{"id":"a7debf92-3ca4-40f3-a984-f5cfeee61dc6","arxiv_id":"1908.05880","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For locally noetherian Grothendieck categories, the injective spectrum (a noncommutative generalization of the Zariski spectrum) is shown to be homeomorphic to the torsion spectrum, and a structure sheaf with two module sheaf functors is introduced.","lead":"This math paper builds a sheaf of rings on the injective spectrum of a noncommutative ring, a space of indecomposable injective modules that generalizes the Zariski spectrum, and relates it to a lattice of torsion theories. It shows when the associated sheaves of modules match, and proves that the injective spectrum and Golan's torsion spectrum are the same topological space for locally noetherian categories.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.3's homeomorphism depends on Lemma 1.9, a black-box Hom-vanishing claim about critical subobjects imported from the self-cited prequel [11]; the paper's compressed proof does not establish it.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing point: Lemma 1.9 is the only unproved, self-cited ingredient needed for the surjectivity half of Theorem 3.3. I reviewed the surrounding proof of Lemma 3.2 and Theorem 3.3 and found no other step that is comparably fragile: the injectivity argument via T0 and the basis correspondence are direct once Lemma 3.2 holds. The paper is otherwise careful, and the later torsion-theoretic results (Theorem 3.10, Corollary 3.11) genuinely follow from the homeomorphism and standard lattice facts. My concern does not force a change in verdict: the paper already receives CONDITIONAL, and the condition should be that Lemma 1.9 be either proved in-line or supplied with a verifiable reference to the prequel. I agree with the reader that, modulo this repair, the central homeomorphism is likely correct.","tokens_in":23837,"tokens_out":14092,"duration_ms":140526,"concrete_test":"Independently re-derive the implication used in Lemma 1.9 without citing [11]: for an α-critical object B in a locally noetherian Grothendieck category, prove that any nonzero map f: B/C → E(B) must have zero image. Concretely, let L = im f, use essentiality of B in E(B) to force L∩B ≠ 0, then use the short-exact-sequence property of Krull dimension to show K(L∩B)=α while L∩B is a submodule of the quotient B/C with K < α, a contradiction. If this derivation reveals that an additional hypothesis is needed (for example, that Krull dimension is exact on short exact sequences in arbitrary Grothendieck categories and that critical subobjects have no lower-dimensional nonzero submodules), then Lemma 1.9 is not established and the proof of Theorem 3.3 must be expanded.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central identification InjSpec(A) ≅ TorSpec(A) in Theorem 3.3 uses Lemma 3.2 to show surjectivity: every indecomposable injective E must cogenerate a prime torsionfree class. That proof needs Lemma 1.9: for a nonzero noetherian object A there is a nonzero subobject B such that every proper quotient B/C has no nonzero morphism to E(B). Lemma 1.9 is not proved here; it is deferred partly to McConnell-Robson [17, §6.2] and mainly to the self-cited prequel [11, 3.1.4 & 3.2.6]. The stated proof is compressed: from K(B/C)<K(B) it immediately asserts (B/C,E(B))=0. The missing argument is substantive: if a nonzero map existed with image L ⊆ E(B), then L∩B ≠ 0 by essentiality, giving a nonzero submodule of B that is a quotient of B/C and hence has Krull dimension < K(B); this contradicts criticality only if every nonzero submodule of a critical object has the same Krull dimension, a fact not stated or derived. Since Theorem 3.3 is the load-bearing result for the entire torsion-spectrum programme (including Ziegler sobriety and Theorem 3.10), an unverified black box at this exact point is the weakest link. If [11, 3.1.4 & 3.2.6] fail, or fail to cover arbitrary locally noetherian Grothendieck categories, the surjectivity half of the homeomorphism collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the injective spectrum of a right noetherian ring and, more generally, of a locally noetherian Grothendieck category. It constructs a structure sheaf of finite-type localisations on the injective spectrum, defines two functors from R-modules to sheaves of modules, and investigates when these functors coincide. The central result is Theorem 3.3, which asserts a homeomorphism between the injective spectrum and Golan's torsion spectrum; from this the paper derives a lattice-theoretic characterisation of prime torsion