{"id":"019c160d-8df7-4165-b714-150e1853deec","arxiv_id":"2508.20281","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Restricted projective and flat dimension classes are finitely deconstructible and satisfy Govorov-Lazard, respectively, over Cohen-Macaulay rings with pointwise dualizing modules and over almost Cohen-Macaulay rings.","lead":"This paper proves that several classes of modules defined by restricted or Gorenstein homological dimensions can be built from finitely generated pieces, either as direct limits or as filtered extensions. The results extend structural theorems from Cohen-Macaulay rings to wider noetherian settings and yield new preenveloping classes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.3's proof reverses both inequality directions: the inclusion C_f⊆RF_n gives f≥f_n, while [18, Prop 4.14] yields f(p)≤f_n(p); as printed the proof of f=f_n is invalid.","rationale":"The paper contains useful and largely well-supported results: Theorem A over Gorenstein rings follows from Lemma 3.1 and standard approximation arguments, and Theorem B(2) via Lemma 4.2 and the trivial extension R⋉Ω is plausible, conditional on [31, Lemma 4.14]. The most fragile part is Section 5, where Theorem 5.3 is used to prove Corollary 5.4 for almost Cohen–Macaulay rings. The reader's conditional verdict focused on the unproved classification of Tor-pairs from the authors' preprint [18]; that is a legitimate concern. My independent check found a sharper, internal problem in the same proof: the two inequality directions in Theorem 5.3 appear reversed. The intended argument can likely be repaired by swapping 'f ≤ f_n' to 'f ≥ f_n' after the inclusion and 'f(p) ≥ f_n(p)' to 'f(p) ≤ f_n(p)' in the first case, so this is not a fatal rebuttal, but it means the proof as written is invalid. Because the flaw is concrete and easily testable, and because it sits exactly at the load-bearing step for Theorem B(3)(i), the conditional verdict is appropriate and should be retained pending the authors' correction or confirmation of the intended inequalities.","tokens_in":15317,"tokens_out":14108,"duration_ms":127871,"concrete_test":"Re-derive the inclusion direction: with C_f = {M | depth M_p ≥ f(p)}, verify that C_f ⊆ C_{f_n} iff f ≥ f_n pointwise. Then recompute the first case of Theorem 5.3: from lim→RP^{<ω}_{n+f_n(p)} = RF_{n+f_n(p)} = (lim→RP^{<ω}_n)^(f_n(p)) and [18, Proposition 4.14], derive max(0, f(p) - f_n(p)) = 0, i.e. f(p) ≤ f_n(p). If these two independent checks confirm the reversal, then the printed proof needs both inequality directions swapped before Corollary 5.4 can be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 5.3 (Section 5), the authors define f by lim→RP^{<ω}_n = C_f = {M | depth M_p ≥ f(p)} via Theorem 5.1, and then assert: 'Since lim→RP^{<ω}_n ⊆ RF_n, we have f ≤ f_n.' But with the definition C_f = {M | depth M_p ≥ f(p)}, containment C_f ⊆ C_{f_n} is equivalent to f(p) ≥ f_n(p) for every p, because a larger depth bound gives a smaller class. The displayed inequality is therefore reversed. In the first case f_n(p) > 0, the proof applies the induction hypothesis and [18, Proposition 4.14] to conclude f(p) ≥ f_n(p), but the same chain of equalities gives (lim→RP^{<ω}_n)^(f_n(p)) = C_{max(0, f - f_n(p))} and RF_{n+f_n(p)} = C_{f_{n+f_n(p)}}; evaluating at p yields max(0, f(p) - f_n(p)) = f_{n+f_n(p)}(p) = 0, hence f(p) ≤ f_n(p), the opposite inequality. The proof of f = f_n thus depends on two reversed inequalities, and as written the key equality is not established. Since Corollary 5.4 relies on Theorem 5.3, the almost Cohen–Macaulay case of Theorem B(3)(i) is not proven as printed. The dependence of Section 5 on the classification of Tor-pairs from [18] is an additional external assumption, but the internal inconsistency in the inequality directions is the more immediate load-bearing defect.