EXTENSIONS TO FILTRATIONS OF SAMUEL NUMBERS ASSOCIATED WITH IDEALS

We prove the existence of the generalized Samuel number w(f)(g)BAR and the equality w(f)(g)BAR = lim(n-->infinity) n w(I)n(g)BAR = lim(n-->infinity) 1/n w(f)(J(n))BAR for any AP filtration f = (I(n)) and any filtration g = (J(n)) on a ring A. It is shown that w(f)(g)BAR greater-than-or-equal-to v(f)(g)BAR for all AP filtrations f and g where f is separated and nonnilpotent. Two real numbers a(f)(g)BAR and b(f)(g)BAR are introduced. It is shown that a(f)(g)BAR = v(f)(g)BAR (resp b(f)(g)BAR = w(f)(g))BAR if v(f)(g)BAR exists (resp if w(f)(g)BAR exists). Several properties of numbers a(f)(g)BAR and b(f)(g)BAR are given. It follows a generalization of Theorem 5.6) and a generalization of ([6], Theorem 2) given by the formula a(f)(g+h)BAR = min (a(f)(g)BAR, a(f)(h))BAR where f, g, h are filtrations on a ring A.