Mathematical Analysis
The present book contains three articles: "Systems of Linear Differential Equations," by V. P. Palamodov; "Fredholm Operators and Their Generalizations," by S. N. Krachkovskii and A. S. Di kanskii; and "Representations of Groups and Algebras in Spaces wi
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R.V. Gamkrelidze Editor
Progress in Mathematics
PROGRESS IN
MATHEMATICS Valurne 10
Mathematical Analysis
PROGRESS IN MATHEMATICS
Translations of Itogi Nauki- Seriya Matematika 1968: Volume 1- Mathematical Analysis Volume 2 - Mathematical Analysis 1969: Volume 3- Probability Theory, Mathematical Statistics, and Theoretical Cybernetics Volume 4 - Mathematical Analysis Volume 5 - Algebra 1970: Volume 6- Topology and Geometry Volume 7- Probability Theory, Mathematical Statistics, and Theoretical Cybernetics Volume 8- Mathematical Analysis 1971: Volume 9 - Algebra and Geometry Volume 10- Mathematical Analysis Volume 11- Probability Theory, Mathematical Statistics, and Theoretical Cybernetics In preparation: Volume 12- Algebra and Geometry Volume 13- Probability Theory, Mathematical Statistics, and Theoretical Cybernetics Volume 14 - Algebra, Geometry, and Topology
PROGRESS IN
MATHEMATICS Volume 10
Mathematical Analysis Edited by
R. V. Gamkrelidze V. A. Steklov Mathematics Institute Academy of Seiences of the USSR, Moscow
Translated from Russian by J. S. Wood
ko. It follows from the regulari ty of p that R k and g k = Ker{Rk- Rk_ 1} arevectorbundlesfor anyk>ko. Thisdefinition has fundamental significance in the theory of Spencer; all the results formulated below pertain exclusively to regular operators. For every pair (k, i) we consider the bundle R~, which is equal to the tensor 'product over C of the hmdle R k and the bundle of exterior differential forms on X of order i; the bundle g~ is constructed analogously. The first ("naive") resolvent of Spencer is the complex of sheavest (3.4)
in which D is a certain first-order differential operator. The opera"We recall the definition of the bundle Jk(E). If x E X and Ux is a sufficieutly small rreighborhood of x, therr the bundle Jk(E) over Ux is isomorphic to the right product of Ux and the space of segmems of the Taylor series to order k at x of the elements f E Ex· The automorphism of a fiber of Jk(E) urrder the transition from Ux to Uy is determined by the automorphism of a fiber of E and the rule for transformation of the coefficients of the Tayior expansion of a scalar function on X due to the corresponding change of Coordinates. The map jk sets the gerrn f E Ex in correspondence with the segments of its Taylor series at points near x. tNote that this cornplex, in general, is notaresolvent per se, i.e., 1t ts not an exact sequence (see below ).
16
V. P. PALAMODOV
tor D takes the subsheaf'gL1 into gt+l and acts accordingly as a zero-order differential operator. Corresponding to this zero-order differential Operator is a bundle map gi - gk+l , which is denoted k+l by o. Consequently, we arrive at the complex of vector bundles 8
I
0-+ gk+n-+ gk+n-1
ö ö ->- · • • -+
g:-+ 0,
(3.5)
which is called the Spencer o -sequence. Spencer in a special case, and later Quillen* in the general case, proved that the o-sequence is exact for k > k 1, where k 1 isaconstantdependingonlyonthedimension of X, E, and the order of p. Sternberg noted a connection between
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