manuscripta math. 27, 81 - 101 (1979) matheinatica

THE ADSEBRAIC AND GEOMETRIC CLASSIFICATION OF ASSOCIATIVE ALGEBRAS Œ DIMENSION FIVE

Guerino MAZZOIA

The scheme Alg^ of associative, unitary algebra structures œï к , к an algebraically closed field with char(k) ?^ 2 , is investigated. We establish the list of GL^-orbits on Alg^ under the action of structural transport. The number olg of irreducible ocnponents of Algc is 10 ; a list of generic structures is included. We exhibit upper and lower bounds for the asynptotic behaviour of the number alg .

In this paper we consider associative, unitary algebras over an algebraically closed field к of characteristic different frcm two, and of k-dimension five. By algebraic classification we mean the tion of the types of isomorj^tiic algebras, vÄiereas geometric tion is the problem of finding generic structural constants in the sense of algebraic gecmetry. These Ью points of view are rather independent: For instance, J. Briançon's theorem [1] on the irreducibility of the Hilbert-scheme of ideals I of colength n < <» in k[[X,Y]] iiiplies the gecmetric classification of algebras of the form k[[X,Y]]/I : the only generic structure is k[X]/(x'^ ) . However - except for small n - an algebraic classification is unknown. - On the other hand, the main effort in our program was the gecmetric classification, the algebraic one being presupposed (it should be mentioned that in the cannrautative case, algebraic classification can be skipped in low dimensions [8]).

To get the algebraic classification, we make use of published and urpublished results of P. Gabriel. For this support, I am grateful to him.

0025 - 2611 / 79 / 0027 / 0081 / $04 . 20