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Govorov–Lazard (GL) and finite deconstructibility (FD) properties for classes of modules of bounded Gorenstein and restricted homological dimensions over commutative noetherian rings. Theorem A establishes GL for Gorenstein flat classes and FD for Gorenstein projective classes over (locally) Gorenstein rings. Theorem B establishes: (1) GL for restricted flat modules and FD for restricted projective modules over finite-dimensional rings; (2) GL for Cohen–Macaulay flat and FD for Cohen–Macaulay projective classes when the ring has a pointwise dualizing module; and (3) GL for restricted flat dimension over almost Cohen–Macaulay finite-dimensional rings. The proofs use cotorsion-pair and Tor-pair techniques, the trivial extension construction, and a classification of hereditary Tor-pairs from the authors' preprint [18].","tokens_in":15628,"tokens_out":9047,"duration_ms":74499,"significance":"If the results are correct, they constitute a significant advance: they extend Holm's work on balanced big Cohen–Macaulay modules, provide a Gorenstein analog of the recent finite-type results for projective and flat dimensions from [24], and give a broad sufficient condition for the Govorov–Lazard property on restricted flat classes. The paper is clearly organized and makes effective use of existing tools, including the Baer criterion and the local-to-global principle of Angeleri Hügel–Trlifaj. A notable strength is the explicit formulation of the properties (GL), (FD), and (LF), which makes the results easy to state and check. However, a load-bearing error in the proof of Theorem 5.3 currently undermines the almost Cohen–Macaulay case of Theorem B.","major_comments":[{"comment":"The proof of (iii)⇒(i) in Theorem 5.3 contains a reversal of the inequality directions. The inclusion lim→RP^{<ω}_n ⊆ RF_n yields C_f ⊆ C_{f_n}; with the definition C_f = {M | depth M_p ≥ f(p)}, this containment is equivalent to f(p) ≥ f_n(p) for every p, not f ≤ f_n as stated. The subsequent argument, intended to prove the 'other inequality', again establishes only f(p) ≥ f_n(p) in the case f_n(p) > 0; indeed, applying [18, Proposition 4.14] to RF_{n+f_n(p)} = (lim→RP^{<ω}_n)^{(f_n(p))} gives max(0, f(p)−f_n(p)) = f_{n+f_n(p)}(p) = 0, hence f(p) ≤ f_n(p), which is the opposite of what the text claims. Thus both displayed inequalities are reversed and the equality f = f_n is not established. Since Corollary 5.4 and Theorem B(3)(i) rely on Theorem 5.3, the Govorov–Lazard property for restricted flat dimension over almost Cohen–Macaulay rings is not proven as printed.","section":"Section 5, Theorem 5.3"}],"minor_comments":[{"comment":"In the proof of the case f_n(p)=0, the text refers to 'the finitely generated module M of assumption (ii)'; since this is inside the (iii)⇒(i) implication, the reference should be to assumption (iii).","section":"Section 5, Theorem 5.3"},{"comment":"The proof of the locality statement is terse; a few more details on why the supremum over maximal ideals coincides with the supremum over all primes (e.g., via the localization isomorphism for Tor and the equality fd_R(N)=fd_{R_p}(N_p)) would improve readability.","section":"Section 2, Proposition 2.1(5)"},{"comment":"The phrase 'it worth mentioning' should be 'it is worth mentioning'.","section":"Introduction, after the definition of (GL)"},{"comment":"The step proving Rpd_R ≤ CMpd_R is very compressed; after citing [21, Lemma 2.12], the inference that Ext^i_R(P,I)=0 for P∈CMP_0 and I∈I^{<∞} should be spelled out, as it is central to the inequality chain.","section":"Section 4, Lemma 4.2"},{"comment":"The definition of RP^{<ω}_n and RF^{<ω}_n is only implicit via the general notation C^{<ω}; it would help to state it explicitly when these classes are first used.","section":"Section 1, Notation"}],"recommendation":"major_revision","confidential_remarks":"The main mathematical concern is the inequality error in Theorem 5.3, which is fixable in principle (the argument appears to work if the two inequality signs are swapped) but as it stands invalidates a key claimed result. Additionally, Section 5 relies heavily on the classification of hereditary Tor-pairs from the authors' preprint [18], which is not proved here; the editor may wish to ensure that [18] has been vetted. The rest of the paper appears sound and well-written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Most of this paper is good. The Gorenstein results (Theorem A) are a natural extension of Hrbek–Le Gros to arbitrary dimension, the n=0 restricted-dimension results over finite-dimensional noetherian rings are clean, and the bridge through trivial extensions for pointwise dualizing modules (Theorem B(2)) is plausible and mostly checks out. The paper also connects these structural properties to preenveloping classes, which gives useful consequences.