Bibliographic record
Abstract
Tilting theory is one of the main tools in the representation theory of algebras. It originated with the study of reflection functors. The first set of axioms for a tilting module is due to Brenner and Butler; the one generally accepted now is due to Happel and Ringel. The main idea of tilting theory is that when the representation theory of an algebra A is difficult to study directly, it may be convenient to replace A with another simpler algebra B and to reduce the problem on A to a problem on B. We then construct an A-module T, called a tilting module, which can be thought of as being close to the Morita progenerators such that, if B = End TA, then the categories mod A and mod B are reasonably close to each other (but generally not equivalent). As will be seen, the knowledge of one of these module categories implies the knowledge of two distinguished full subcategories of the other, which form a torsion pair and thus determine up to extensions the whole module category. Because this procedure can be seen as generalising Morita theory, it is reasonable to give special attention to the full subcategory Gen TA of all A-modules generated by T and to use the adjoint functors HomA(T, −) and −⊗BT to compare mod A and mod B.
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How this classification was reachedexpand
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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".