{"id":"74d6f3aa-0400-4385-aca1-f151e2b9e83d","arxiv_id":"2411.13706","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Closed subcategories of a quotient category X/Y correspond bijectively to Y-closed subcategories of the base category, and closed subcategories of any category with compact projective generators correspond to ideals in those generators.","lead":"This mathematics paper proves that the closed subcategories of a quotient category X/Y are described exactly by the closed subcategories of X that are stable under Y-torsion, when X has exact products. It gives the first bijection of this kind that preserves closed subcategories, and shows with examples that the needed hypotheses are not removable.","discovery_kind":"extension","skeptic_critique":null,"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies closed subcategories of Grothendieck categories, viewed as noncommutative quasi-schemes. Section 3 gives a description, for a Grothendieck category with a small set of compact projective generators, of weakly closed subcategories in terms of compatible filter systems on the generators (Theorem 3.4), and of closed subcategories in terms of ideals in the set of generators (Corollary 3.6). Section 4 analyzes quotient categories X/Y: Theorem 4.3 establishes a bijection between Y-weakly closed subcategories of X and weakly closed subcategories of X/Y, and Theorem 4.4, under the AB4* hypothesis, restricts this to a bijection between Y-closed subcategories of X and closed subcategories of X/Y. The paper then gives several examples showing that the hypotheses in Theorem 4.4 are necessary: Y-essential stability is needed in part (1) (Examples 5.7, 5.2, 5.4), AB4* is needed in part (2) (Example 5.9), and unions of closed subcategories need not remain closed (Example 6.7). A counterexample to a distributivity question of Smith is also provided (Example 6.8).","tokens_in":25139,"tokens_out":7028,"duration_ms":70195,"significance":"If the main theorems are correct, the paper provides a clean and useful dictionary for closed subcategories of quotient categories, generalizing earlier work of Rosenberg, Kanda, and Smith. The two main innovations are the generator-ideal correspondence for closed subcategories and the identification of the precise stability conditions and exactness hypotheses needed to transfer closedness to quotient categories. The paper is unusually careful about its hypotheses: it supplies explicit examples, including quantum-plane examples, showing that Y-essential stability and AB4* cannot simply be dropped. The proofs in the text are detailed and appear to be sound, and the paper is honest about the limitations of the theory, including the fact that the bijection is only guaranteed for exact-products base categories. These features make the paper a solid contribution to noncommutative algebraic geometry and categorical ring theory.","major_comments":[],"minor_comments":[{"comment":"The characterization of localizing subcategories as Gabriel filter systems is stated with the proof left to the reader. Since this result is not used elsewhere in the paper, this is not an obstacle, but it would be helpful to indicate explicitly that it is only included for completeness.","section":"§3 (Theorem 3.4(4))"},{"comment":"The description of products in X/Y as π(∏ ω(M_α)) is used in the proofs of Theorems 4.3 and 4.4; a one-sentence derivation from the adjunction π⊣ω and πω≅id would improve readability.","section":"§2"},{"comment":"The proof of Proposition 4.2(1) uses the notation overline{M}=π(M) without a formal definition; defining it just before the proof would remove a minor ambiguity.","section":"§4 (Proposition 4.2)"},{"comment":"The conclusion that X/Y1 fails AB4* is phrased informally ('the only way to make sense'); since it is a direct contrapositive of Theorem 4.4(2), stating it that way would make the argument easier to verify.","section":"Example 5.9"}],"recommendation":"accept","confidential_remarks":"The paper is polished and self-contained, with the main theorems proved in detail and with explicit examples demonstrating sharpness. The only suggestions are presentational. I see no obstacle to publication in math.RA."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dan,\n\nThis is a genuinely useful paper for anyone working with Grothendieck categories as quasi-schemes, especially the noncommutative projective scheme construction Qgr-B. Rogalski proves a clean dictionary between closed subcategories of a quotient category X/Y and Y-closed subcategories of X, with Theorem 4.4 as the centerpiece. The weakly closed version (Theorem 4.3) needs no AB4* and no essential stability, and it complements Kanda's earlier bijection rather than duplicating it.