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Introduction. Let C be a field of characteristic p 0, and let K be a finite algebraic extension field of C such that the pth power of every element of K lies in C. Then K/C is called a purely inseparable extension of exponent 1. It was shown by N. Jacobson that there is a Galois theory for such extensions in which the place of the Galois group is taken by the derivation algebra of K/C. In particular, if K is any field of characteristic p, the purely inseparable extensions K/C of exponent 1 are precisely those in which C is the field of constants of a restricted K-Lie ring of derivations of K which is of finite dimension over K. In the classical theory of simple algebras, it is shown that, if K/C is a Galois extension, the Brauer similarity classes of the simple algebras with center C and split by K constitute a group which is canonically isomorphic with the group of equivalence classes of the group extensions of the multiplicative group of K by the Galois group of K/C. The present paper provides the answer to a question put to me by J-P. Serre, of whether one could establish an analogous result, for K/ C purely inseparable of exponent 1, in which restricted Lie algebra extensions [2] of K by the derivation algebra of K/C take the place of the group extensions. Not only is the answer to this question affirmative, but it provides an excellent illustration of the theory of restricted Lie algebra extensions. It turns out, in fact, that the Lie algebra extensions which arise from simple algebras are trivial extensions when regarded as ordinary extensions, so that the essential structural elements are here precisely those which differentiate the restricted extensions from the ordinary ones. ?1 contains the field theoretical background of our problem. In particular, it gives a simple proof of the main theorem of Jacobson's Galois theory [4] which we include here because it gives us the connection, on which many of our subsequent arguments are based, between the structure of the field extension K/C and that of the derivation algebra of K/C. Theorem 2, which is not needed in the sequel, is the analogue for the present situation of a well known result in the classical Galois theory and is significant for the cohomology theory of derivation algebras. In ?2 we give a proof of a theorem of Jacobson's on derivations (in a slightly generalized form) which is fundamental for the crossed product theory that follows, in the same way as the analogous theorem for isomorphisms is the source of the classical theory of crossed products. In ?3 we discuss the special type of restricted Lie algebra

Introduction. Let F be a field of characteristic p, and let C be a finite algebraic extension field of F such that CPCF. It has been shown in [4] that there is a one to one correspondence (up to isomorphisms) between the simple finite dimensional algebras with center F and containing C as a maximal commutative subring and the regular restricted Lie algebra extensions of C by the derivation algebra of C/F. This led to a very simple description of the group of similarity classes of these algebras. The main task of the present paper is to investigate the connection between restricted Lie algebras and simple associative algebras of characteristic p quite generally. For this purpose, we generalize the notion of a regular extension of C by the derivation algebra of C/F to that of a regular extension of a simple algebra A with center C by the derivation algebra of C/F. The correspondence mentioned above is then generalized to a one to one correspondence between these extensions and the simple algebras with center F which contain A as the commutator algebra of C (Theorem 3). We shall then show how every simple algebra containing a purely inseparable extension field of its center as a maximal commutative subring can be built up in a series of steps from regular Lie algebra extensions. With a suitable composition of regular extensions, which we define in ?2, this reduces the structure of the group of algebra classes with a fixed purely inseparable splitting field to the structural elements of restricted Lie algebras, at least in principle. However, it does not yield a direct description of this group, as was the case for splitting fields of exponent one. In ?3 we show that the tensor multiplication with a purely inseparable extension field C of F maps the Brauer group of algebra classes over F onto the Brauer group over C (Theorem 5). Although this result seems not to have been stated before, it may well be regarded as one of the culminating points of Albert's theory of p-algebras. In fact, Chapter VII of [1] contains all the essential points of a proof along classical lines, which is the first proof we give in ?3. Here, the main tools are Galois theory and the theory of cyclic p-extensions. Our second proof is entirely different. It proceeds within the framework of the first part of this paper, replacing the field theory with the theory of Lie algebras. In particular, we use the theory of restricted Lie algebra kernels, [3], in order to show that if A is a simple algebra with center C and CPC F then there always exists a regular extension of A by the derivation algebra of C/F, whence (by Theorem 3) A can be imbedded as the

