{"id":"917c66b7-e28e-432c-a3a3-7323e6745bff","arxiv_id":"1908.02424","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"In affine type, every basic two-term silting complex of a complete preprojective algebra is exactly one of two Coxeter-group-indexed families, and two-term silting complexes coincide with two-term tilting complexes.","lead":"For a family of infinite-dimensional algebras built from affine graphs, the authors classify all two-term tilting complexes: each one comes from an element of the associated reflection group, in one of two disjoint families. The paper completes the affine case of a classification program and links the algebra to the geometry of Coxeter groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 3.1 silently uses the equality C(R_w)=wC_- , which is never stated or proved; the cone-cover argument depends on it.","rationale":"The reader identified Proposition 3.6 as the weakest assumption, focusing on the affine Coxeter cone cover. That proposition itself is standard and likely correct. However, the actual application of the cone cover in the proof of Theorem 3.1 requires an additional identity that the paper never states: C(R_w) = wC_-. This is precisely the step that transfers the abstract geometric cover of V^* by wC_+ and wC_- to the cover by the cones of the two constructed families P_w and R_w. Without it, the proof has a gap at the central dichotomy. The gap is probably fillable through the duality between Λ and Λ^op, and the claim is consistent with the known classification, so I do not recommend rejection. But the written proof should supply this argument or explicitly cite it; hence a conditional acceptance is the appropriate verdict. My concern is adjacent to but more specific than the reader's: the weakest link is not the Coxeter geometry itself, but the unproved correspondence between the geometric chambers and the cones of the second family R_w.","tokens_in":13729,"tokens_out":29113,"duration_ms":305505,"concrete_test":"For the affine type Δ = ^A_1 (where W is the infinite dihedral group and n = 2), compute the g-vectors of R_w for w = s_1, s_1 s_2, s_2 s_1, and a few longer elements, using the minimal projective resolution of I_w as a Λ^op-module and the duality R_w = Hom_{Λ^op}(Q_w, Λ[1]). Then compare the cone spanned by these g-vectors with wC_- = -wC_+. If every tested R_w satisfies C(R_w) = wC_-, the missing step is confirmed in the smallest nontrivial case; if some R_w has a g-vector outside wC_-, then Theorem 3.1(b) would fail and the cover argument must be revised.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 3.1, the authors combine Proposition 3.6 with Theorem 3.4(b) to assert an equality ⋃_w C(P_w) ∪ ⋃_w C(R_w) = V^*. This step requires both C(P_w) = wC_+ (Theorem 3.4(b)) and C(R_w) = wC_-. The first equality is proved. The second is not stated or proved anywhere: C_- is defined only as C(R_id), and the paper does not justify that the cone of R_w is the w-translate of C_-. Yet this equality is load-bearing: if C(R_w) ≠ wC_- for some w, the family {R_w} would not cover the opposite half-space, and a two-term silting complex whose g-vector lies in the uncovered region would escape the dichotomy “S ≅ P_w or S ≅ R_w.” The missing equality is plausible and can be derived by applying Theorem 3.4(a) to the opposite algebra Λ^op and then using the duality Hom_{Λ^op}(-,Λ) together with the anti-isomorphism Λ ≅ Λ^op; this should give [R_w] = -w[P_id] and hence C(R_w) = -wC_+ = wC_-. But this derivation is not included, and the current text leaves the cone-cover argument incomplete at the exact point where the two families are matched with the two halves of V^*.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two-term tilting and silting complexes over the complete preprojective algebra Λ of a non-Dynkin graph ∆. It constructs two families P_w and R_w indexed by the Coxeter group W, shows they are disjoint families of two-term tilting complexes, and proves in affine type that every basic two-term silting complex is isomorphic to exactly one member of these families; in particular every two-term silting complex is tilting. The proof uses g-vector cones, identifies C(P_w) with a Weyl chamber, proves an affine Weyl group cone cover, and imports g-vector separation results from [DIJ, P, Hi]. The paper also proves that K^b(projΛ) is Krull-Schmidt when Λ is a complete ring satisfying condition (F), and includes an appendix by Iyama showing that all silting complexes of affine type are tilting.","tokens_in":13978,"tokens_out":15711,"duration_ms":156449,"significance":"If the missing cone identification is supplied, the main theorem provides a complete and elegant classification of two-term silting complexes for affine preprojective algebras in terms of two copies of the affine Coxeter group, reinforcing the connection between representation theory of preprojective