Commutative Endomorphism Algebras of Torsion-Free, Rank-Two Kronecker Modules with Singular Height Functions
Bibliographic record
Abstract
As is the case with Abelian groups, rank-1, torsion-free Kronecker modules are characterized by height functions.A height function is called singular provided it never assumes the value infinity.The endomorphism algebras of singular, rank-1, torsion-free Kronecker modules are trivial.Here we consider the endomorphism algebras of torsion-free, rank-2 modules that are extensions of finite-dimensional modules by modules of rank-1.If K is the ground field and K(X) the field of rational functions, the endomorphism algebras of the rank-2, indecomposable modules are known to be commutative K-subalgebras of the matrix ring M 2 (K(X)).When the height function is nonsingular the resulting endomorphism algebras can be varied, including, for example, coordinate rings of elliptic curves.This paper examines the possibilities for the singular case, which we show are more limited.Yet their endomorphism algebras offer examples from an important class of commutative rings, namely, zero-dimension, local rings.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".