\n\nThe last section has a real flaw. In Theorem 5.3, the proof of (iii)=>(i) defines f via lim→ RP_n^{<ω} = C_f and claims that the containment lim→ RP_n^{<ω} ⊆ RF_n gives f ≤ f_n. That is backwards: C_f ⊆ C_{f_n} gives f ≥ f_n pointwise. Later, in the case f_n(p)>0, the chain using [18, Prop 4.14] actually yields max(0, f(p)−f_n(p)) = f_{n+f_n(p)}(p) = 0, so f(p) ≤ f_n(p); the proof asserts the opposite. As printed, neither inequality is established, so the equality f = f_n is not proven. The stress-test note is correct on both counts, and Corollary 5.4 / Theorem B(3)(i) are not proven as printed.\n\nThat said, the error looks mechanical. Flipping both inequalities makes the argument go through: containment gives the upper bound on f, the shift computation gives the lower bound. So the result may well be true, and the proof is likely repairable. Still, the authors need to rewrite that part carefully; right now a reader cannot verify the main almost Cohen–Macaulay claim.\n\nThe other caveat is that Section 5 imports the classification of Tor-pairs from the authors' own preprint [18]. That is appropriate if the preprint is correct, but it makes the proof hostage to an unrefereed source. The reader's conditional verdict is fair.\n\nWho is this for: commutative algebraists working on cotorsion/Tor pairs, Gorenstein dimensions, and approximation theory. The paper deserves a serious referee; the core ideas are good and the Gorenstein and Cohen–Macaulay cases appear sound. But I would not accept it as is; the Section 5 proof needs a careful rewrite, with either corrected inequality directions or a different argument for the almost Cohen–Macaulay case. Send it to review with the expectation of revision.","headline":"Solid Gorenstein and restricted-dimension theorems, but Theorem 5.3's proof has reversed inequalities that must be fixed before the almost Cohen-Macaulay claim is established.","tokens_in":16248,"tokens_out":6445,"would_cite":true,"duration_ms":52501,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13C60","13C14","13D05","13D07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Over Cohen–Macaulay rings with a pointwise dualizing module, modules of bounded restricted projective dimension are finitely deconstructible and modules of bounded restricted flat dimension satisfy the Govorov–Lazard property.","keywords":["Govorov–Lazard property","finite deconstructibility","restricted homological dimensions","Gorenstein projective dimension","Gorenstein flat dimension","Cohen–Macaulay rings","pointwise dualizing module","Tor-pairs"],"falsifier":"Exhibit a commutative noetherian almost Cohen–Macaulay ring of finite Krull dimension and an $n\\ge0$ for which some module of restricted flat dimension at most $n$ is not a direct limit of finitely generated modules of restricted projective dimension at most $n$; such a module would directly refute Corollary 5.4. A smaller check is to compute, on a concrete non-Gorenstein almost Cohen–Macaulay ring such as $k[[x,y]]/(xy)$, the Tor-pair generated by the finitely generated part of $\\mathrm{RF}_n$ and compare it with the Tor-pair assigned by the classification to the function $f_n(p)=\\max(0,\\operatorname{depth}R_p-n)$.","tokens_in":15063,"feed_emoji":"🧩","tokens_out":12193,"duration_ms":100470,"temperature":0.7,"pith_summary":"This paper asks when a class of modules defined by a bound on a relative homological dimension can be rebuilt from its finitely generated members. The two structural properties are the Govorov–Lazard property, meaning every module is a direct limit of finitely generated modules from the class, and finite deconstructibility, meaning every module is a direct summand of a transfinite extension of finitely generated modules from the class. The main results establish these properties for