algebras and Coxeter group combinatorics. The Krull-Schmidt theorem for homotopy categories of complete rings is a useful standalone contribution, and the appendix strengthens the result by showing all silting complexes are tilting in affine type. The paper is clearly written and the overall strategy is convincing.","major_comments":[{"comment":"The proof of Theorem 3.1 uses the equality ⋃_{w∈W} C(R_w) = ⋃_{w∈W} wC_−, but this equality is never stated or proved. The text defines C_− := C(R_id) = −C_+ and Proposition 3.6 concerns the cones wC_+ and wC_−; it does not identify C(R_w) with wC_−. Since Theorem 3.4(b) only covers C(P_w) = wC_+, the second half of the cover argument is not justified as written. This is load-bearing because, without C(R_w) = wC_−, the family {R_w} is not shown to cover the opposite half-space. Please add a lemma (or a reference) establishing C(R_w) = wC_−, for example by applying Theorem 3.4(a) to Λ^op and using the duality Hom_{Λ^op}(−,Λ), which should give C(R_w) = −wC_+ = wC_−.","section":"Section 3, before Proposition 3.6 and in the proof of Theorem 3.1"},{"comment":"The final inference 'Thus Theorem 3.8(b) shows that S ≅ P_w or S ≅ R_w' is too compressed. Please spell out the chamber argument: since S is silting, C(S) is a full-dimensional simplicial cone; the interiors of the cones C(P_w) and C(R_w) are disjoint open convex sets covering V*; hence the interior of C(S) lies in a single one of these interiors, so C(S) is contained in the corresponding closed cone; Theorem 3.8(b) then forces S to be isomorphic to that P_w or R_w.","section":"Section 3, proof of Theorem 3.1"}],"minor_comments":[{"comment":"The phrase 'belongs one of them' should be 'belongs to one of them'.","section":"Abstract"},{"comment":"The sentence 'By the definitions of orderings of tilting modules and tilting complexes, the second one is a poset isomorphism' is terse; a one-line justification or a reference would help the reader.","section":"Proposition 2.5(b)"},{"comment":"The proof says 'From a duality Hom_Λ(−,Λ) : K^b(projΛ) ≃ K^b(projΛ^op), we have the assertion'; it would be helpful to state explicitly that this duality is an anti-equivalence on the poset of tilting complexes.","section":"Proposition 2.6"},{"comment":"The phrase 'σ*_{s_i} acts as an orthogonal reflection relative to E_i' should clarify which bilinear form is used for orthogonality.","section":"Proposition 3.6"},{"comment":"The figures for Examples 3.7(a) and 3.7(b) appear to be missing or not labeled in the text; please ensure the diagrams are included in the published version.","section":"Example 3.7"}],"recommendation":"major_revision","confidential_remarks":"The missing equality C(R_w)=wC_− is the only substantive gap I found; it is local and fixable. I recommend major revision rather than rejection. The paper would benefit from a slightly expanded proof of Theorem 3.1. The appendix by Iyama is a valuable addition."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper gives the first complete classification of two-term silting complexes for affine preprojective algebras, and the main structural proof is sound. The reader's accept verdict is about right, with one caveat that matches the stress-test note.\n\nWhat's new: the two families P_w and R_w, each parametrized by the Coxeter group, were known from [BIRS], but the completeness statement for affine type—that every two-term silting complex is isomorphic to exactly one of these—is new. The authors also prove silting equals tilting for two-term complexes, and they include Iyama's appendix showing the stronger statement for all silting complexes. The Krull-Schmidt section (Section 4) is a genuinely useful and reusable technical result for homotopy categories over complete rings satisfying condition (F). The paper is clearly written and the citations to the load-bearing external results look appropriate.\n\nWhere the soft spots are. The proof of Theorem 3.1 has a compressed step that is easy to fill but should not be left implicit. After Proposition 3.6, the authors write\n    ⋃_w C(P_w) ∪ ⋃_w C(R_w) = V*\nand attribute this to Proposition 3.6. That proposition gives the cover by wC₊ and wC₋, and Theorem 3.4(b) identifies C(P_w) with wC₊. What is missing is the matching identity C(R_w) = wC₋. It is never stated or proved anywhere in the text. The equality is true: applying Theorem 3.4(a) to the opposite algebra and using the duality Hom_{Λ^op}(−,Λ) gives [R_w] = −σ*_w[Λ], hence C(R_w) = −wC₊ = wC₋. But the paper needs to say this. As written, the cone-cover argument has a small hole at exactly the point where the two families are matched to the two half-spaces.\n\nThere is also a minor rapidity in Proposition 3.6 itself: the claim that ⋃_w wC₊ is exactly a half-space is imported from affine Weyl group geometry in a few lines. This is standard and I think correct, but a referee should check the Humphreys references carefully.