theories, Ziegler sobriety, and consequences for spectral spaces. The paper is presented as largely independent of the author's prequel [11], with a few results imported as black boxes.","tokens_in":24162,"tokens_out":31843,"duration_ms":305829,"significance":"If the main claims hold, the identification of InjSpec and TorSpec in Theorem 3.3 is significant: it shows that the Zariski-type injective spectrum and Golan's torsion spectrum carry exactly the same topological information, and it gives a purely lattice-theoretic description of the points of the spectrum via Theorem 3.10. The sheaf-theoretic part provides a noncommutative analogue of the Zariski structure sheaf and a useful criterion, Theorem 2.11, for when the tensor and torsion sheaf functors agree. The paper is honest about its dependence on the prequel and explicitly records open questions in Sections 3.3 and 4.1. However, the central proof chain passes through Lemma 1.9 and an unproved stalk identification in Theorem 2.11; these points need to be supplied before the main theorem can be regarded as established.","major_comments":[{"comment":"The proof asserts without argument that 'the stalks are the localisations at torsionfree classes cogenerated by a single indecomposable injective; i.e., at prime torsion theories.' The stalk at a point E is a colimit of the localisations R_T(A) over all finitely presented A with (A,E)≠0, and it is not automatic that this colimit is the single localisation at T F(E); one must show that the torsion classes T(A) form a cofinal family generating T F(E) and that localisation commutes with the resulting colimit. Moreover, the identification with prime torsion theories is only proved later in Lemma 3.2, so as written the proof of Theorem 2.11 depends on a future result. Please supply the missing stalk computation or reorder the paper so that Theorem 2.11 is proved after Section 3.2.","section":"§2.2, Theorem 2.11"},{"comment":"The proof of Lemma 1.9 is too compressed and leaves a load-bearing gap. From K(B/C)<K(B) the text immediately concludes (B/C,E(B))=0 by citing [11, 3.1.4 & 3.2.6]; this conclusion needs the additional fact that every nonzero submodule of a critical object has the same Krull dimension as the critical object and is itself critical, and that fact is neither stated nor proved in the present paper. In addition, Lemma 1.9 is stated for an arbitrary Grothendieck category, while the cited references [17, §6.2] concern modules over a ring; the extension to Grothendieck categories is not explained. Since Lemma 1.9 is used in Lemma 3.2 to prove the surjectivity half of the homeomorphism in Theorem 3.3, this missing argument directly affects the main theorem.","section":"§1.3, Lemma 1.9"},{"comment":"The sentence 'the associated torsionfree class is F([M]), cogenerated by the indecomposable injectives in [M]' is false as written. By Lemma 1.3, F(M)=F(E(M)), and the indecomposable summands of E(M) are precisely those indecomposable injectives E with (M,E)≠0, i.e., the elements of (M), not of [M]. For M=R over a commutative domain, for instance, [R]={0}, so F([R])=0, which is not the torsionfree class cogenerated by R. The well-definedness argument for T(M) can be repaired by replacing [M] with (M), but the construction of the structure sheaf should be corrected.","section":"§2.1, structure sheaf construction"}],"minor_comments":[{"comment":"The example states that the two indecomposable injectives of kA2 with orientation 1→2 are (k→0) and (k→k), but the indecomposable injectives for this orientation are (k→k) and (0→k); the representation (k→0) is not injective. Consequently the computed stalks should be k and k rather than k and M2(k), and the ring of global sections is k⊕k, not k⊕M2(k). The conclusion of the example, that the global sections are not Morita equivalent to R, still survives with this correction, but the computation needs to be fixed.","section":"§2.2, Example 2.2"},{"comment":"There is a typesetting artifact in the proof: 'InjSpec(A)/integerdivideC(F)' should be the set difference InjSpec(A)\\C(F). This should be corrected, and the surrounding text searched for similar artifacts.","section":"§3.2, Lemma 3.6"},{"comment":"There are several typographical errors, including 'torison' in Lemma 3.13, 'the the family' in §4.1, and 'Respectively' capitalization in Theorem 4.2. A careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The review process should ensure access to the author's prequel [11], since both Lemma 1.9 and Theorem 2.11 rely on results imported from it. The computational error in Example 2.2 is not fatal, but it