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- W2922364022 abstract "The aim of this note is to prove various general properties of a generalization of the full module of first order differential operators on a commutative ring - a $operatorname{D}$-Lie algebra. A $operatorname{D}$-Lie algebra $tilde{L}$ is a Lie-Rinehart algebra over $A/k$ equipped with an $Aotimes_k A$-module structure that is compatible with the Lie-structure. It may be viewed as a simultaneous generalization of a Lie-Rinehart algebra and an Atiyah algebra with additional structure. Given a $operatorname{D}$-Lie algebra $tilde{L}$ and an arbitrary connection $(rho, E)$ we construct the universal ring $tilde{U}^{otimes}(tilde{L},rho)$ of the connection $(rho, E)$. The associative unital ring $tilde{U}^{otimes}(tilde{L},rho)$ is in the case when $A$ is Noetherian and $tilde{L}$ and $E$ finitely generated $A$-modules, an almost commutative Noetherian sub ring of $operatorname{Diff}(E)$ - the ring of differential operators on $E$. It is constructed using non-abelian extensions of $operatorname{D}$-Lie algebras. The non-flat connection $(rho, E)$ is a finitely generated $ tilde{U}^{otimes}(tilde{L},rho)$-module, hence we may speak of the characteristic variety $operatorname{Char}(rho,E)$ of $(rho, E)$ in the sense of $D$-modules. We may define the notion of holonomicity for non-flat connections using the universal ring $ tilde{U}^{otimes}(tilde{L},rho)$. This was previously done for flat connections. We also define cohomology and homology of arbitrary non-flat connections. The cohomology and homology of a non-flat connection $(rho,E)$ is defined using $operatorname{Ext}$ and $operatorname{Tor}$-groups of a non-Noetherian ring $operatorname{U}$. In the case when the $A$-module $E$ is finitely generated we may always calculate cohomology and homology using a Noetherian quotient of $operatorname{U}$. This was previously done for flat connections." @default.
- W2922364022 created "2019-03-22" @default.
- W2922364022 creator A5078537584 @default.
- W2922364022 date "2019-03-11" @default.
- W2922364022 modified "2023-10-16" @default.
- W2922364022 title "Lie algebras of differential operators: The universal ring" @default.
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