Gorenstein projective and flat dimensions over Gorenstein rings, and for restricted projective and flat dimensions over Cohen–Macaulay rings with a pointwise dualizing module; an almost Cohen–Macaulay version covers restricted flat dimension for every bound. These structural descriptions matter because a class that is the direct limit closure of its finitely presented part and is closed under products yields preenvelopes in the module category, and because they extend the known structure theorem for balanced big Cohen–Macaulay modules from bound zero to arbitrary bounds and to non-local rings.","feed_headline":"Restricted flat modules are direct limits of finitely generated ones","feed_subtitle":"Over Cohen–Macaulay rings, bounded restricted homological dimensions behave like flat modules.","key_machinery":"The main mechanism is a bridge from restricted dimensions to Gorenstein dimensions over the trivial extension $R\\ltimes\\Omega$. When $R$ is Cohen–Macaulay with a pointwise dualizing module $\\Omega$, Lemma 4.1 makes $R\\ltimes\\Omega$ into a Gorenstein ring, and Lemma 4.2 identifies $\\operatorname{Rfd}_R(M)$ with $\\operatorname{Gfd}_{R\\ltimes\\Omega}(M)$ and $\\operatorname{Rpd}_R(M)$ with $\\operatorname{Gpd}_{R\\ltimes\\Omega}(M)$, reducing the Cohen–Macaulay case to the Gorenstein case. On the Gorenstein side the argument runs through the cotorsion pair generated by finitely generated Gorenstein projectives, together with the direct-limit and Tor-pair calculus that converts Govorov–Lazard statements into generation by finitely presented modules, to pass from $n=0$ to all $n$. For the almost Cohen–Macaulay result, the second machine is the classification of hereditary Tor-pairs whose right class lies in the modules locally of finite flat dimension: such pairs correspond to functions $f:\\operatorname{Spec}R\\to\\mathbb{Z}_{\\ge0}$ with $f(p)\\le\\operatorname{depth}R_p$, with the restricted flat Tor-pair $(\\mathrm{RF}_n,D_n)$ attached to $f_n(p)=\\max(0,\\operatorname{depth}R_p-n)$.","core_discovery":"The paper's central claim is that the classical dichotomy between flat modules, which are direct limits of finitely generated projectives, and projective modules, which are direct summands of free modules, persists for the relative dimensions that recover Gorenstein dimensions. Over a Gorenstein ring, the class $\\mathrm{GP}_n$ of modules of Gorenstein projective dimension at most $n$ is finitely deconstructible and the class $\\mathrm{GF}_n$ of modules of Gorenstein flat dimension at most $n$ satisfies the Govorov–Lazard property, for every $n\\ge 0$. The same pair of conclusions is then transferred to restricted projective and flat dimensions, $\\mathrm{RP}_n$ and $\\mathrm{RF}_n$, over Cohen–Macaulay rings with a pointwise dualizing module, using the identification of restricted dimensions with Cohen–Macaulay dimensions and a trivial-extension bridge; over almost Cohen–Macaulay rings of finite Krull dimension, the Govorov–Lazard claim for $\\mathrm{RF}_n$ holds for every $n$ via a classification of hereditary Tor-pairs. A direct corollary is that the finitely generated modules in these classes are preenveloping in $\\mathrm{mod}\\,R$ in the relevant settings.","pith_inferences":["One extension the paper does not pursue is the same bridge in the language of complexes: since the trivial-extension identification of Lemma 4.2 is dimension-level, a derived-category reformulation might transfer Gorenstein deconstructibility statements to restricted dimensions for complexes, provided the relevant homotopy categories behave well.","The criterion in Theorem 5.3 suggests a testable sufficient condition for arbitrary noetherian rings: if every prime ideal $p$ admits a finitely generated module of restricted projective dimension at most $\\operatorname{depth}R_p$ with $p$ among its associated primes, then all restricted flat classes $\\mathrm{RF}_n$ should satisfy the Govorov–Lazard property; checking this on rings of infinite Kru","A further consequence of the preenveloping corollaries is that approximation theory for restricted dimensions is now available outside the dualizing-module setting; one could ask whether these