simple (finite-dimensional) algebras. Since the meaning of the designation "Galois theory" has begun to change quite considerably, it may be appropriate to point out that we study the relationships between certain subalgebras of an algebra A on the one hand, and certain groups of automorphisms of A on the other. The ordinary theory of central simple algebras covers the case of subalgebras containing the center of A. Another special case, in which the subalgebras are subfields of the center, has been discussed from a somewhat different point of view by Teichmueller [7] and by Eilenberg and MacLane [4](1). Some of the recent generalizations of Galois theory to far more general rings-in particular J. Dieudonne's [3]-are concerned chiefly with Galois correspondences between certain subrings of a given ring A on the one hand, and certain rings of endomorphisms on the other. These rings of endomorphisms are intimately related to groups of automorphisms of A in the case of division rings, but in the general case this theory is quite far removed from a theory of automorphisms of A. The theory given here resembles the Galois theory for division rings as given by H. Cartan [2] and N. Jacobson [6], and covers the special case of division algebras in the same way. Some of our results are related also to those of Dieudonne [3]. However, our method is independent of the new techniques and consists simply in combining the well known results of the theory of simple algebras with those of the ordinary Galois theory for fields, a knowledge of which we shall assume(2). 1. Auxiliary results.

Let F be a field of characteristic 0, and let V be a finite dimensional vector space over F. Let E denote the algebra of all endomorphisms of V, and let L be any Lie subalgebra of E. Among the algebraic Lie algebras contained in E and containing L, there is one that is contained in all of them, and this is called the algebraic hull of L in E. Here, an algebraic Lie algebra is defined as the Lie algebra of an algebraic group. It is an easy consequence of the definitions that if A and B are algebraic groups of automorphisms of V such that A CB then the Lie algebra of A is contained in the Lie algebra of B. Hence the existence of the algebraic hull of L is an immediate consequence of the following basic result: let G be the intersection of all algebraic groups of automorphisms of V whose Lie algebras contain L. Then the Lie algebra of G contains L. This theorem reduces at once to the case where L is one dimensional. For any xCE, let G. be the intersection of all algebraic groups of automorphisms of V whose Lie algebras contain x. Then we have GZCG, whenever xCEL, and it suffices to show that the Lie algebra of G, contains x. This is part of [1, Theorem 10, p. 165], but it is not clear from the proof given in [1 ] that, although this result is not as obvious as it might seem at first sight, it can be proved quite directly without invoking any special knowledge of algebraic groups. The proof we give here is based on the simple idea of recovering G, from its generic point exp(tx). We use an auxiliary variable t over F and introduce the ring F {t of the integral power series in t with coefficients in F. Let E* denote the dual space of E, and let P be the algebra of all polynomial functions on E. The elements of E* are canonically extended to become F { t } -linear maps of E 0 FF { t } into F { t }, and the elements of P are extended accordingly to become F { t } -valued functions on E 0 FF { t }. With xCE, we interpret the formal power series exp(tx) as the element of E0FF{t} that is determined by the conditions

Introduction. The main purpose of this paper is to draw attention to certain functors, exactly analogous to the functors "Tor" and "Ext" of Cartan-Eilenberg [2], but applicable to a module theory that is relativized with respect to a given subring of the basic ring of operators. In particular, we shall show how certain relative cohomology theories for groups, rings, and Lie algebras can be subsumed under the theory of the relative Ext functor, just as (in [2]) the ordinary cohomology theories have been subsumed under the theory of the ordinary Ext functor. Among the various relative cohomology groups that have been considered so far, some can be expressed in terms of the ordinary Ext functor; these have been studied systematically within the framework of general homological algebra by M. Auslander (to appear). A typical feature of these relative groups is that they appear naturally as terms of exact sequences whose other terms are the ordinary cohomology groups of the algebraic system in question, and of its given subsystem. There is, however, another type of relative cohomology theory whose groups are not so intimately linked to the ordinary cohomology groups and exhibit a more individualized behaviour. Specifically, the relative cohomology groups for Lie algebras, as defined (in [3]) by Chevalley and Eilenberg, and the relative cohomology groups of groups, defined and investigated by I. T. Adamson [1], are of this second type. It is these more genuinely "relative" cohomology theories that fall in our present framework of relative homological algebra. Our plan here is to sketch the general features of the relative Tor and Ext functors (?2) and to illustrate some of their possible uses or interpretations by a selection of unelaborated examples. Thus, ?3 illustrates the use of the relative Ext functor in extending the cohomology theory for algebras. ?4 deals with relative homology and relative cohomology of groups, and involves both the relative Tor functor and the relative Ext functor. ?5 discusses the role played by the relative Ext functor in the cohomology theory for Lie algebras. Since this paper is intended to serve as a preliminary survey, and since the topics dealt with are supplementary to the corresponding topics of the nonrelative theory (contained in [2]), we feel justified in presupposing that the

