Galois extensions of differential algebras
type: Item
instance of: scholarly article
title: Galois extensions of differential algebras
zbMATH Open document ID: 0903.12004
author: D. Massart
published in: Pacific Journal of Mathematics
publication date: 1998-05-24T00:00:00Z
review text: Let \(A\) be a commutative ring containing a field of characteristic zero, and let \(T_A\) be a finitely generated \(A\) submodule of \(\text{Der}(A,A)\) closed under Lie bracket. The pair \((A,T_A)\) is then called a differential algebra. An ideal of \(A\) is a differential ideal if it is preserved by every derivation in \(T_A\), and \((A,T_A)\) is said to be simple if its only differential ideals are \(0\) and \(A\). Let \(\mathcal D_A\) be the algebra of differential operators on \(A\) (the algebra generated by \(A\) and \(T_A\)). \(\mathcal D_A\) is naturally filtered and, under appropriate hypotheses, called regularity, \(T_A\) is projective and \(\text{gr} \mathcal D_A\) is the symmetric algebra on it. If \(M\) is a \(\mathcal D_A\) module, \(C(M) = \{ m \in M \mid \partial m =0 \quad \forall \partial \in T_A \}\). If \((A,T_A)\) is simple then the natural map \(A \otimes_{C(A)} C(M) \to M\) is injective; if it is bijective then \(M\) is called trivial. A morphism \((A,T_A) \to (B,T_B)\) of differential algebras is a pair \((\phi,\phi')\) where \(\phi: A \to B\) is a ring homomorphism, \(\phi': T_A \to T_B\) is an \(A\) module homomorphism, satisfying compatibility conditions and such that \(\phi(C(A)) \subset C(B)\). \((B,T_B)\) is a differential algebra over \((A,T_A)\) if further \(T_B= B\phi'(T_A)\); it is a trivial differential algebra if the canonical morphism \(A \otimes_{C(A)} C(B) \to B\) is an isomorphism. Now assume \((A,T_A)\) is simple. A left ideal \(I\) of \(\mathcal D_A\) is called Picard-Vessiot (P.V.) if \(\mathcal D_A/I\) is \(A\) finitely generated and projective. If \((B,T_B)\) is a differential algebra over \((A,T_A)\) and \(B \otimes_A \mathcal D_A/I\) is trivial, then \(I\) is said to be completely solvable in \(B\). For any \(I\), \(\text{Sol}(I,B)\) denotes \(C(\text{Hom}_A(\mathcal D_A/I, B))\). The author proves that if \(M\) is a finite \(A\) rank \(A\) free \(\mathcal D_A\) module, and \(C(A)\) is an algebraically closed field, th…
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 13N99
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 13H99
zbMATH DE Number: 1136645
zbMATH Keywords: differential algebra
zbMATH Keywords: Joyal extension
zbMATH Keywords: Picard-Vessiot extension
zbMATH Keywords: Galois extension
reviewed by: Andy R. Magid
MaRDI profile type: Publication
full work available at URL: pjm.1997.180.7
OpenAlex ID: W2037879246
DOI: 10.2140/PJM.1997.180.7
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description: scientific article; zbMATH DE number 1136645