. They show the criterion is automatic for noetherian rings, yielding a global noetherian statement (Corollary 5.2) that recovers the local theorem of Benson--Iyengar--Krause--Pevtsova. The proof proceeds by an abstract tt-completion theorem (Theorem 2.16), a comparison of torsion and completion categories via Koszul complexes (Theorem 3.20), and a technical characterization of dualizable objects in the classically complete case (Theorem 4.17 and Corollary 4.26).

Significance. If the main theorem holds, this is a valuable contribution: it gives a necessary and sufficient condition in the non-noetherian setting, clarifies the distinction between tt-completion and classical I-adic completion, and provides a global noetherian statement with a canonical commutative diagram. The paper is careful and detailed, with explicit counterexamples showing failure outside the criterion, and the structure of the proof is transparent. The main results are supported by a chain of reductions rather than by circular reasoning, and the noetherian corollary is a genuine extension of prior work. The paper also makes a useful conceptual point about wanting to Ind-complete the dualizable objects rather than the whole Y-complete category.

major comments (1)

[Section 2, Theorem 2.16] The hypothesis (S2) is mis-stated: conservativity of f_* is not equivalent to S^c being generated, as a thick subcategory, by the image of f^*(T^c). For example, take T = D(k) and S = D(k) × D(k) for a field k, with f^* : x ↦ (x,x). This is a geometric tt-functor, and f_*(A,B) = A ⊕ B is conservative. Yet the thick subcategory generated by f^*(T^c) consists of pairs (U,W) with U ≅ W and does not contain (k,0). The proof of (ii)⇒(iii) uses the generation condition, so the statement should take that condition as (S2) or add it as a separate hypothesis; the current 'or equivalently' makes Theorem 2.16 false as stated. The application in Theorem 3.20 verifies the stronger generation condition directly, so the main theorem is not affected, but the abstract theorem needs repair.

minor comments (4)

[Theorem 5.1 statement] In the first line, 'Y = V(s_1, \ldots, s_n)' should read 'Y = V(s_1, \ldots, s_r)'.
[Proof of Theorem 5.1, (ii)⇒(i)] The text says that e_Y ⊗ − is the right adjoint of the inclusion T_Y ↣ T; e_Y ⊗ − is the left adjoint (the right adjoint is [e_Y, −] under the usual identifications). The intended retraction statement is correct, but the wording should be fixed.
[Lemma 4.15(b)] The proof cites the homology long exact sequence of the homotopy-limit triangle, but that sequence alone gives boundedness only up to one extra degree at the top. The stated interval [b,a−1] uses the fact that the transition maps on the top homology are pro-zero (for instance, because the Koszul powers kill the relevant multiplication by the generators). Please add a sentence making this Mittag-Leffler/pro-zero point explicit.
[After (4.21) in Theorem 4.17] The equality (T_Y)^c = (T^c)_Y is attributed to 'by construction of T_Y'; this is Neeman's theorem recalled in Recollection 2.1 and should be cited there, not described as a construction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central equivalence is derived from structural results, not assumed or fitted.

full rationale

The paper's main theorem (Theorem 5.1) establishes an equivalence between Koszul-completeness of a generating sequence and a tt-equivalence between dualizable Y-complete complexes and perfect complexes over the completion. The proof does not assume this equivalence. It proceeds through the abstract completion criterion of Theorem 2.16, the ring-theoretic translation in Theorem 3.20, and the structural characterization of dualizable objects in the complete case in Theorem 4.17 and Corollary 4.26. The noetherian case is handled by Proposition 3.17, which proves Koszul-completeness from noetherian flatness of completion, independently of the target equivalence. Earlier work by the same authors, such as BF11 and BDS16, is cited for standard tools like idempotent triangles, recollements, and projection formulas; these are background results, not the paper's conclusion. The BIKP23 result is recovered as a special case after the proof, not assumed as an input. The only substantially external load-bearing citation is Neeman's theorem [Nee92] that the compact objects of the localizing tensor-ideal T_Y are exactly (T^c)_Y, which is used after equation (4.21) to show that kos(s) ⊗ d is compact in T_Y when d is dualizable; this is a standard theorem in triangulated category theory, not a restatement of the paper's claims. There are no fitted parameters, no quantities defined in terms of the target result, and no renaming of a known result as a new one. The derivation chain is self-contained once the cited standard recollections are granted, so the paper does not exhibit circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters: the paper's condition (Koszul-completeness) is a hypothesis, not a fitted quantity. No invented entities: the paper introduces a new property and notation, not new objects. The axioms are standard structural theorems of tensor-triangulated geometry and commutative algebra, all cited.