\n\nWhat is actually new: the ideal-in-generators parameterization of closed subcategories for categories with compact projective generators (Corollary 3.6), which directly generalizes Rosenberg's description for Mod-R, and the correct quotient statement with the right hypotheses. The paper is unusually honest about its assumptions. Examples 5.7 and 5.9 show that essential stability and AB4* are each genuinely necessary. Example 6.7 shows that the union of two closed subcategories in a quotient can fail to be closed, and Example 6.8 answers a question of Smith negatively. Those examples are the most valuable part; they tell you exactly where naive statements break.\n\nSoft spots are minor. The abstract says the description holds when X has compact projective generators, but the quotient theorem also needs AB4*, and the body is precise about this while the abstract is not. A few peripheral proofs are left to the reader (the Gabriel filter characterization in Theorem 3.4(4), the converse half of Example 5.3). The main proofs are detailed and I saw no gaps. The category theory is not machine-checked, but the write-up is careful enough that a normal referee pass should be enough.\n\nWho is this for? People who study closed subcategories as noncommutative closed subschemes, and anyone who wants to move between a module category and a Gabriel quotient. I would definitely cite it. It deserves a serious referee, and my own verdict would be accept, with a request to fix the abstract.","headline":"Solid and honest: the closed-subcategory quotient bijection is real, the hypotheses are stress-tested with examples, and the only real flaw is an overbroad abstract.","tokens_in":25757,"tokens_out":3118,"would_cite":true,"duration_ms":75102,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18E10","18E35","14A22"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that in a Grothendieck category with exact products, the closed subcategories of any quotient category X/Y are in bijection with the Y-closed subcategories of X, and gives the matching ideal description on the generator…","keywords":["closed subcategories","quotient categories","Grothendieck categories","noncommutative projective schemes","quasi-schemes","filter systems","AB4*","ideals in a set of generators"],"falsifier":"Compute, in any AB4* Grothendieck category $X$ with localizing $Y$, whether the equality $\\pi((\\pi^{-1}(\\overline{Z}))')=\\overline{Z}$ holds for every closed subcategory $\\overline{Z}$ of $X/Y$; a single $\\overline{Z}$ for which one side is strictly smaller than the other would refute the bijection. The paper's own quantum-plane example (Example 5.9) supplies the analogous failure in a non-AB4* quotient, so checking that the product-exactness hypothesis is really violated there---by exhibiting a product of epimorphisms that is not an epimorphism in $X/Y_1$---would directly test the boundary of the theorem.","tokens_in":24940,"feed_emoji":"📐","tokens_out":9333,"duration_ms":85696,"temperature":0.7,"pith_summary":"This paper gives a structural answer to how closed subcategories behave when one passes to a quotient category. In the setting of a Grothendieck category $X$ with a localizing subcategory $Y$, it proves that if products are exact in $X$, then the closed subcategories of $X/Y$ are in bijection with the $Y$-closed subcategories of $X$---those generated by $Y$-torsionfree objects and stable under the operation $\\omega\\pi$. It also completes the affine side of the dictionary: in any Grothendieck category with a small set of compact projective generators, closed subcategories correspond exactly to ideals in the generator set, generated by quotients $O_\\alpha/I_\\alpha$. These two results together describe closed subcategories of noncommutative projective schemes $\\mathrm{Qgr}\\text{-}B$, and more generally of any quotient category arising in this way. The exactness-of-products condition is essential: an example over a quantum plane shows the bijection can fail without it.","feed_headline":"Closed subcategories of quotient categories have a complete dictionary","feed_subtitle":"The bijection runs between Y-closed subcategories of X and closed subcategories of X/Y, provided