{"id":"133c01b3-4719-456e-8221-3bd6f28f4487","arxiv_id":"2605.30262","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Semi-Bousfield classes generalize Bousfield classes and tensor-compatible t-structures, with a bijection assigning (nonmonotone) perversities on Noetherian schemes X to semi-Bousfield classes in D_qc(X) that stratifies the full lattice when X is regular.","lead":"This paper defines semi-Bousfield classes in tensor triangulated categories using vanishing of tensor products in positive degrees relative to a t-structure. It shows these classes generalize Bousfield classes and certain t-structures, then extends a stratification result from monotone to nonmonotone perversities on Noetherian schemes.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the dependence on a reasonable t-structure and the Noetherian D_qc(X) properties, but the abstract already states the claim is conditional on those hypotheses and recovers prior work. With the full manuscript available, no additional load-bearing gap appears that would alter the UNVERDICTED verdict.","tokens_in":1803,"tokens_out":345,"duration_ms":21553,"concrete_test":"Pick a concrete non-monotone perversity p on a regular Noetherian scheme X (e.g., X = Spec k[[x,y]] or P^1) and compute the associated semi-Bousfield class via the assignment in the specialization section; check whether the resulting class lies outside the image of monotone perversities and whether the inverse map (from class back to perversity) recovers p.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim introduces semi-Bousfield classes via vanishing of tensor products in positive degrees w.r.t. a fixed reasonable t-structure in rigidly-compactly generated tensor triangulated categories, then extends the known stratification bijection from monotone perversities to all (not necessarily monotone) perversities on Noetherian X in D_qc(X), with the image being the full semi-Bousfield lattice when X is regular and precisely the finite-Tor-dimension classes when X is singular. The argument is scoped precisely to these hypotheses; the recovery of Dubey-Sahoo for the monotone case is stated explicitly. No internal inconsistency, hidden assumption on monotonicity, or unsupported leap from the general definition to the D_qc(X) specialization is visible.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces semi-Bousfield classes in rigidly-compactly generated tensor triangulated categories, defined via vanishing of tensor products in positive degrees with respect to a fixed reasonable t-structure. These classes are shown to generalize both Bousfield classes and compactly generated tensor-compatible t-structures. Specializing to D_qc(X) for a Noetherian scheme X, the stratification bijection is extended to an assignment sending (not necessarily monotone) perversities to semi-Bousfield classes; for regular X this stratifies the full semi-Bousfield lattice, while for singular X the image consists precisely of the finite-Tor-dimension classes. Restriction to monotone perversities recovers the Dubey-Sahoo classification.","tokens_in":1901,"tokens_out":261,"duration_ms":18012,"significance":"If the results hold, the work supplies a common generalization that unifies Bousfield classes with tensor-compatible t-structures and extends the known stratification to the nonmonotone case in a controlled way. The precise distinction between the regular and singular cases, together with the explicit recovery of the prior monotone classification, strengthens the contribution and provides a falsifiable framework for further study of the semi-Bousfield lattice.","major_comments":[],"minor_comments":[],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of the manuscript and for recommending acceptance. The report correctly captures the main results on semi-Bousfield classes and their relation to perversities.","responses":[],"tokens_in":1287,"tokens_out":55,"duration_ms":6880,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core advance is the definition of semi-Bousfield classes in any rigidly-compactly generated tensor triangulated category, using vanishing of tensor products in positive degrees against a fixed reasonable t-structure. This unifies Bousfield classes with compactly generated tensor-compatible t-structures. The authors then restrict to D_qc(X) for Noetherian X and show the stratification assignment works for nonmonotone perversities as well, recovering the Dubey-Sahoo classification when restricted to the monotone case. When X is regular the image fills the whole semi-Bousfield lattice; when singular it hits exactly the finite-Tor-dimension classes.\n\nThe paper does the generalization cleanly and states the recovery of the earlier result explicitly. The scoping to the stated hypotheses looks deliberate and avoids overclaiming.\n\nThe main soft spot is that the general definition depends on what counts as a reasonable t-structure, and the abstract gives no further detail on that condition. In the D_qc(X) specialization the argument appears to rest on the usual properties of that category, but without the full proofs it is hard to see exactly how the bijection is built for nonmonotone perversities. That said, the stress-test finds no internal inconsistency or hidden monotonicity assumption, so the gap is mainly one of missing verification steps rather than a load-bearing flaw.