Pseudofunctorial behavior of Cousin complexes on formal schemes
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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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".