부산대학교 도서관

This paper introduces a notion of a free proalgebraic group on a givenset $X$, which can be considered as a direct limit of linear algebraicgroups with a universal mapping property in much the same way as a freeprofinite group can be considered as a direct limit of finite groupswith a universal mapping property, and also the notion of a saturatedproalgebraic group relative to a formation $\Cal{C}$ of linearalgebraic groups, where, roughly speaking, the property of beingsaturated is a universal embedding property. The notion of a formationshall be clarified below.\par Knowledge of profinite groups and Galois theory, for both finite andinfinite extensions, as well as knowledge of proalgebraic groups anddifferential Galois theory, is certainly necessary for understandingthis paper.\par The ambition of the paper is to give clear structure to the concept ofa free proalgebraic group, and the concept of a saturated proalgebraicgroup, and to establish a lemma characterizing free proalgebraic groupsby means of the concept of saturated proalgebraic groups in order tomake progress towards a proof of Matzat's conjecture, which is ageneralization of Shafarevic's conjecture to differential fields. Letus briefly indicate the statements of Shafarevic's conjecture andMatzat'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 groupon 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, thenthe absolute differential Galois group of $K\coloneq k(x)$ is the freeproalgebraic group on a set of cardinality $|K|$.\par To give a clear indication of the content of the paper it will benecessary to state the definition of when a class $\Cal{C}$ of linearalgebraic groups is a formation.\par Let us suppose that $\Cal{C}$ is a non-empty class of linear algebraicgroups defined over a field $k$ with the property that any linearalgebraic $k$-group which is $k$-isomorphic to a group in $\Cal{C}$ isalso a member of $\Cal{C}$. A group which belongs to $\Cal{C}$ is saidto be a $\Cal{C}$-group. We say that $\Cal{C}$ is a formation if thefollowing 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 takingsubdirect 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 subdirectproduct of $G_1, G_2$ if the projections $H \rightarrow G_i$ areepimorphisms for ${i=1, 2}$.\par The following ten classes of linear algebraic groups are all examplesof 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 Section2 what it means for a proalgebraic group to be a free proalgebraicgroup on a set $X$, and in Section 3, what it means for a proalgebraicgroup to be a saturated proalgebraic group relative to a formation$\Cal{C}$.\par As someone with only a modest background in number theory this paperwas extremely helpful to me in getting acquainted with some newconcepts.