## Universal Covers and Category Theory in Polynomial and Differential Galois Theory

### BibTeX

@MISC{Magid_universalcovers,

author = {Andy R. Magid},

title = {Universal Covers and Category Theory in Polynomial and Differential Galois Theory},

year = {}

}

### OpenURL

### Abstract

Abstract. The category of finite dimensional modules for the proalgebraic differential Galois group of the differential Galois theoretic closure of a differential field F is equivalent to the category of finite dimensional F spaces with an endomorphism extending the derivation of F. This paper presents an expository proof of this fact modeled on a similar equivalence from polynomial Galois theory, whose proof is also presented as motivation. 1

### Citations

27 |
An outline of differential Galois theory
- Singer
- 1990
(Show Context)
Citation Context ...urre’s account of it [9] which is still an excellent exposition. Differential Galois theory is the work of Ellis Kolchin [4]. For a comprehensive modern introduction, see M. van der Put and M. Singer =-=[10]-=-. The survey article by Singer [11] is also a good introduction. Less advanced are the author’s introductory expository lectures on the subject [6], which is a reference for much of the terminology us... |

25 |
Lectures on an Introduction to Grothendieck’s Theory of the Fundamental Group
- Murre
- 1967
(Show Context)
Citation Context ... Section 2 is basically an account of the special (one point) case of A. Grothendieck’s theory of Galois categories and the fundamental group. I learned this material from J. P. Murre’s account of it =-=[9]-=- which is still an excellent exposition. Differential Galois theory is the work of Ellis Kolchin [4]. For a comprehensive modern introduction, see M. van der Put and M. Singer [10]. The survey article... |

23 |
Galois theories, Cambridge
- Borceux, Janelidze
- 2001
(Show Context)
Citation Context ...articles4 Andy R. Magid [1]. Information about the Picard–Vessiot closure is found in [7] and [8]. And of course the standard reference for Galois theory is categories is F. Borceaux and G. Janelidze =-=[2]-=-. 2 The Galois Group of the Separable Closure The Fundamental Theorems recalled above were called “Correspondence Theorem Galois Theory”. Even in their infinite forms, they are just special cases of C... |

18 | Lectures on Differential Galois Theory
- Magid
- 1997
(Show Context)
Citation Context ...dern introduction, see M. van der Put and M. Singer [10]. The survey article by Singer [11] is also a good introduction. Less advanced are the author’s introductory expository lectures on the subject =-=[6]-=-, which is a reference for much of the terminology used here. Section 3 is an account of a version of the Tannakian Categories methods in differential Galois Theory. This originated in work of P. Deli... |

10 | Direct and inverse problems in differential Galois theory
- Singer
- 1999
(Show Context)
Citation Context ...till an excellent exposition. Differential Galois theory is the work of Ellis Kolchin [4]. For a comprehensive modern introduction, see M. van der Put and M. Singer [10]. The survey article by Singer =-=[11]-=- is also a good introduction. Less advanced are the author’s introductory expository lectures on the subject [6], which is a reference for much of the terminology used here. Section 3 is an account of... |

4 |
Review of “Lectures on differential Galois theory”, by
- Bertrand
- 1996
(Show Context)
Citation Context ...are explanations of this in both [11] and [10]. A compact explanation of the theory as well as how to do the Fundamental Theorem in this context is also found in D. Bertrand’s articles4 Andy R. Magid =-=[1]-=-. Information about the Picard–Vessiot closure is found in [7] and [8]. And of course the standard reference for Galois theory is categories is F. Borceaux and G. Janelidze [2]. 2 The Galois Group of ... |

3 |
Pro-algebraic groups and the Galois theory of differential fields
- Kovacic
- 1973
(Show Context)
Citation Context ...infinite Picard–Vessiot extension, then the restriction map G → G(K/F ) is a surjection with kernel G(E/K). If H is (Zariski closed and) normal in G, then K H is an infinite Picard–Vessiot extension. =-=[5]-=- We shall call these theorems (and their finite dimensional versions stated previously) “Correspondence Theorem Galois Theory”. These theorems are about the pair consisting of the base field and the e... |

3 | The Picard-Vessiot Antiderivative Closure
- Magid
(Show Context)
Citation Context ...various reasons, the direct analogues of “algebraic closure” and its properties for differential Galois extensions do not hold. However, the following notion is of interest, and can be shown to exist =-=[7]-=-: A Picard–Vessiot closure E ⊃ F of F is a differential field extension which is a union of Picard–Vessiot extensions of F and such that every such Picard–Vessiot extension of F has an isomorphic copy... |

2 |
The Picard–Vessiot Closure in Differential Galois Theory
- Magid
- 2002
(Show Context)
Citation Context ...of the theory as well as how to do the Fundamental Theorem in this context is also found in D. Bertrand’s articles4 Andy R. Magid [1]. Information about the Picard–Vessiot closure is found in [7] and =-=[8]-=-. And of course the standard reference for Galois theory is categories is F. Borceaux and G. Janelidze [2]. 2 The Galois Group of the Separable Closure The Fundamental Theorems recalled above were cal... |

1 |
Catégories tannakiennes in Carties P., et. al, eds
- Deligne
- 1990
(Show Context)
Citation Context ...ich is a reference for much of the terminology used here. Section 3 is an account of a version of the Tannakian Categories methods in differential Galois Theory. This originated in work of P. Deligne =-=[3]-=-; there are explanations of this in both [11] and [10]. A compact explanation of the theory as well as how to do the Fundamental Theorem in this context is also found in D. Bertrand’s articles4 Andy R... |