products are exact.","key_machinery":"The argument runs on two matching descriptions. On the side of the base category, Theorem 3.4 pairs weakly closed subcategories of $X$ with compatible systems of filters inside each compact projective generator, and closed subcategories with principal filter systems; equivalently, with ideals $\\{I_\\alpha\\}$ such that $f(I_\\alpha)\\subseteq I_\\beta$ for every morphism $f:O_\\alpha\\to O_\\beta$, the closed subcategory being generated by the quotients $O_\\alpha/I_\\alpha$. On the quotient side, the key object is the operation $\\omega\\pi$: a closed subcategory $Z$ is $Y$-essentially stable when $M\\in Z$ forces $\\omega\\pi(M)\\in Z$, and $Y$-torsionfree generated when its $Y$-torsionfree objects generate it. A $Y$-closed subcategory is one satisfying both, and the inverse bijection sends a closed subcategory $\\overline{Z}$ of $X/Y$ to the subcategory generated by all $Y$-torsionfree $M$ with $\\pi(M)\\in\\overline{Z}$. AB4* enters precisely to make products of such generators pass through the quotient in the proof of Theorem 4.4(2).","core_discovery":"The central claim is Theorem 4.4: for an AB4* Grothendieck category $X$ and a localizing subcategory $Y$, with quotient category $\\overline{X}=X/Y$, quotient functor $\\pi$, and section functor $\\omega$, the maps $Z\\mapsto \\pi(Z)$ and $\\overline{Z}\\mapsto(\\pi^{-1}(\\overline{Z}))'$ are inverse bijections between the $Y$-closed subcategories of $X$ and the closed subcategories of $\\overline{X}$. Here $Z$ is $Y$-closed when it is closed under subquotients, products, and $\\omega\\pi$, and is generated by its $Y$-torsionfree objects. The weakly closed version of the bijection (Theorem 4.3) needs no AB4* hypothesis and no essential-stability condition; the closed version requires both. The paper further proves (Corollary 3.6) that if $X$ has compact projective generators $\\{O_\\alpha\\}$, its closed subcategories are exactly the categories generated by $\\{O_\\alpha/I_\\alpha\\}$ where $\\{I_\\alpha\\}$ is an ideal in the generator set. Examples show the hypotheses are sharp: without essential stability the image $\\pi(Z)$ can be only weakly closed, and without AB4* the inverse construction can also fail.","pith_inferences":["If the dictionary is right, a practical route to computing closed subcategories of any Grothendieck category is to represent it as a quotient of a module category (where products are exact) and then compute which ideals survive saturation and essential stability; the paper notes this route but does not develop it.","The failure of unions to be closed suggests that the lattice of closed subcategories of a quasi-scheme is not a topology; a worthwhile test is whether some intermediate class of subcategories, larger than closed but smaller than weakly closed, restores distributivity in the examples of Section 6.","One could probe whether a weaker hypothesis than full AB4*---for example, exactness of products over countable index sets---still yields the bijection for the specific quotient categories that arise in noncommutative projective geometry."],"forward_implications":["Closed subcategories of a noncommutative projective scheme $\\mathrm{Qgr}\\text{-}B$ are governed by $Y$-closed subcategories of $\\mathrm{Gr}\\text{-}B$, so the problem reduces to computing ideals in the set of graded shifts $B(n)$ and checking stability under $\\omega\\pi$.","In the commutative case, every closed subcategory is automatically $Y$-essentially stable, so the quotient dictionary reduces to the classical bijection between closed subschemes of $\\mathrm{Proj}\\,B$ and saturated graded ideals of $B$.","For Ore localizations of a noetherian ring, the general machinery recovers the standard correspondence between ideals of $RS^{-1}$ and $S$-saturated ideals of $R$.","The union of two closed subcategories of a quotient category can fail to be closed even when each summand is closed, because the union may fail to be $Y$-essentially stable; Example 6.7 realizes this in a quantum polynomial ring.","Without AB4*, the correspondence can fail in both directions, and Example 5.9 shows the failure is genuine: a weakly closed image in one quotient becomes a closed image after passing to a larger localizing subcategory."],"supporting_citations":[{"why":"Establishes the closed-subscheme correspondence for quasi-projective schemes and introduces the weakly open/complement viewpoint whose quotient description is compared