Homological and combinatorial aspects of virtually Cohen–Macaulay sheaves
Bibliographic record
Abstract
When studying a graded module M over the Cox ring of a smooth projective toric variety X, there are two standard types of resolutions commonly used to glean information: free resolutions of M and vector bundle resolutions of its sheafification.Each approach comes with its own challenges.There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques.Recently, Berkesch, Erman and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study.The theory of virtual resolutions includes a notion of a virtually Cohen-Macaulay property, though tools for assessing which modules are virtually Cohen-Macaulay have only recently started to be developed.In this article, we continue this research program in two related ways.The first is that, when X is a product of projective spaces, we produce a large new class of virtually Cohen-Macaulay Stanley-Reisner rings, which we show to be virtually Cohen-Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure.The second is that, for an arbitrary smooth projective toric variety X, we develop homological tools for assessing the virtual Cohen-Macaulay property.Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions.We also use these tools to establish relationships among the arithmetically, geometrically and virtually Cohen-Macaulay properties.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".