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Bibliographic record
Abstract
We consider the general circumstance of an Azumaya algebra A of degree n over a locally ringed topos (\mathbf{X}, \mathcal O_\mathbf{X}) where the latter carries a (possibly trivial) involution, denoted \lambda . This generalizes the usual notion of involutions of Azumaya algebras over schemes with involution, which in turn generalizes the notion of involutions of central simple algebras. We provide a criterion to determine whether two Azumaya algebras with involutions extending \lambda are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the question of when an Azumaya algebra A is Brauer equivalent to an algebra carrying an involution extending \lambda , by giving a cohomological condition. We remark that these results are novel even in the case of schemes, since we allow ramified, non-trivial involutions of the base object. We observe that, if the cohomological condition is satisfied, then A is Brauer equivalent to an Azumaya algebra of degree 2n carrying an involution. By comparison with the case of topological spaces, we show that the integer 2n is minimal, even in the case of a nonsingular affine variety X with a fixed-point free involution. As an incidental step, we show that if R is a commutative ring with involution for which the fixed ring S is local, then either R is local or R/S is a quadratic étale extension of rings.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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 it