{"id":"f9ea529e-1688-4ff7-8bf0-c1259b5b3357","arxiv_id":"2411.14761","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Perfect complexes over an I-adic completion are equivalent to dualizable I-complete complexes precisely when the ring's Koszul homology is unchanged by completion.","lead":"This paper pins down exactly when the perfect complexes over the I-adic completion of a ring can be recovered from the dualizable objects in the derived category of I-complete complexes, using a new condition called Koszul-completeness. The condition always holds for noetherian rings, so the recovery works globally, not just in the local case solved earlier by Benson, Iyengar, Krause and Pevtsova.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof is internally consistent and the central equivalence is supported; the most external reliance is a standard theorem.","rationale":"The reader identified Neeman's compact-generation theorem as the weakest assumption. I agree that it is the most external input to the proof, but it is a true theorem for the categories in question, so it does not create a correctness risk. I checked the other potentially fragile steps: the use of Lemma 4.15 to bound [eY,d], the induction in Theorem 4.17, and the passage from Koszul-completeness to the tt-equivalences in Theorem 5.1. Each is supported by the text or by standard arguments. The paper gives a clean counterexample in the non-noetherian case and does not overclaim. Therefore I see no reason to change the reader's ACCEPT verdict. The concrete test above would confirm the one genuinely external categorical input, giving a sharper basis for the accepted verdict.","tokens_in":20213,"tokens_out":41103,"duration_ms":404689,"concrete_test":"Verify the step after (4.21) in an independent formalism: in the tt-category TY, for a compact object k and a dualizable object d, derive Hom_TY(k⊗d, ⊕_i y_i) ≅ ⊕_i Hom_TY(k⊗d, y_i) using d∨⊗− as a left adjoint; if this fails for non-noetherian R, the boundedness premise of Lemma 4.15 would not be established by the stated route.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After reading the proof in good faith, I find no load-bearing flaw. The central assertion is Theorem 5.1, and it is established through the chain: Theorem 3.20 reduces Koszul-completeness to the abstract completion conditions of Theorem 2.16, while Theorem 4.17 and Corollary 4.26 identify dualizable objects with perfect complexes when the ring is classically complete. The noetherian case then follows from Proposition 3.17. The most externally dependent step is the citation of Neeman [Nee92] just after (4.21), where the equality (TY)^c = (Tc)_Y is used to conclude that kos(s) ⊗ d is compact in TY when d is dualizable. This is a standard theorem and is valid for D(R) with Thomason subsets, so it is not a vulnerability. I also scrutinized Lemma 4.15(a), where the authors assert that cone(s^{ℓ+1}) is an extension of cone(s^{ℓ}) and cone(s); while the justification is terse, the needed boundedness conclusion also follows from the long exact homology sequence associated with the triangle d → d → K(s)⊗d, so the lemma is not a hidden counterexample. In short, the proof is rigorous enough to support ACCEPT, with only the usual caveat that category-theoretic details are not machine-checked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":20400,"tokens_out":34688,"duration_ms":340850,"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":[{"comment":"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.","section":"Section 2, Theorem 2.16"}],"minor_comments":[{"comment":"In the first line, 'Y = V(s_1, \\ldots, s_n)' should read 'Y = V(s_1, \\ldots, s_r)'.","section":"Theorem 5.1 statement"},{"comment":"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.","section":"Proof of Theorem 5.1, (ii)⇒(i)"},{"comment":"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.","section":"Lemma 4.15(b)"},{"comment":"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.","section":"After (4.21) in Theorem 4.17"}],"recommendation":"major_revision","confidential_remarks":"The central theorem appears sound, and the proof of Theorem 5.1 is detailed and coherent. The only substantive issue is the false equivalence in the statement of Theorem 2.16. Since the application in Theorem 3.20 checks the stronger generation condition, the main results can be preserved by correcting the abstract theorem's hypothesis. I would be willing to accept after that revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nSoft 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.\n\nWho 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.\n\nI'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.","headline":"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.","tokens_in":20989,"tokens_out":2420,"would_cite":true,"duration_ms":23449,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18F99"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["perfect complexes","I-adic completion","derived complete complexes","dualizable objects","Koszul complexes","tensor-triangular category","tt-equivalence","noetherian rings"],"falsifier":"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.","tokens_in":19952,"feed_emoji":"🔁","tokens_out":8743,"duration_ms":75037,"temperature":0.7,"pith_summary":"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.","feed_headline":"Koszul-complete sequences make completion match perfect complexes","feed_subtitle":"A single condition relates I-adic completion to derived complete complexes; it always holds for noetherian rings.","key_machinery":"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 $\\hat R$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the local noetherian maximal-ideal case whose insight this paper generalizes and recovers as a special case.","marker":"[BIKP23]"},{"why":"Supplies the compact-generation fact that compact objects of the localizing subcategory generated by Y-supported compacts are exactly those compacts, used in the proof of Theorem 4.17.","marker":"[Nee92]"},{"why":"Gives the criterion that a module is classically I-complete iff it is I-separated and derived I-complete, used to verify completeness of the completed ring.","marker":"[Sta20, Proposition 15.91.5]"},{"why":"Classifies thick subcategories of perfect complexes and identifies Koszul complexes as generators of the Y-supported compact objects, used in Theorem 3.20.","marker":"[Tho97]"},{"why":"Provides the generalized tensor idempotents and functoriality of the tt-spectrum on which the completion construction and Theorem 2.16 rest.","marker":"[BF11]"},{"why":"Gives the torsion-complete equivalence in derived categories, which underlies the identification of $T_Y$ with its double orthogonal.","marker":"[DG02]"}],"fun_headline_variants":["Koszul-complete sequences unify completion and perfect complexes","Noetherian rings always satisfy completion-perfect equivalence","One condition links completion's derived category to perfect complexes","When completion's perfect complexes equal dualizable objects"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Koszul-complete sequences unify completion and perfect complexes","Noetherian rings always satisfy completion-perfect equivalence","One condition links completion's derived category to perfect complexes","When completion's perfect complexes equal dualizable objects"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000329,"raw_usage":{"total_tokens":1824,"prompt_tokens":921,"completion_tokens":903,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":537,"completion_tokens_details":{"reasoning_tokens":839}},"tokens_in":537,"tokens_out":903,"duration_ms":8651,"temperature":1.0,"reasoning_tokens":839,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:57:38.499768+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}