By Olga Ladyzhenskaya

Contributions are dedicated to questions of the habit of trajectories for semi-groups of nonlinear bounded non-stop operators in a in the community non-compact metric house and for suggestions of summary evolution equations.

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Let k be a eld and R a positively graded a ne k-algebra. Set n = dim R. (a) The following are equivalent for homogeneous elements x1 . . xn : (i) x1 . . xn is a homogeneous system of parameters (ii) R is an integral extension of k x1 . . xn] (iii) R is a nite k x1 . . xn ]-module. (b) There exist homogeneous elements x1 . . xn satisfying one, and therefore all, of the conditions in (a). Moreover, such elements are algebraically independent over k. (c) If R is a homogeneous k-algebra and k is in nite, then such x1 .

Ym of homogeneous elements of positive degree generating R over k. If (i) holds, then I is -primary, and there exists an e such that z = y1e ymem 2 I whenever deg z e (deg is the degree in R ). Let M be the S -submodule of R generated by those monomials z with deg z < e. We claim that R = M . In fact, every r 2 R is a k-linear combination of monomials y1e ymem , and thus it is enough that s = y1e ymem 2 M for all ei 2 N. If deg s < e, thenPs 2 M for trivial reasons. So assume deg s e. Then s 2 I , and s = ni=1 fixi with elements fi 2 R .

Thus, if P is a projective resolution of M in M0 (R ), then . ExtiR (M N ) = H i ( HomR (P N )) for all i 0. It is immediate from this de nition and the above remarks that ExtiR (M N ) = ExtiR (M N ) for Noetherian R and nite M . Nevertheless we shall use the notation ExtiR (M N ) to emphasize that these modules are graded. 9. Let R be a Noetherian graded ring, M a nite graded R-module, and 2 Supp M a non-graded prime ideal. Then depth M = depth M + 1 and r(M ) = r(M ): . p p p p p In order to compute the depths and types of M and M we may consider both modules as modules over the homogeneous localization R( ) of R with respect to .