\n\nNeither issue is load-bearing: the missing identity is a routine derivation from results already cited, and the half-space argument is standard. This is a good paper that deserves a proper referee, not a desk reject. The fix is a short added lemma in Section 3. I would be happy to see it in print after that revision.","headline":"A genuinely new affine classification of two-term silting complexes with an essentially sound proof, though one cone-identification is silently assumed and should be spelled out.","tokens_in":14531,"tokens_out":6184,"would_cite":true,"duration_ms":64465,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","16G10","18E30","20F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for a complete preprojective algebra of affine type, every basic two-term silting complex is isomorphic to exactly one of two families $P_w$ and $R_w$ indexed by the Coxeter group $W$, and is automatically tilting.","keywords":["preprojective algebras","two-term tilting complexes","silting complexes","Coxeter groups","affine type","g-vectors","Weyl chambers","Krull-Schmidt categories"],"falsifier":"The claim can be tested directly in the smallest affine case ($\\tilde{A}_1$): compute all basic two-term silting complexes of the complete preprojective algebra and check whether their classes in the real Grothendieck group cover exactly the two half-space families predicted by the two chambers. Finding even one such complex whose class lies outside both families, or two distinct complexes whose cones overlap in an interior point, would refute Theorem 3.1.","tokens_in":13487,"feed_emoji":"📐","tokens_out":9822,"duration_ms":94505,"temperature":0.7,"pith_summary":"This paper studies two-term tilting and silting complexes over complete preprojective algebras of non-Dynkin type, where the algebra is infinite-dimensional and classical tilting-module techniques do not directly apply. It establishes two disjoint families of two-term tilting complexes, $\\{P_w\\}_{w\\in W}$ and $\\{R_w\\}_{w\\in W}$, each indexed by the elements of the associated Coxeter group. In affine type, the paper gives a complete classification: every basic two-term silting complex belongs to exactly one of these families, and hence every two-term silting complex is tilting. A sympathetic reader should care because this describes all of two-term silting theory for these infinite-dimensional algebras in terms of Coxeter combinatorics, extending the Dynkin-type classification and connecting g-vector cones to Weyl chambers.","feed_headline":"Coxeter-group families classify all affine two-term silting complexes","feed_subtitle":"Every basic two-term silting complex is tilting and belongs to one of two indexed families.","key_machinery":"The argument is carried by the cone construction $C(T)$: the cone in the real Grothendieck space $K_0(\\Lambda)_{\\mathbb{R}}$ spanned by the classes of the indecomposable summands of a two-term complex $T$, together with the identification $C(P_w) = wC_+$, where $C_+$ is the cone over the projectives and $w$ acts through the contragradient geometric representation. The paper's Proposition 3.6 supplies the covering fact: for an affine Coxeter group, the closures of the cones $wC_+$ over all $w$ fill one half-space and the cones $wC_- = -wC_+$ fill the opposite half-space, so their union covers all of $K_0(\\Lambda)_{\\mathbb{R}}$. Against this, the injectivity of the g-vector map from two-term silting complexes into $K_0$ and the fact that distinct two-term silting cones meet only along boundaries force every silting complex's cone to coincide with one of the $wC_+$ or $wC_-$ chambers. A separate Krull-Schmidt theorem for the homotopy category of finitely generated projectives over complete rings satisfying condition (F) ensures that the basic decomposition language is legitimate.","core_discovery":"On the paper's own terms, the central discovery is Theorem 3.1: for a complete preprojective algebra $\\Lambda$ of affine type with Coxeter group $W$, one has $2\\operatorname{-silt}\\Lambda = 2\\operatorname{-tilt}\\Lambda$ and $2\\operatorname{-tilt}\\Lambda = \\{P_w\\}_{w\\in W} \\amalg \\{R_w\\}_{w\\in W}$. Here $P_w$ is the two-term complex given by a minimal projective resolution of the tilting ideal $I_w$, and $R_w$ is obtained by applying the duality $\\operatorname{Hom}_{\\Lambda^{\\mathrm{op}}}(-, \\Lambda[1])$ to the analogous resolution of $I_w$ as a $\\Lambda^{\\mathrm{op}}$-module. The two families are disjoint; the first is order-reversing in $w$ and the second order-preserving, and together they exhaust all basic two-term silting complexes. The conclusion that silting implies tilting at the two-term level is a corollary of the classification, since both families consist of tilting complexes.","pith_inferences":["The chamber-cover strategy suggests a template for