Homological Dimensions of Local (Co)homology Over Commutative DG-rings
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Bibliographic record
Abstract
Abstract Let A be a commutative noetherian ring, let a ⊆ A be an ideal, and let I be an injective A -module. A basic result in the structure theory of injective modules states that the A -module Γ a ( I ) consisting of ɑ-torsion elements is also an injective A -module. Recently, de Jong proved a dual result: If F is a flat A -module, then the ɑ-adic completion of F is also a flat A -module. In this paper we generalize these facts to commutative noetherian DG-rings: let A be a commutative non-positive DG-ring such that H 0 ( A ) is a noetherian ring and for each i < 0, the H 0 ( A )-module H i ( A ) is finitely generated. Given an ideal ⊆ H 0 ( A ), we show that the local cohomology functor R associated with does not increase injective dimension. Dually, the derived -adic completion functor LΛ does not increase flat dimension.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.013 | 0.002 |
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