preenvelopes are minimal or whether the same method produces precovers for the corresponding right classes, which would yield new cotorsion pairs."],"forward_implications":["Over any Gorenstein ring, for every $n\\ge0$, every module of Gorenstein flat dimension at most $n$ is a direct limit of finitely generated modules of Gorenstein projective dimension at most $n$, and every module of Gorenstein projective dimension at most $n$ is a direct summand of a transfinite extension of such finitely generated modules.","Over a Cohen–Macaulay ring with a pointwise dualizing module, the same two structural conclusions hold for restricted projective and flat dimensions, equivalently Cohen–Macaulay projective and flat dimensions, at every bound $n$.","Over an almost Cohen–Macaulay ring of finite Krull dimension, the class of modules of restricted flat dimension at most $n$ satisfies the Govorov–Lazard property for every $n$, and the finitely generated members form a preenveloping class in $\\mathrm{mod}\\,R$.","For any commutative noetherian ring of finite Krull dimension, the restricted flat modules satisfy the Govorov–Lazard property and the restricted projective modules are finitely deconstructible, covering the $n=0$ case in full generality.","The paper extends the known structure theorem for balanced big Cohen–Macaulay modules from bound zero to arbitrary bounds and to non-local Cohen–Macaulay rings with a pointwise dualizing module."],"supporting_citations":[{"why":"Establishes the finite-deconstructibility and Govorov–Lazard characterizations for ordinary bounded projective and flat dimensions in terms of Serre's conditions; this is the base case that the Gorenstein theorems build on and is used directly in the proofs of Theorems 3.3 and 3.4.","marker":"[24]"},{"why":"Introduces restricted projective and flat dimensions, gives their basic properties such as $\\mathrm{RP}_n^{<\\omega}=\\mathrm{RF}_n^{<\\omega}$, local determination, and coincidence with Gorenstein dimensions, and supplies the fact that maximal Cohen–Macaulay modules are $\\mathrm{RP}_0^{<\\omega}$.","marker":"[10]"},{"why":"Proves the lemma identifying restricted flat and projective dimensions over $R$ with Gorenstein dimensions over the trivial extension $R\\ltimes C$, the key bridge used in Lemma 4.2 and in the comparison with dualizing-free Cohen–Macaulay flat dimensions.","marker":"[31]"},{"why":"Provides the classification of hereditary Tor-pairs with right class contained in locally finite flat dimension and the shift formula $(\\mathrm{RF}_n)^{(i)}=\\mathrm{RF}_{n+i}$; this is the basis for the Theorem 5.3 criterion and Corollary 5.4.","marker":"[18]"},{"why":"Supplies the theorem that direct-limit-closed classes form Tor-pairs and the equivalence between the Govorov–Lazard property and generation by finitely presented modules; used throughout to convert GL statements into Tor-pair statements.","marker":"[1]"},{"why":"Gives the counterexample showing that outside Gorenstein rings the Gorenstein Govorov–Lazard property can fail; used in Lemma 1.4 and Corollary 3.5 to show the failure is genuine and to motivate passing to restricted dimensions.","marker":"[23]"},{"why":"Defines Cohen–Macaulay projective and flat dimensions via semidualizing modules, and supplies the lemma used in Lemma 4.2 to relate restricted dimensions to Gorenstein dimensions over the trivial extension.","marker":"[22]"},{"why":"Supplies the correspondence between preenveloping classes in $\\mathrm{mod}\\,R$ and closure of their direct limit closure under products, used in Corollaries 4.4 and 5.5 to derive the preenveloping conclusions.","marker":"[12]"}],"fun_headline_variants":["Restricted flat modules: direct limits of finitely generated ones over CM rings","Restricted dimensions: deconstructibility and Govorov–Lazard on CM rings","Restricted projective classes deconstructible; flat classes direct limits","Govorov–Lazard and deconstructibility for restricted homological dimensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The almost Cohen–Macaulay part of the main theorem depends on a classification of Tor-pairs imported from a companion preprint and not proved in this paper; if that