On Amitsur's complex and restricted Lie algebras

48 Pages·1962·1.2 MB·English

by  Astrid J. Berkson

This dissertation has been 63-1267
m icrofilm ed exactly as received
BERKSON, Astrid J., 1936-
ON AMITSUR'S COMPLEX AND RESTRICTED
LIE ALGEBRAS.
Northwestern University, Ph.D., 1962
Mathematics
University Microfilms, Inc., Ann Arbor, Michigan
NORTHWESTERN UNIVERSITY
ON AMITSUR'S COMPLEX AND RESTRICTED LIE ALGEBRAS
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
for the degree
DOCTOR OF PHILOSOPHY
Field of Mathematics
by
Astrid J. Berkson
Evanston. Illinois
June 1962
ACKNOWLEDGEMENTS
With gratitude and pleasure, I wish to acknowledge
my debt to Professor A. Rosenberg for his encouragement,
guidance, and assistance in the writing of this dissertation.
TABLE OF CONTENTS
Section Page
I. Introduction.
1
IL Notations. 3
III. The first cohomology group. 5
IV. Split regular extensions. 7
V. All cohomology groups but one are aero. 25
VI. A class of Frobenius algebras. 33
Footnotes. 42
Bibliography. 43
1
SECTION I. INTRODUCTION
Given a commutative ring C and a commutative C-algebra A,
Amitsur introduced a complex which will be described in Section II. We
will denote the cohomology groups of this complex by H^A).
In Section III we prove that if C is a field, then for any C-
algebra A, H*(A) = 0. It was shown in [9, Theorem 1, p. 330] and
[ 1, Theorem . 1, p. 99] that if C is a field and A is a finite normal
6
separable extension field with Galois group G, then H^A) is isomorphic
to H ^G, A - 1 0 j ), the n cohomology group of the Galois group of
**1
A over C with coefficients in the m ultiplicative group A - i.0J. Sec­
tion 111 thus gives a generalisation of H ilbert's Theorem 90.
If F is a separable field extension of C, the groups ^ ( F )
have been described in [9] • However, in the inseparable case no gen­
eral results on H ^F ) were known. Our main results deal with that case.
Let C be an im perfect field of characteristic p and F a finite
purely inseparable extension field of exponent one. Hochschild ( 7 ]
introduced the notion of a regular restricted Lie algebra extension of F
by T, where T is the restricted Lie algebra of derivations of F over
C. He also showed that the group of regular restricted Lie algebra
extensions of F by T is equivalent to the Brauer group of F over C.
In [ 9 ] Rosenberg and Zelinsky introduced for each Amitsur 2- cocycle
t of F a regular restricted extension of F by T, denoted by 0 (t),
_
2
and showed that under the correspondence induced by t -•■0(t}, H (F)
is isomorphic to the group of extensions introduced by Hochschild. In
Section IV we define regular restricted Lie algebra extensions of M by
T where M is an abelian restricted Lie algebra satisfying some addi­
tional hypotheses. We also define for each n a restridjbd Lie algebra Kn
which serves as an M. We then show that for certain classes of kernels,
including the K° for n > 2, any regular restricted Lie algebra extension
of the kernels by T must split. In Section V, by a definition sim ilar to
0
that of the (t), we define for each n - cocycle t of the Amitsur com­
0n(O
plex, a restricted Lie algebra extension °f Kn by T. This ex­
tension is regular in the sense of Section IV and has the property that if
Q n(t) splits then t must be a coboundary. Then we use the results of
Section IV to prove that for F as above and n > 2, h V ) = .
0
Finally, in Section VI, we show that the restricted universal
associative algebra of a finite restricted Lie algebra is a Frobeniua
algebra.
3
SECTION II . NOTATIONS
Let A be a commutative C-algebra with unit, where C is a field.