assumptions (6)

standard math The compactly generated big tensor-triangulated category framework: Tc=Td for T=D(R), and the idempotent triangle eY→1→fY exists for each Thomason subset Y.
Section 2 setup; standard in HPS97 and BF11. Used to define T^wedge_Y and the completion functor.
standard math Thomason's classification: Spec(R) is homeomorphic to Spc(Dperf(R)), and the Koszul complex of a generating sequence has support Y=V(I), generating the thick ideal Tc_Y.
Used in Theorem 3.20 and Theorem 5.1 to pass between closed subsets Y and tt-ideals; cited [Tho97].
standard math Dwyer-Greenlees equivalence TY≅(TY)^⊥⊥ and the identification of (TY)^⊥⊥ with derived I-complete complexes.
Definition 2.5 and Recollection 3.2; central to the statement that (D(R)^wedge_Y)_d is the category of dualizable I-complete complexes. Cited [DG02].
standard math Neeman's theorem that the compact objects of the localizing ideal TY=Loc(Tc_Y) are exactly Tc_Y.
Used in Section 4, proof of Theorem 4.17, to show compactness of kos(s)⊗d and to test isomorphisms via compact objects of TY. Cited [Nee92].
domain assumption For a noetherian ring, I-adic completion satisfies M⊗_R Rhat ≅ Mhat_I for finitely generated modules M.
Proposition 3.17 proves every sequence is Koszul-complete in the noetherian case; standard commutative algebra.
standard math The classical completion Rhat is always classically I'-complete for the extended finitely generated ideal I'=I*Rhat (Stacks 05GG).
Remark 3.19, used to verify Hypothesis (S1) in Theorem 3.20.

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Cite this review

Pith. "Pith review of Perfect complexes and completion." pith.science (2026). https://pith.science/paper/S7XUHPJW

@misc{pith2026241114761,
  author       = {Pith},
  title        = {Pith review of: Perfect complexes and completion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S7XUHPJW}},
  note         = {Machine review of arXiv:2411.14761}
}

abstract

Let $\hat{R}$ be the $I$-adic completion of a commutative ring $R$ with respect to a finitely generated ideal $I$. We give a necessary and sufficient criterion for the category of perfect complexes over $\hat{R}$ to be equivalent to the subcategory of dualizable objects in the derived category of $I$-complete complexes of $R$-modules. Our criterion is always satisfied when $R$ is noetherian. When specialized to $R$ local and noetherian and to $I$ the maximal ideal, our theorem recovers a recent result of Benson, Iyengar, Krause and Pevtsova.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 12 canonical work pages

[1]

Grothendieck-- N eeman duality and the W irthm\"uller isomorphism

Paul Balmer, Ivo Dell'Ambrogio, and Beren Sanders. Grothendieck-- N eeman duality and the W irthm\"uller isomorphism. Compos. Math. , 152(8):1740--1776, 2016

work page 2016
[2]

Generalized tensor idempotents and the telescope conjecture

Paul Balmer and Giordano Favi. Generalized tensor idempotents and the telescope conjecture. Proc. Lond. Math. Soc. (3) , 102(6):1161--1185, 2011

2011
[3]

Local dualisable objects in local algebra

Dave Benson, Srikanth B. Iyengar, Henning Krause, and Julia Pevtsova. Local dualisable objects in local algebra. Preprint, 15 pages, available online at arXiv:2302.08562 https://arxiv.org/abs/2302.08562, 2023

work page Pith review arXiv 2023
[4]

W. G. Dwyer and J. P. C. Greenlees. Complete modules and torsion modules. Amer. J. Math. , 124(1):199--220, 2002

work page 2002
[5]

J. P. C. Greenlees and J. P. May. Derived functors of I -adic completion and local homology. J. Algebra , 149(2):438--453, 1992

work page 1992
[6]

J. P. C. Greenlees. Tate cohomology in axiomatic stable homotopy theory. In Cohomological methods in homotopy theory ( B ellaterra, 1998) , volume 196 of Progr. Math. , pages 149--176. Birkh\"auser, Basel, 2001

work page 1998
[7]

Palmieri, and Neil P

Mark Hovey, John H. Palmieri, and Neil P. Strickland. Axiomatic stable homotopy theory. Mem. Amer. Math. Soc. , 128(610), 1997

1997
[8]

Notes on prismatic cohomology

Kiran Kedlaya. Notes on prismatic cohomology. https://kskedlaya.org/prismatic/prismatic.html, 2024

work page 2024

Show all 19 references

[9]

