The Dixmier-Moeglin equivalence for extensions of scalars and Ore extensions
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Bibliographic record
Abstract
An algebra <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> satisfies the Dixmier-Moeglin equivalence if we have the equivalences: <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P primitive long left right double arrow upper P rational long left right double arrow upper P locally tilde closed tilde for upper P element-of Spec left-parenthesis upper A right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mtext> </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>primitive</mml:mtext> </mml:mrow> <mml:mspace width="thickmathspace"/> <mml:mo stretchy="false"> ⟺ </mml:mo> <mml:mspace width="thickmathspace"/> <mml:mi>P</mml:mi> <mml:mtext> </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>rational</mml:mtext> </mml:mrow> <mml:mspace width="thickmathspace"/> <mml:mo stretchy="false"> ⟺ </mml:mo> <mml:mspace width="thickmathspace"/> <mml:mi>P</mml:mi> <mml:mtext> </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>locally~closed~</mml:mtext> </mml:mrow> <mml:mspace width="2em"/> <mml:mtext> </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>for</mml:mtext> </mml:mrow> <mml:mtext> </mml:mtext> <mml:mi>P</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Spec</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P~\textrm {primitive}\iff P~\textrm {rational}\iff P ~\textrm {locally~closed~}\qquad ~\textrm {for}~P\in \textrm {Spec}(A).</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> We study the robustness of the Dixmier-Moeglin equivalence under extension of scalars and under the formation of Ore extensions. In particular, we show that the Dixmier-Moeglin equivalence is preserved under base change for finitely generated complex noetherian algebras. We also study Ore extensions of finitely generated complex noetherian algebras <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> . If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper A right-arrow upper A"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">T:A\to A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra automorphism or a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -linear derivation 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> , we say that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>frame-preserving</italic> if there exists a finite-dimensional subspace <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V subset-of-or-equal-to upper A"> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo> ⊆ </mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">V\subseteq A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that generates <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> as an algebra such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis upper V right-parenthesis subset-of-or-equal-to upper V"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⊆ </mml:mo> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">T(V)\subseteq V</mml:annotation> </mml:semantics>
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.
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 | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| 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 it