REGULAR LOCAL ALGEBRAS OVER VALUATION DOMAINS: WEAK DIMENSION AND REGULAR SEQUENCES
Bibliographic record
Abstract
A local ring O is called regular if every finitely generated ideal I ◃ O possesses finite projective dimension. In the article localizations O = Aq, q ∈ Spec A, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdim O. A formula to compute wdim O is provided. Furthermore regular sequences within the maximal ideal M ◃ O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim O. If q ∩ R has finite height, then this sequence can be chosen such that the radical of the ideal generated by its members equals M. As a consequence it is proved that if O is regular, then the factor ring O/(q ∩ R)O, which is noetherian, is Cohen–Macaulay. If in addition (q ∩ R)Rq ∩ R is not finitely generated, then O/(q ∩ R)O itself is regular.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".