The connection between the K -theory localization theorem of T homason, T robaugh and Y ao and the smashing subcategories of B ousfield and R avenel

Amnon Neeman. The connection between the K -theory localization theorem of T homason, T robaugh and Y ao and the smashing subcategories of B ousfield and R avenel. Ann. Sci. \'Ecole Norm. Sup. (4) , 25(5):547--566, 1992

1992
[10]

The G rothendieck duality theorem via B ousfield's techniques and B rown representability

Amnon Neeman. The G rothendieck duality theorem via B ousfield's techniques and B rown representability. J. Amer. Math. Soc. , 9(1):205--236, 1996

1996
[11]

Triangulated categories , volume 148 of Annals of Mathematics Studies

Amnon Neeman. Triangulated categories , volume 148 of Annals of Mathematics Studies . Princeton University Press, 2001

2001
[12]

A symmetric monoidal fracture square, 2024

Niko Naumann, Luca Pol, and Maxime Ramzi. A symmetric monoidal fracture square, 2024. Preprint, 40 pages, available online at arXiv:2411.05467 https://arxiv.org/abs/2411.05467

2024 arXiv
[13]

Remarks on derived complete modules and complexes

Leonid Positselski. Remarks on derived complete modules and complexes. Math. Nachr. , 296(2):811--839, 2023

2023
[14]

On the homology of completion and torsion

Marco Porta, Liran Shaul, and Amnon Yekutieli. On the homology of completion and torsion. Algebr. Represent. Theory , 17(1):31--67, 2014

2014
[15]

Proregular sequences, local cohomology, and completion

Peter Schenzel. Proregular sequences, local cohomology, and completion. Math. Scand. , 92(2):161--180, 2003

2003
[16]

S tacks P roject

The Stacks Project Authors . S tacks P roject . http://stacks.math.columbia.edu, 2020

2020
[17]

Disconnecting spectra via localization sequences

Greg Stevenson. Disconnecting spectra via localization sequences. J. K-Theory , 11(2):324--329, 2013. Appendix to: Module categories for group algebras over commutative rings, by D.\ Benson, H.\ Krause and S.\ Iyengar

2013
[18]

R. W. Thomason. The classification of triangulated subcategories. Compositio Math. , 105(1):1--27, 1997

1997
[19]

On flatness and completion for infinitely generated modules over N oetherian rings

Amnon Yekutieli. On flatness and completion for infinitely generated modules over N oetherian rings. Comm. Algebra , 39(11):4221--4245, 2011

2011

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Grothendieck-- N eeman duality and the W irthm\"uller isomorphism

Paul Balmer, Ivo Dell'Ambrogio, and Beren Sanders. Grothendieck-- N eeman duality and the W irthm\"uller isomorphism. Compos. Math. , 152(8):1740--1776, 2016

work page 2016

[2] [2]

Generalized tensor idempotents and the telescope conjecture

Paul Balmer and Giordano Favi. Generalized tensor idempotents and the telescope conjecture. Proc. Lond. Math. Soc. (3) , 102(6):1161--1185, 2011

2011

[3] [3]

Local dualisable objects in local algebra

Dave Benson, Srikanth B. Iyengar, Henning Krause, and Julia Pevtsova. Local dualisable objects in local algebra. Preprint, 15 pages, available online at arXiv:2302.08562 https://arxiv.org/abs/2302.08562, 2023

work page Pith review arXiv 2023

[4] [4]

W. G. Dwyer and J. P. C. Greenlees. Complete modules and torsion modules. Amer. J. Math. , 124(1):199--220, 2002

work page 2002

[5] [5]

J. P. C. Greenlees and J. P. May. Derived functors of I -adic completion and local homology. J. Algebra , 149(2):438--453, 1992

work page 1992

[6] [6]

J. P. C. Greenlees. Tate cohomology in axiomatic stable homotopy theory. In Cohomological methods in homotopy theory ( B ellaterra, 1998) , volume 196 of Progr. Math. , pages 149--176. Birkh\"auser, Basel, 2001

work page 1998

[7] [7]

Palmieri, and Neil P

Mark Hovey, John H. Palmieri, and Neil P. Strickland. Axiomatic stable homotopy theory. Mem. Amer. Math. Soc. , 128(610), 1997

1997

[8] [8]

Notes on prismatic cohomology

Kiran Kedlaya. Notes on prismatic cohomology. https://kskedlaya.org/prismatic/prismatic.html, 2024

work page 2024