\n\nThis is for people already working on t-structures and tensor-triangular geometry in algebraic geometry. A reader who cares about classifications of these objects will find the extension useful. It is worth sending to peer review; the claims are scoped, the prior result is recovered, and the new pieces look independent of the monotone case.","headline":"The paper introduces semi-Bousfield classes to unify two notions and extends the known stratification bijection from monotone to all perversities in D_qc(X).","tokens_in":2404,"tokens_out":410,"would_cite":false,"duration_ms":14202,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Semi-Bousfield classes generalize Bousfield classes and tensor t-structures, and correspond bijectively to perversities on D_qc(X) for Noetherian schemes X.","keywords":["semi-Bousfield classes","perversities","tensor triangulated categories","derived categories","stratification","t-structures","Noetherian schemes","Bousfield classes"],"falsifier":"Exhibit a regular Noetherian scheme X together with two distinct perversities whose associated semi-Bousfield classes coincide, or produce a semi-Bousfield class in D_qc(X) that does not arise from any perversity.","tokens_in":2683,"feed_emoji":"","tokens_out":852,"duration_ms":19197,"temperature":0.7,"pith_summary":"The paper defines semi-Bousfield classes in any rigidly-compactly generated tensor triangulated category by the condition that an object's tensor product with a fixed reasonable t-structure vanishes in positive degrees. These classes unify Bousfield classes (recovered from the standard t-structure) and compactly generated tensor-compatible t-structures. Specializing to the unbounded derived category of a Noetherian scheme, the authors construct a map sending any perversity, monotone or otherwise, to a semi-Bousfield class. When the scheme is regular this map is a bijection that stratifies the entire semi-Bousfield lattice; otherwise its image is precisely the classes coming from objects of finite Tor-dimension. The monotone case recovers an earlier classification of t-structures.","feed_headline":"Perversities classify semi-Bousfield classes on D_qc(X)","feed_subtitle":"The assignment from any perversity to a semi-Bousfield class is bijective precisely when the Noetherian scheme is regular.","key_machinery":"semi-Bousfield classes, defined by the positive-degree vanishing of tensor products with respect to a fixed reasonable t-structure, acting as the common generalization that carries the stratification bijection.","core_discovery":"In a rigidly-compactly generated tensor triangulated category equipped with a reasonable t-structure, the semi-Bousfield class of an object is the collection of all objects whose tensor product with it vanishes in positive degrees. These classes simultaneously generalize Bousfield classes and compactly generated tensor-compatible t-structures. Specializing to D_qc(X) for a Noetherian scheme X yields an assignment from (not necessarily monotone) perversities on X to semi-Bousfield classes; when X is regular the assignment is a stratification of the whole semi-Bousfield lattice, while for singular X the image consists exactly of those classes arising from finite-Tor-dimension objects. Restrict","pith_inferences":["The same construction may classify analogous objects in other rigidly-compactly generated tensor triangulated categories that are not derived categories of schemes.","Nonmonotone perversities may produce t-structures whose hearts have properties not visible from the monotone case alone.","Explicit computation of the assignment on low-dimensional singular schemes could reveal whether the finite-Tor-dimension restriction is sharp.","The unification suggests that further invariants of tensor triangulated categories might be stratified by suitable generalizations of perversities."],"forward_implications":["When X is regular the perversity assignment stratifies every semi-Bousfield class.","For singular X the image of the assignment is exactly the semi-Bousfield classes coming from finite-Tor-dimension objects.","Restricting the assignment to monotone perversities recovers the known classification of compactly generated tensor-compatible t-structures.","The definition and generalization properties hold in any rigidly-compactly generated tensor triangulated category with a reasonable t-structure."],"fun_headline_variants":["Semi-Bousfield classes from nonmonotone perversities on D_qc(X)","Perversities classify semi-Bousfield classes beyond monotone cases","Nonmonotone perversities stratify the semi-Bousfield lattice","Bijection between perversities and semi-Bousfield classes for regular X"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The underlying category must be rigidly-compactly generated and the t-structure must be reasonable for the vanishing condition to define well-behaved classes and for the bijection with perversities to hold in D_qc(X).","fun_headline_variants_meta":{"raw":{"variants":["Semi-Bousfield classes from nonmonotone perversities on D_qc(X)","Perversities classify semi-Bousfield classes beyond monotone cases","Nonmonotone perversities stratify the semi-Bousfield lattice","Bijection between perversities and semi-Bousfield classes for regular X"]},"model":"grok-4.3","cost_usd":0.005598,"raw_usage":{"total_tokens":2626,"prompt_tokens":720,"num_sources_used":0,"completion_tokens":76,"cost_in_usd_ticks":55978000,"prompt_tokens_details":{"text_tokens":720,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1830,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":720,"tokens_out":76,"duration_ms":11258,"temperature":1.0,"reasoning_tokens":1830,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T23:25:17.525775+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Exhibit a regular Noetherian scheme X together with two distinct perversities whose associated semi-Bousfield classes coincide, or produce a semi-Bousfield class in D_qc(X) that does not arise from any perversity.","supporting_citations":[],"review_version":1}