Abstract
This paper originates from the work of H. Hironaka on the equivalence of singularities and the work of M. Gerstenhaber on the deformations of algebras. Hironaka ([6], [7]) shows, in short, that the formal completion along the singular locus of a point in an algebraic variety is entirely determined by a finite truncation, whereas M. Gerstenhaber ([3]) shows that there is no non-trivial deformation if an appropriate cohomology group vanishes. This makes one to suspect that their works are somewhat related. Namely if an appropriate cohomology group is small enough then the whole structure may be determined by a finite datum. Now let k be a fixed commutative noetherian ring and consider a pair (A,I) where A is a commutative algebra of essentially finite type over k and I is an ideal in A. For such pair (A, I) we set A (n) IvlIv+n+l v
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