suggests that the explicit examples should be independently checked. The main structural claims are plausible, but the gaps in Lemma 1.9 and in the stalk identification in Theorem 2.11 are load-bearing and should be addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version: this is a real paper, with a genuinely new categorical homeomorphism and two module-sheaf functors that aren't in the earlier literature, but there are three spots a referee should push on. The load-bearing one is Theorem 3.3, InjSpec(A) ≅ TorSpec(A). The proof is short and the idea is right, but it leans on Lemma 1.9, imported from the self-cited prequel, and the compressed argument doesn't show why K(B/C) < K(B) gives Hom(B/C, E(B)) = 0. I think the gap is fillable — standard Krull-dimension exactness plus criticality should do it — but the paper currently just points to [11, 3.1.4 & 3.2.6]. Since Theorem 3.3 carries the rest of Section 3, the referee should ask for a proof or a precise statement of what is being imported.\n\nThe sheaf part is the most original. The structure sheaf on InjSpec is Gabriel's, but the tensor sheaf and torsion sheaf functors, the natural transformation between them, and the perfection criterion in Theorem 2.11 are new. The kA2 example showing that global sections can differ from R is clean and useful. Corollary 2.14 has a false line: classical localization at a multiplicative set is not full — try R = k[x], D = {x^n}; the induced map on Hom(R,R) misses x^{-1}. The conclusion is still true, since the quotient category is equivalent to Mod-D^{-1}R via the fully faithful restriction adjunction, so this is a repair, not a collapse.\n\nTheorem 2.11's claim that the stalks are prime localisations is asserted, not proved. It looks right in locally noetherian categories because every torsion theory is finite type and the neighbourhood system is cofinal in the prime torsion class, but that identification needs to be written down. Section 4's spectrality results are honestly conditional, and the isolation of closed sets is a nice application.\n\nWho is this for? Someone working on noncommutative spectra, torsion theories, or representation theory of noetherian rings. It deserves a serious referee: the main structural claims are likely correct and significant, but the paper needs revision on the points above before I would rely on it.","headline":"A genuinely useful paper whose central homeomorphism is probably right, but with a false fullness claim and two black-box imports that need opening up before I'd cite it.","tokens_in":24667,"tokens_out":8323,"would_cite":true,"duration_ms":86911,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18E15","16P40","16S90","18F20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The injective spectrum and the torsion spectrum of a locally noetherian Grothendieck category are the same topological space.","keywords":["injective spectrum","torsion spectrum","Grothendieck categories","torsion theories","structure sheaf","Ziegler topology","localization","noetherian rings"],"falsifier":"Find a locally noetherian Grothendieck category with two distinct indecomposable injective objects $E$ and $F$ such that $F(E)=F(F)$; the homeomorphism would force $E\\cong F$, so a single such example would refute Theorem 3.3. Equivalently, find an indecomposable injective $E$ whose cogenerated torsionfree class is not prime, contradicting Lemma 3.2.","tokens_in":23640,"feed_emoji":"📐","tokens_out":4685,"duration_ms":40024,"temperature":0.7,"pith_summary":"This paper establishes that for any locally noetherian Grothendieck category — in particular, the category of modules over a right noetherian ring — the injective spectrum and Golan's torsion spectrum are the same topological space. The identification sends each indecomposable injective object $E$ to the torsionfree class it cogenerates, $F(E)$. Because the two spectra are defined from very different data, the homeomorphism unifies two research threads and lets torsion-theoretic machinery prove topological facts about the injective spectrum, such as sobriety in the Ziegler topology. The paper also builds a sheaf of rings on the injective spectrum and shows that the two natural ways to turn a module into a sheaf agree exactly when every prime torsion theory is perfect. A reader should care because the result suggests that the injective spectrum is the noncommutative analogue of the Zariski spectrum, with prime torsion theories playing the role of prime ideals.","feed_headline":"Injective and torsion spectra are the same space","feed_subtitle":"For locally noetherian categories, the