Pseudofunctorial behavior of Cousin complexes on formal schemes
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Bibliographic record
Abstract
On a suitable category of formal schemes equipped with codimen- sion functions we construct a canonical pseudofunctor (−) ♯ taking values in the corresponding categories of Cousin complexes. Cousin complexes on such a formal scheme X functorially represent derived-category objects F by the local cohomologies H codim(x) x F (x ∈ X) together with residue maps from the cohomology at x to that at each immediate specialization of x; this rep- resentation is faithful when restricted to F which are Cohen-Macaulay (CM), i.e., H iF = 0 whenever i 6 codim(x). Formal schemes provide a framework for treating local and global duality as aspects of a single theory. One motivation has been to gain a better understanding of the close relation between local properties of residues and global variance properties of dualizing complexes (which are CM). Our construction, depending heavily on local phenomena, is inspired by, but generalizes and makes more concrete, that of the classical pseudofunctor (−) � taking values in residual complexes, on which the proof of Grothendieck's (global) Theorem in Hartshorne's Residues and Duality is based. Indeed, it is shown in the following paper by Sastry that (−) ♯ is a good concrete approximation to the fundamental duality pseudo- functor (−)!. The pseudofunctor (−)♯ takes residual complexes to residual complexes, so contains a canonical representative of (−)�; and it generalizes as well several other functorial (but not pseudofunctorial) constructions of residual complexes which appeared in the 1990s.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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