classifying two-term silting complexes of other infinite-dimensional or complete algebras: whenever the g-vector cones form a fan whose closures cover the Grothendieck space and distinct cones meet only at boundaries, the same injection argument should force a Coxeter-group-style enumeration.","The two families are separated by a hyperplane in the Grothendieck space, so one may ask whether mutation across that boundary can produce new silting complexes in wild-type preprojective algebras, or whether the two-family pattern persists beyond affine type.","A testable extension is to compute explicitly all two-term silting complexes for the smallest affine example, the $\\tilde{A}_1$ preprojective algebra; the classification predicts exactly the two half-space families, so an exhaustive homological computation would either confirm the cover or reveal a missing cone."],"forward_implications":["Every basic two-term silting complex over an affine preprojective algebra is tilting; there are no silting-but-not-tilting two-term complexes.","The two-term tilting complexes are exactly two disjoint copies of the Coxeter group $W$, with the weak order giving an anti-isomorphism on the $P$-family and an isomorphism on the $R$-family.","The g-vector cones of two-term tilting complexes realize the closures of all Weyl chambers, and their union covers the whole real Grothendieck space.","The homotopy category $K^b(\\operatorname{proj}\\Lambda)$ and the category $\\operatorname{fp}\\Lambda$ of finitely presented modules are Krull-Schmidt categories, so the classification into indecomposables is well-defined.","In the affine case, the appendix goes further: all silting complexes of $\\Lambda$, not only the two-term ones, are tilting."],"supporting_citations":[{"why":"Supplies the bijection between $W$ and the ideals $I_w$ and the fact that each $I_w$ is a tilting module, from which the $P_w$ family is built.","marker":"[BIRS]"},{"why":"Provides the identification of the geometric representation action with tensoring by $I_w$, yielding $C(P_w)=wC_+$.","marker":"[IR1, BIRS]"},{"why":"Supplies the affine Weyl group alcove geometry used in Proposition 3.6 to show that the cones $wC_+$ and $wC_-$ cover $V^*$.","marker":"[Hu]"},{"why":"Gives the injection from two-term silting complexes into $K_0$ via g-vectors and the boundary-intersection property of their cones.","marker":"[DIJ]"},{"why":"Extends the injectivity argument to the infinite-dimensional setting needed for complete preprojective algebras.","marker":"[P]"},{"why":"Supplies the cone-volume and boundary result used in Theorem 3.8(b) for distinct two-term silting cones.","marker":"[Hi]"},{"why":"Provides the foundational silting-complex definitions and the fact that $[S]$ is a basis of $K_0$, used to place $[S]$ inside the covering cones.","marker":"[AI]"},{"why":"Supplies the Krull-Schmidt argument for homotopy categories of complete path algebras, which the paper generalizes to complete rings with condition (F).","marker":"[KeY]"}],"fun_headline_variants":["Two Coxeter-indexed families classify affine two-term tilting complexes","Every affine two-term silting complex is tilting, and classified","Affine preprojective two-term silting complexes are all tilting","All affine two-term silting complexes are tilting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the geometric covering statement used in the proof: the closures of the translated positive and negative Weyl chambers fill the entire Grothendieck vector space. If some point of that space were missed, a two-term silting complex could exist outside the two Coxeter-group-indexed families, and the classification would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Two Coxeter-indexed families classify affine two-term tilting complexes","Every affine two-term silting complex is tilting, and classified","Affine preprojective two-term silting complexes are all tilting","All affine two-term silting complexes are tilting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000469,"raw_usage":{"total_tokens":2288,"prompt_tokens":851,"completion_tokens":1437,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":1373}},"tokens_in":467,"tokens_out":1437,"duration_ms":11290,"temperature":1.0,"reasoning_tokens":1373,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:44:55.906971+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The claim can be tested directly in the smallest affine case ($\\tilde{A}_1$): compute all basic two-term silting complexes of the complete preprojective algebra and check whether their classes in the real Grothendieck group cover exactly the two half-space families predicted by the two chambers. Finding even one such complex whose class lies outside both families, or two distinct complexes whose cones overlap in an interior point, would refute Theorem 3.1.","supporting_citations":[],"review_version":1}