two spectra carry exactly the same topology.","key_machinery":"The load-bearing object is the map $E \\mapsto F(E)$, the torsionfree class cogenerated by an indecomposable injective $E$, i.e. the class of objects embedding into some product of copies of $E$. The argument that this map is surjective rests on torsion-critical objects: objects whose every proper quotient is torsion for the torsionfree class they cogenerate. Lemma 1.9, imported from Krull dimension theory, guarantees that every non-zero noetherian object in a Grothendieck category has a critical subobject $B$ such that no proper quotient of $B$ maps to $E(B)$; this produces a torsion-critical subobject inside any indecomposable injective. The sheaf machinery uses torsion-theoretic localisation: to each basic open set $[M]$ one attaches the endomorphism ring of the localisation of the ring at the torsion class generated by $M$, and then sheafifies.","core_discovery":"The central claim is Theorem 3.3: for a locally noetherian Grothendieck category $\\mathcal{A}$, the assignment $E \\mapsto F(E)$ is a homeomorphism between $\\mathrm{InjSpec}(\\mathcal{A})$ and $\\mathrm{TorSpec}(\\mathcal{A})$. Here $\\mathrm{InjSpec}(\\mathcal{A})$ is the space of indecomposable injective objects with the Zariski topology, and $\\mathrm{TorSpec}(\\mathcal{A})$ is the space of prime torsion theories with the finitary order topology. The proof shows surjectivity by proving that every prime torsionfree class is cogenerated by a single indecomposable injective, and injectivity by observing that two indecomposable injectives cogenerating the same torsionfree class specialise to each other, which forces equality because the injective spectrum is $T_0$. The homeomorphism is proved by checking that it sends basic open sets to basic open sets. This identification is then used to prove that the Ziegler topology on the injective spectrum is sober, and to characterise prime torsion theories as exactly the join-irreducible torsionfree classes, equivalently the meet-irreducible torsion classes.","pith_inferences":["A testable extension of the paper's identification is that sobriety of the Zariski topology on the injective spectrum might be provable purely from the lattice of torsion classes: if every irreducible Zariski-closed set corresponds to a join of prime torsionfree classes with a unique maximal element, then the Zariski topology would be sober.","The paper's equivalence suggests that computability transfers between the two spectra: torsion-theoretic invariants such as Gabriel dimension or the lattice-theoretic structure of torsion classes could be read off the injective spectrum, and vice versa.","One could check in explicit noncommutative examples, such as the first Weyl algebra or other right hereditary rings, whether the perfectness condition for all prime torsion classes has a module-theoretic shadow that determines when the tensor and torsion sheaf constructions agree."],"forward_implications":["Topological questions about the injective spectrum can be translated into lattice-theoretic questions about torsion classes, since the points are exactly the meet-irreducible torsion classes and a basis of open sets is given by the compact elements of that lattice.","Sobriety of the injective spectrum in its Ziegler topology follows directly from the homeomorphism and the lattice characterisation of prime torsion theories.","For a right noetherian domain, the ring of global sections of the structure sheaf on the injective spectrum is precisely the original ring, recovering the commutative expectation for a large class of noncommutative rings.","For a commutative noetherian ring, the injective spectrum homeomorphism recovers the usual Zariski spectrum via the Matlis bijection, and the structure sheaf constructed here is the classical Zariski structure sheaf.","The tensor sheaf and torsion sheaf functors from modules to sheaves on the injective spectrum are naturally isomorphic precisely when every prime torsion class is perfect, which holds for commutative noetherian rings and for right noetherian right hereditary rings."],"supporting_citations":[{"why":"The prequel that defines the injective spectrum, its Zariski and Ziegler topologies, and supplies the black-box results on specialisation, $T_0$-ness, and irreducible domains used throughout.","marker":"[11]"},{"why":"Gabriel's work supplies the original construction of the injective spectrum and the structure sheaf, and the theorem that for commutative noetherian rings it is homeomorphic to the prime spectrum.","marker":"[6]"},{"why":"Golan's paper is the