This method is based on the Cartan-Leray spectral sequence, [3; 1 ], and can be generalized to other algebraic situations, as will be shown in a forthcoming paper of Cartan-Eilenberg [2]. Since the details of the Cartan-Leray technique have not been published (other than in seminar notes of limited circulation), we develop them in Chapter I. The auxiliary theorems we need for this purpose are useful also in other connections. In Chapter II, which is independent of Chapter I, we obtain a spectral sequence quite directly by filtering the group of cochains for G. This filtration leads to the same group E2=H(G/K, H(K)) (although we do not know whether or not the succeeding terms are isomorphic to those of the first spectral sequence) and lends itself more readily to applications, because one can identify the maps which arise from it. This is not always the case with the first filtration, and it is for this reason that we have developed the direct method in spite of the somewhat lengthy computations which are needed for its proofs. Chapter III gives some applications of the spectral sequence of Chapter II. Most of the results could be obtained in the same manner with the spectral sequence of Chapter I. A notable exception is the connection with the theory of simple algebras which we discuss in ?5. Finally, let us remark that the methods and results of this paper can be transferred to Lie Algebras. We intend to take up this subject in a later paper.

Introduction. It has been shown in a previous paper [311 that every algebra A with radical R, such that AIR is separable, is a homomorphic image of a certain maximal algebra which is determined to within an isomorphism by A/R, the A /R-module (two-sided) R/R2, and the index of nilpotency of R. Furthermore, some indication was given of how the structure of maximal algebras can be determined in simple cases. Here, we wish to give a further illustration by describing a rather wide class of maximal and primary algebras whose structure will be shown to resemble that of crossed products, in certain respects. In fact, we shall impose a certain normality condition and then trace the consequences of a few simple facts of the noncommutative Galois theory. An algebra B over the field F, with radical R, is called primary if it has an identity element and if BIR is simple. As is well known,2 B is then isomorphic with a Kronecker product FmXC, where Fm denotes the full matrix algebra of degree m over F, and where C is completely primary, in the sense that it has an identity element and that the quotient of C by its radical is a division algebra over F. We are concerned with primary algebras B for which this division algebra (which is determined to within an isomorphism by B) is normal over F, in the sense of the noncommutative Galois theory.3 This will be the case if and only if the center Z of BIR is a separable normal extension field of F and every automorphism of Z over F is induced by an automorphism of BIR. A completely primary algebra C with radical S will henceforth be called quasinormal if C/S is normal over F. If 4 is an isomorphism of Fm X C onto B then 4 maps the radical Fm XS onto the radical R of B, and B/R is isomorphic with Fm X C/S. Therefore, if C is quasinormal over F, then B/;R is automatically separable over F. By [3], B has then a maximal related extension B* which is evidently primary. Moreover, it is easily seen that B* FmXC*, where C* is the maximal related extension of C, and is quasinormal over F. Finally, the natural extension to B* of a homomorphism of C* onto C is a homomorphism of B* onto B. From these facts it is evident

Let K and L be Lie algebras over a field F. Let D(K) denote the derivation algebra of K, and I(K) the ideal consisting of the inner derivations of K. If q5 is a homomorphism of L into D(K)/I(K) then 4) defines what is called the structure of an L-kernel on K. The L-kernel K is said to be extendible if there exists a Lie algebra extension with kernel K and image L which induces the given L-kernel structure on K in the natural fashion. The L-kernels with a fixed L-module C as common center can be partitioned into equivalence classes, two kernels being equivalent if they differ (in the sense of a certain composition of kernels) by an extendible kernel. It has been shown in [2] that these equivalence classes of L-kernels constitute a vector group over F which is canonically isomorphic with the 3-dimensional cohomology group H3(L, C). In particular, every L-kernel determines a 3-dimensional cohomology class which is called the obstruction of the kernel and whose vanishing is equivalent to the extendibility of the kernel. A cohomology class u for the finite dimensional Lie algebra L in the finite dimensional L-module C is said to be effaceable if there exists a finite dimensional L-module C' containing C and such that the canonical image of u in H(L, C') is 0. It is known from [2] and [3] that every effaceable 3-dimensional cohomology class for the finite dimensional Lie algebra L in a finite dimensional L-module C is the obstruction of a finite dimensional L-kernel K with center C. Moreover, in the case of characteristic 0, it has been shown in [3] that, conversely, the obstruction of a finite dimensional kernel is effaceable. The proof depends heavily on the structure and representation theory of Lie algebras of characteristic 0 and therefore breaks down completely in characteristic pH 0. The purpose of this note is to prove this result in the case of characteristic p #0. When this is combined with the known results we have just mentioned there results the following theorem for arbitrary characteristic.

