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**Differential Galois theory of infinite dimension.**
*(English)*
Zbl 0878.12002

Kolchin introduced the notion of strongly normal extensions of differential fields and developed an elegant theory of finite-dimensional differential Galois extensions which extended the Picard-Vessiot theory. However, strongly normal extensions do not generalize Galois extensions of abstract fields (when considered as fields with trivial derivation).

In this paper, the author introduces several functors from the category of algebras over a commutative ring \(R\) to the category of groups. For example the Lie-Ritt functor \(\Gamma_{nR}\), which associates to each commutative \(R\)-algebra \(A\) a group \(\Gamma_n(A)\) of all infinitesimal coordinate transformations of \(n\)-variables defined over \(A\), the functor \({\mathcal F}\), which associates with each algebra \(A\) over an abstract field \(L\) the set of all differential homomorphisms from \({\mathcal L}\), the subfield of the formal Laurant series over \(L\) (namely contained in \(L[[t]] [t^{-1}]\) generated by \(L^*\) and this is shown to be independent of the choice of the transcendental basis of \(L\) over \(K\) (as abstract fields)). There is another functor \(\text{Inf-diff bir}_K(L)\), which is a group functor from the category of \(L\)-algebras to the category of groups, which is in fact a Lie-Ritt functor. The main result (Theorem 5.15) shows that when \(L\) is a strongly normal extension of \(K\) with Galois group \(G\), then \(\text{Inf-diff bir}_K(L)\) is the formal group associated with the algebraic group scheme \(G\) and that this group ignores algebraic extensions and extensions generated by constants.

In this paper, the author introduces several functors from the category of algebras over a commutative ring \(R\) to the category of groups. For example the Lie-Ritt functor \(\Gamma_{nR}\), which associates to each commutative \(R\)-algebra \(A\) a group \(\Gamma_n(A)\) of all infinitesimal coordinate transformations of \(n\)-variables defined over \(A\), the functor \({\mathcal F}\), which associates with each algebra \(A\) over an abstract field \(L\) the set of all differential homomorphisms from \({\mathcal L}\), the subfield of the formal Laurant series over \(L\) (namely contained in \(L[[t]] [t^{-1}]\) generated by \(L^*\) and this is shown to be independent of the choice of the transcendental basis of \(L\) over \(K\) (as abstract fields)). There is another functor \(\text{Inf-diff bir}_K(L)\), which is a group functor from the category of \(L\)-algebras to the category of groups, which is in fact a Lie-Ritt functor. The main result (Theorem 5.15) shows that when \(L\) is a strongly normal extension of \(K\) with Galois group \(G\), then \(\text{Inf-diff bir}_K(L)\) is the formal group associated with the algebraic group scheme \(G\) and that this group ignores algebraic extensions and extensions generated by constants.

Reviewer: N.Sankaran (Bangalore)

### Keywords:

differential Galois theory of infinite dimension; differential fields; Lie-Ritt functor; algebraic group scheme
Full Text:
DOI

### References:

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