부산대학교 도서관

This paper introduces a notion of a free proalgebraic group on a given set $X$, which can be considered as a direct limit of linear algebraic groups with a universal mapping property in much the same way as a free profinite group can be considered as a direct limit of finite groups with a universal mapping property, and also the notion of a saturated proalgebraic group relative to a formation $\Cal{C}$ of linear algebraic groups, where, roughly speaking, the property of being saturated is a universal embedding property. The notion of a formation shall be clarified below. \par Knowledge of profinite groups and Galois theory, for both finite and infinite extensions, as well as knowledge of proalgebraic groups and differential Galois theory, is certainly necessary for understanding this paper. \par The ambition of the paper is to give clear structure to the concept of a free proalgebraic group, and the concept of a saturated proalgebraic group, and to establish a lemma characterizing free proalgebraic groups by means of the concept of saturated proalgebraic groups in order to make progress towards a proof of Matzat's conjecture, which is a generalization of Shafarevic's conjecture to differential fields. Let us briefly indicate the statements of Shafarevic's conjecture and Matzat's conjecture. \par The statement of Shafarevic's conjecture is: \par The absolute Galois group of the maximal abelian extension $\Bbb{Q}^{ab}$ of $\Bbb{Q}$ is isomorphic to the free profinite group on a countably infinite set of generators. \par The statement of Matzat's conjecture is: \par If $k$ is an algebraically closed field of characteristic zero, then the absolute differential Galois group of $K\coloneq k(x)$ is the free proalgebraic group on a set of cardinality $|K|$. \par To give a clear indication of the content of the paper it will be necessary to state the definition of when a class $\Cal{C}$ of linear algebraic groups is a formation. \par Let us suppose that $\Cal{C}$ is a non-empty class of linear algebraic groups defined over a field $k$ with the property that any linear algebraic $k$-group which is $k$-isomorphic to a group in $\Cal{C}$ is also a member of $\Cal{C}$. A group which belongs to $\Cal{C}$ is said to be a $\Cal{C}$-group. We say that $\Cal{C}$ is a formation if the following properties hold. \par (i) $\Cal{C}$ is closed under taking quotients; that is to say, if $G \twoheadrightarrow H$ is an epimorphism of algebraic groups and $G \in \Cal{C}$, then $H \in \Cal{C}$. (ii) $\Cal{C}$ is closed under taking subdirect products; i.e., if $H$ is a subdirect product of $G_1, G_2 \in \Cal{C}$ then $H \in \Cal{C}$, where $H$ is said to be a subdirect product of $G_1, G_2$ if the projections $H \rightarrow G_i$ are epimorphisms for ${i=1, 2}$. \par The following ten classes of linear algebraic groups are all examples of formations. \par (i) The class of all algebraic groups; \par (ii) The class of all abelian algebraic groups; \par (iii) The class of all nilpotent algebraic groups; \par (iv) The class of all solvable algebraic groups; \par (v) The class of all unipotent algebraic groups; \par (vi) The class of all étale algebraic groups; \par (vii) The class of all diagonalizable algebraic groups; \par (viii) The class of all infinitesimal algebraic groups; \par Moreover, over a field of characteristic zero \par (ix) The class of all semisimple algebraic groups, and \par (x) The class of all reductive linear algebraic groups. \par Using this concept of a formation, the paper characterizes in Section 2 what it means for a proalgebraic group to be a free proalgebraic group on a set $X$, and in Section 3, what it means for a proalgebraic group to be a saturated proalgebraic group relative to a formation $\Cal{C}$. \par As someone with only a modest background in number theory this paper was extremely helpful to me in getting acquainted with some new concepts.