Dénombrabilité des classes d’équivalences dérivées de variétés algébriques
Bibliographic record
Abstract
Soient <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> un schéma affine, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X long right-arrow upper S"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> ⟶ </mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X \longrightarrow S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> une famille miniverselle de schémas projectifs et lisses, et <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> une catégorie triangulée fixée. On démontre que les points <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s element-of upper S"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">s\in S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tels que la catégorie dérivée de la fibre en <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript c o h Superscript b Baseline left-parenthesis upper X Subscript s Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>c</mml:mi> <mml:mi>o</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">D_{coh}^{b}(X_{s})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , soit équivalente à <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , forment un ensemble au plus dénombrable. Nous déduisons de cela que l’ensemble des classes d’isomorphisme des variétés complexes lisses et projectives qui possèdent une catégorie dérivée fixée est au plus dénombrable. Notre démonstration passe par la construction d’un certain préchamp classifiant les dg-catégories saturées et connexes, ainsi qu’une application des périodes allant du champ des variétés lisses et projectives vers ce préchamp, et qui à une variété associe un dg-modèle pour sa catégorie dérivée.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.007 | 0.018 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.007 |
| Bibliometrics | 0.007 | 0.004 |
| Science and technology studies | 0.006 | 0.009 |
| Scholarly communication | 0.016 | 0.015 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.006 | 0.016 |
| Insufficient payload (model declined to judge) | 0.030 | 0.009 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".