Bibliographic record
Abstract
Abstract Auslander and Smalø (Auslander, M., Smalø, S. O. (1980). Preprojective modules over artin algebras. J. Algebra 66:61–122) introduced and studied extensively preprojective modules and preinjective modules over an artin algebra. We now call a module hereditarily preprojective or hereditarily preinjective if its submodules are all preprojective or its quotient modules are all preinjective, respectively. Coelho (Coelho, F. U. (1993). Components of Auslander-Reiten quivers with only preprojective modules. J. Algebra 157: 472–488) studied Auslander-Reiten components containing only hereditarily preprojective modules and gave a number of characterizations of such components. We shall study further these modules by using the description of shapes of semi-stable Auslander-Reitan components; see Liu (Liu, S. (1992). Degrees of irreducible maps and the shapes of Auslander-Reiten quivers. J. London Math. Soc. 45(2):32–54; Liu, S. (1993). Semi-stable components of an Auslander-Reiten quiver. J. London Math. Soc. 47(2):405–416). Our results will imply the result of Coelho (Coelho, F. U. (1993). Components of Auslander-Reiten quivers with only preprojective modules. J. Algebra 157: 472–488, (1.2)) and that of Auslander-Smalø (Auslander, M. and Smalø, S. O. (1980). Preprojective modules over artin algebras, J. Algebra 66:61–122, (9.16)). As an application, moreover, we shall show that a stable Auslander-Reiten component with “few” stable maps in TrD-direction is of shape ℤA ∞.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".