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Let S be the polynomial ring over a field K in a finite set of variables, and let m be the graded maximal ideal of S. For a finitely generated graded Smodule M and all integers k ≫ 0, we show that m k M is componentwise linear, we describe the pattern of the Betti-diagram of m k M when M is an ideal and char(K) = 0, and show that m k M has linear quotients if M is a monomial ideal.

Let K be a field and let S = K [x 1 , . . . , x n ] be a standard polynomial ring over a field K . We characterize the extremal Betti numbers, values as well as positions, of a t-spread strongly stable ideal of S. Our approach is constructive. Indeed, given some positive integers a 1 , . . . , a r and some pairs of positive integers (k 1 , 1 ), . . . , (k r , r ), we are able to determine under which conditions there exists a t-spread strongly stable ideal I of S with β k i ,k i + i (I ) = a i , i = 1, . . . , r , as extremal Betti numbers, and then to construct it.