1. Let G be an algebraic linear group over a field F. If G acts by linear automorphisms on some vector space M over F we say that M is a rational G-module if it is the sum of finite-dimensional G-stable subspaces V such that the representation of G on each V is a rational representation of G. The rational G-module M is said to be rationally injective if, whenever U is a rational G-module and q5 is a G-module homomorphism of a G-submodule V of U into M, 4) can be extended to a G-module homomorphism of U into M. This notion is basic for the cohomology theory of algebraic linear groups, as developed in [I1]. If K is a normal algebraic subgroup of G it is of interest to secure the spectral sequence relation between the cohomologies of G, K and G/K. In order to do this, one needs to know that a rationally injective rational G-module is cohomologically trivial as a K-module. Although this remains unsettled over arbitrary fields, we shall see here that an even stronger result holds over fields of characteristic 0. We shall also obtain convenient criteria for injectivity that are valid over arbitrary fields.

It was proved by Jacobson [2] that every finite-dimensional Lie algebra of characteristic p has a finite-dimensional representation space which is not semisimple. In this note we prove a similar result for Jacobson's restricted Lie algebras [1]. The structure of a restricted Lie algebra L over a field F of characteristic p comprises, in addition to the ordinary Lie algebra structure, a map x-x[P' of L into itself, with formal properties corresponding to those of the map x-+xP in an associative algebra of characteristic p. Accordingly, one studies the "restricted" representations of L, in which the operator corresponding to x[P] is the pth power of the operator corresponding to x. Our result is the following.

For the suggestion that this nilpotency property of p might be secured I am indebted to Leonard Ross who used the characteristic 0 case of Theorem 1 in his proof of Ado's Theorem for graded Lie algebras (Thesis, Cohomology of graded lie algebras, University of California, Berkeley, 1964). In the case of characteristic 0, it is known that there exists a faithful finite-dimensional representation of L whose restriction to the maximum nilpotent ideal of L is nilpotent [1, pp. 202-203]. Hence, in order to establish Theorem 1 in the case of characteristic 0, it suffices to make the following observation: Let L be a finite-dimensional Lie algebra over a field of characteristic 0, and let M be a finite-dimensional L-module on which the maximum nilpotent ideal N of L is nilpotent. Let x be an element of L whose adjoint image a(x) is nilpotent. Then x is nilpotent on M. PROOF. Write L = S+R, where R is the radical of L and S is a semisimple subalgebra of L. Accordingly, write x=s+r, with s in S and r in R. Since a(x) is nilpotent, it is clear that the adjoint representation of S sends s onto a nilpotent derivation of S. Since S

Introduction. The group A (G) of all continuous and open automorphisms of a locally compact topological group G may be regarded as a topological group, the topology being defined in the usual fashion from the compact and the open subsets of G (see ?1). In general, this topological structure of A (G) is somewhat pathological. For instance, if G is the discretely topologized additive group of an infinite-dimensional vector space over an arbitrary field, then A (G) already fails to be locally compact. On the other hand, if G is a connected Lie group, we shall show without any difficulty that the compact-open topology of A (G) coincides with the topology obtained by identifying A (G) with a closed subgroup of the linear group of automorphisms of the Lie algebra of G, as was done by Chevalley (in [1]) in order to make A (G) into a Lie group. We shall then deduce that A (G) is a Lie group whenever the group of its components, G/Go, is finitely generated('), where Go denotes the component of the identity element in G. The other questions with which we shall be concerned are the following: Let I(Go) denote the group of the inner automorphisms of Go, and let E(Go, G) denote the natural image in A (Go) of A (G). Regard I(Go) and E(Go, G) as