with Theorem 4.3.","marker":"[11]"},{"why":"Provides the standard background on Grothendieck categories, quotient categories, section functors, and exact products used throughout.","marker":"[8]"},{"why":"Defines the noncommutative projective scheme Qgr-B = Gr-B/Tors-B that motivates the quotient-category problem.","marker":"[1]"},{"why":"Proves that closed subcategories of Mod-R are exactly the categories Mod-(R/I) for ideals I, the affine case the generator-ideal result generalizes.","marker":"[10]"},{"why":"Gives the earlier bijection for weakly closed subcategories of quotient categories which does not restrict to closed subcategories; the paper compares its bijection with this one.","marker":"[5]"},{"why":"Shows that images Qgr-(B/I) are closed in Qgr-B for noetherian B, the noncommutative result the quotient dictionary explains.","marker":"[12]"},{"why":"Originates the filter description of localizing and weakly closed subcategories in module categories that Section 3 extends to a set of compact projective generators.","marker":"[2]"}],"fun_headline_variants":["Bijection maps Y-closed subcategories to quotient closed subcategories","Closed subcategories of quotient categories: exact dictionary","Quotient category closed subcategories: full classification","Y-closed and quotient closed subcategories: bijective"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole dictionary rests on the base category $X$ having exact products (AB4*); if that property fails, the inverse construction can send a closed subcategory of the quotient to only a weakly closed subcategory, so the bijection breaks.","fun_headline_variants_meta":{"raw":{"variants":["Bijection maps Y-closed subcategories to quotient closed subcategories","Closed subcategories of quotient categories: exact dictionary","Quotient category closed subcategories: full classification","Y-closed and quotient closed subcategories: bijective"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001052,"raw_usage":{"total_tokens":4434,"prompt_tokens":977,"completion_tokens":3457,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":593,"completion_tokens_details":{"reasoning_tokens":3391}},"tokens_in":593,"tokens_out":3457,"duration_ms":27796,"temperature":1.0,"reasoning_tokens":3391,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:00:03.830915+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, in any AB4* Grothendieck category $X$ with localizing $Y$, whether the equality $\\pi((\\pi^{-1}(\\overline{Z}))')=\\overline{Z}$ holds for every closed subcategory $\\overline{Z}$ of $X/Y$; a single $\\overline{Z}$ for which one side is strictly smaller than the other would refute the bijection. The paper's own quantum-plane example (Example 5.9) supplies the analogous failure in a non-AB4* quotient, so checking that the product-exactness hypothesis is really violated there---by exhibiting a product of epimorphisms that is not an epimorphism in $X/Y_1$---would directly test the boundary of the theorem.","supporting_citations":[{"cited_title":"Maps between non-commutat ive spaces","cited_arxiv_id":null,"evidence_quote":"Shows that images Qgr-(B/I) are closed in Qgr-B for noetherian B, the noncommutative result the quotient dictionary explains."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the closed-subscheme correspondence for quasi-projective schemes and introduces the weakly open/complement viewpoint whose quotient description is compared with Theorem 4.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the standard background on Grothendieck categories, quotient categories, section functors, and exact products used throughout."},{"cited_title":"Artin and J","cited_arxiv_id":null,"evidence_quote":"Defines the noncommutative projective scheme Qgr-B = Gr-B/Tors-B that motivates the quotient-category problem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that closed subcategories of Mod-R are exactly the categories Mod-(R/I) for ideals I, the affine case the generator-ideal result generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the earlier bijection for weakly closed subcategories of quotient categories which does not restrict to closed subcategories; the paper compares its bijection with this one."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Originates the filter description of localizing and weakly closed subcategories in module categories that Section 3 extends to a set of compact projective generators."}],"review_version":1}