Homological dimensions for co-rank one idempotent subalgebras
Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an algebraically closed field and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a (left and right) Noetherian associative <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra. Assume further that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either positively graded or semiperfect (this includes the class of finite dimensional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras that are finitely generated modules over a Noetherian central Henselian ring). Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e"> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding="application/x-tex">e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a primitive idempotent of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which we assume is of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is positively graded. We consider the idempotent subalgebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma equals left-parenthesis 1 minus e right-parenthesis upper A left-parenthesis 1 minus e right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mi>e</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mi>e</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma = (1-e)A(1-e)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript e"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>e</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">S_e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the simple right <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -module <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript e Baseline equals e upper A slash e rad upper A"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>e</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>e</mml:mi> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>rad</mml:mtext> </mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">S_e = eA/e\textrm {rad}A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rad upper A"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>rad</mml:mtext> </mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\textrm {rad}A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Jacobson radical of <inline-formula content-type="math/mathml"> <mml:math xmln
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 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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.013 | 0.003 |
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".