Let An = A m ... e A, (n factors). We define C-algebra mono-
C W
morphisms C ^ : A*1-*- An+l by
L
£ .(a. • ... e a ) = a, • ... e a. . e l e a . « . . . ea for i= l,. . . . n+ .
1
i n i i n
1 - 1
n*
As usual we let A be the multiplicative group of units of the
algebra An. We define
A : An* .n-f *
1
An A -*■ A
by
AJx) = Cx (x) c (x"l) ... ^ (x*1)
2 n +1
Clearly A^ is a homomorphism and it was shown in [ 1. Theorem 5.1.
p. 93] by Amitsur that the sequence of groups and homomorphisms
(A , A ) form a complex, i. e. A A = 1 . We shall call this the
n n n
+1
Amitsur complex over A and denote it by Q(A). The cohomology group
Ker(A ) / will be denoted by H^A). For the significance of
'lm (A )
0 +1
n
these groups see [ I ] and [ 9 ].
We shall have use for a variant of Q(A). Let £ n * An— A°+*
be given by
^ n * L l~ Z Z + -
^n *
+1
It is clear that £ is an additive homomorphism. It was proved in [9.
n
Lemma 4. 1, p. 347] that the sequence (An, & ) of additive groups and
homomorphisms is exact. This is easily seen:
Let X be a C-linear map of A onto C such that *f (1) = 1.
Then define s ‘ An-*An * by
n '
s (a, • ... e a ) = [ (t.)a , t ... • a and s. = .
0
n ' n n
1 1 2 0
Since s . . f ... = t .s for i ^ 1, and • is the identity
n tl x+i x n n+i i
n .,.
+1
map, then s .. C, + C , s = ■ . , £ , + (- )
r n ° n d n-l n n i ' 1' n x
+1 +1 1 »2 +1
4^- l+l
+ i= 1 (-1) € mI n* = the identity map, so that a n is a contracting
homotopy.
Since the operator £ = £ ^ - ... + £ n+j can be
2
thought of as 1 • £ . for the complex A •_ A and since • _ is
n- x w Wi
an exact functor, it follows that the sequence (An, £ ) is exact also.
n
SECTION III. THE FIRST COHOMOLOGY GROUP
Let A be a commutative C*algebra with unit. We shall show that
H*(A) « 0. To begin with we prove
Lemma 3.1 Let V and W be vector spaces over the field C. Then if
x * 0 ia an element of V V/, we have
X = V. • w.
i
1 1
where the v^ and w. are linearly independent in V and W respectively
(cf. [ 2, $ Exercise ] ).
1 6
Proof. From the definition of V W we know that every element
n
of V sr W can be written as -ZT f • g , with f. in V and g. in W. We
V i~l
1 1 1 1
define s ® rank of x, as the sm allest positive integer such that x is the
a
sum of s tensor products f • g. Then if x has rank af x = ZZ v. e w, .
i=l * i
Suppose the were linearly dependent. Then we could write
v * c. v. with c. in C.
r i^ r i i i
But this would make x = ■£— v. e(w. + c. w ) of rank less than s, con*
i* r i ' i i r
tradicting the definition of s. Similarly, the w^'s m ust be linearly
independent.
Theorem 1 Let C be a field and let A be any commutative C*algebra
with unit. Then H*(A) » 0.
2*
Proof. Let x be a cocycle of rank s in A . We may then write
x * ZT v^ • with the v^, w^ elements of A and the seta f v^ j and £
linearly independent over C. Since x is a cocycle we have ^(x) ■ 1 , or
* 8 8
• v. • w.) (£ v. e w. e ) b ZZ v, • • w,
( 1 1 1
* X J J 1
which can be written as
(3. 1) a <Vj wj - y) • w. = 0 (Kronecker delta).
Since fo rfa ^ any basis of A, the elements • a^ • w ^are linearly
Independent over C [ 2, § Corollary 2 to Proposition 7, p. 11* used
1
tw ice], then if » : ^ * we have that
2
v “k , w i *
Hence each c ^ must be 0* and so v^Wj - ^ ^ « 0. Thus each v^ is a
unit with inverse w^. But i*j implies v^ b 0, hence s « 1 and
x s v e w with v w b l. But Aj(w) b (1 e w)(w e 1)’ * e v e w = x, so
that H*(A) b 0.
In case A is a finite normal separable extension field with Galois
group G. we have H*(A) b H*(G,A*)« [9* Theorem 1, p. 330] and [ 1,
Theorem . 1, p. 99] * so that our Theorem 1 is a generalisation of
6
H ilbert's Theorem 90.