source of the torsion spectrum, prime torsion theories, and the finitary order topology that the paper modifies and then identifies with the injective spectrum.","marker":"[8]"},{"why":"McConnell and Robson's treatment of Krull dimension provides the critical subobject lemma (Lemma 1.9) that is used to show every indecomposable injective cogenerates a prime torsionfree class.","marker":"[17]"},{"why":"Stenström's monograph supplies the torsion-theoretic localisation machinery, Gabriel filters, and the characterisation of perfect torsion classes (Theorem 2.10) used for the sheaf functors.","marker":"[25]"},{"why":"Prest's book provides the Ziegler topology background, compactness of the injective spectrum, and the lattice bijections and localisation results that underpin the torsion-theoretic arguments.","marker":"[20]"},{"why":"Wisbauer's theory of the subcategory $\\sigma[M]$ supports the isolation of closed sets in Section 4, where the injective spectrum of $\\sigma[M]$ is identified with a basic closed set of the original injective spectrum.","marker":"[27]"}],"fun_headline_variants":["Injective and torsion spectra are homeomorphic","For locally noetherian, injective equals torsion spectra","Two spectra, one topology: the injective and torsion","Injective spectrum and torsion spectrum: same topology","Spectra coincide: injective and torsion for noetherian"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the classical fact that every non-zero noetherian object in a Grothendieck category contains a subobject $B$ such that no proper quotient of $B$ has a nonzero map into the injective hull of $B$; if that fact ever failed, the identification of the injective and torsion spectra would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Injective and torsion spectra are homeomorphic","For locally noetherian, injective equals torsion spectra","Two spectra, one topology: the injective and torsion","Injective spectrum and torsion spectrum: same topology","Spectra coincide: injective and torsion for noetherian"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1408,"prompt_tokens":919,"completion_tokens":489,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":412}},"tokens_in":535,"tokens_out":489,"duration_ms":4732,"temperature":1.0,"reasoning_tokens":412,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:03:17.249841+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a locally noetherian Grothendieck category with two distinct indecomposable injective objects $E$ and $F$ such that $F(E)=F(F)$; the homeomorphism would force $E\\cong F$, so a single such example would refute Theorem 3.3. Equivalently, find an indecomposable injective $E$ whose cogenerated torsionfree class is not prime, contradicting Lemma 3.2.","supporting_citations":[{"cited_title":"Gulliver","cited_arxiv_id":null,"evidence_quote":"The prequel that defines the injective spectrum, its Zariski and Ziegler topologies, and supplies the black-box results on specialisation, $T_0$-ness, and irreducible domains used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gabriel's work supplies the original construction of the injective spectrum and the structure sheaf, and the theorem that for commutative noetherian rings it is homeomorphic to the prime spectrum."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Golan's paper is the source of the torsion spectrum, prime torsion theories, and the finitary order topology that the paper modifies and then identifies with the injective spectrum."},{"cited_title":"McConnell and J.C","cited_arxiv_id":null,"evidence_quote":"McConnell and Robson's treatment of Krull dimension provides the critical subobject lemma (Lemma 1.9) that is used to show every indecomposable injective cogenerates a prime torsionfree class."},{"cited_title":"Stenstr¨ om.Rings of Quotients","cited_arxiv_id":null,"evidence_quote":"Stenström's monograph supplies the torsion-theoretic localisation machinery, Gabriel filters, and the characterisation of perfect torsion classes (Theorem 2.10) used for the sheaf functors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prest's book provides the Ziegler topology background, compactness of the injective spectrum, and the lattice bijections and localisation results that underpin the torsion-theoretic arguments."},{"cited_title":"Wisbauer","cited_arxiv_id":null,"evidence_quote":"Wisbauer's theory of the subcategory $\\sigma[M]$ supports the isolation of closed sets in Section 4, where the injective spectrum of $\\sigma[M]$ is identified with a basic closed set of the original injective spectrum."}],"review_version":1}