classification, or its identification of restricted flat dimension with the function $f(p)=\\max(0,\\operatorname{depth} R_p-n)$, were incorrect, the Govorov–Lazard conclusion would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Restricted flat modules: direct limits of finitely generated ones over CM rings","Restricted dimensions: deconstructibility and Govorov–Lazard on CM rings","Restricted projective classes deconstructible; flat classes direct limits","Govorov–Lazard and deconstructibility for restricted homological dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001418,"raw_usage":{"total_tokens":5729,"prompt_tokens":953,"completion_tokens":4776,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":4696}},"tokens_in":569,"tokens_out":4776,"duration_ms":31151,"temperature":1.0,"reasoning_tokens":4696,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:48:25.493880+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a commutative noetherian almost Cohen–Macaulay ring of finite Krull dimension and an $n\\ge0$ for which some module of restricted flat dimension at most $n$ is not a direct limit of finitely generated modules of restricted projective dimension at most $n$; such a module would directly refute Corollary 5.4. A smaller check is to compute, on a concrete non-Gorenstein almost Cohen–Macaulay ring such as $k[[x,y]]/(xy)$, the Tor-pair generated by the finitely generated part of $\\mathrm{RF}_n$ and compare it with the Tor-pair assigned by the classification to the function $f_n(p)=\\max(0,\\operatorname{depth}R_p-n)$.","supporting_citations":[{"cited_title":"The finite type of modules of bounded projective dimension and Serre’s conditions","cited_arxiv_id":null,"evidence_quote":"Establishes the finite-deconstructibility and Govorov–Lazard characterizations for ordinary bounded projective and flat dimensions in terms of Serre's conditions; this is the base case that the Gorenstein theorems build on and is used directly in the proofs of Theorems 3.3 and 3.4."},{"cited_title":"Restricted homological dimensions and Cohen– Macaulayness","cited_arxiv_id":null,"evidence_quote":"Introduces restricted projective and flat dimensions, gives their basic properties such as $\\mathrm{RP}_n^{<\\omega}=\\mathrm{RF}_n^{<\\omega}$, local determination, and coincidence with Gorenstein dimensions, and supplies the fact that maximal Cohen–Macaulay modules are $\\mathrm{RP}_0^{<\\omega}$."},{"cited_title":"Cohen–Macaulay homological dimensions","cited_arxiv_id":null,"evidence_quote":"Proves the lemma identifying restricted flat and projective dimensions over $R$ with Gorenstein dimensions over the trivial extension $R\\ltimes C$, the key bridge used in Lemma 4.2 and in the comparison with dualizing-free Cohen–Macaulay flat dimensions."},{"cited_title":"Direct limits of modules of finite projective dimension.Rings, Modules, Algebras, and Abelian Groups, LNPAM, 236:27–44, 2004","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that direct-limit-closed classes form Tor-pairs and the equivalence between the Govorov–Lazard property and generation by finitely presented modules; used throughout to convert GL statements into Tor-pair statements."},{"cited_title":"Rings without a Gorenstein analogue of the Govorov–Lazard theorem","cited_arxiv_id":null,"evidence_quote":"Gives the counterexample showing that outside Gorenstein rings the Gorenstein Govorov–Lazard property can fail; used in Lemma 1.4 and Corollary 3.5 to show the failure is genuine and to motivate passing to restricted dimensions."},{"cited_title":"Cohen–Macaulay homological dimensions","cited_arxiv_id":null,"evidence_quote":"Defines Cohen–Macaulay projective and flat dimensions via semidualizing modules, and supplies the lemma used in Lemma 4.2 to relate restricted dimensions to Gorenstein dimensions over the trivial extension."},{"cited_title":"Locally finitely presented additive categories","cited_arxiv_id":null,"evidence_quote":"Supplies the correspondence between preenveloping classes in $\\mathrm{mod}\\,R$ and closure of their direct limit closure under products, used in Corollaries 4.4 and 5.5 to derive the preenveloping conclusions."}],"review_version":1}