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Coactions of Hopf Algebras on Algebras in Positive Characteristic

Marilena CrupiGaetana Restuccia — 2010

Bollettino dell'Unione Matematica Italiana

Let K be a field of positive characteristic p > 0 . We study the coactions of the Hopf algebra of the multiplicative group H m with underlying algebra H = K [ X 1 , , X n ] / ( X 1 p s 1 , , X n p s n ) , n 1 , s 1 s n 1 on a K -algebra A . We give the rule for the set of additive endomorphism of A , that define a coaction of H m on A commutative. For s 1 = = s n = 1 , we obtain the explicit expression of such coactions in terms of n derivations of A .

Fitting Conditions for Symmetric Algebras of Modules of Finite Projective Dimension

Cristodor IonescuGaetana RestucciaRosanna Utano — 2007

Bollettino dell'Unione Matematica Italiana

Let E be a finitely generated R -module, having finite projective dimension. We study the acyclicity of the approximation complex 𝒵 ( E ) of E in terms of certain Fitting conditions F k ( i ) on the Fitting ideals of the i -th module of a projective resolution of E . We deduce some good properties of the symmetric algebra of E .

On the symmetric algebra of certain first syzygy modules

Gaetana RestucciaZhongming TangRosanna Utano — 2022

Czechoslovak Mathematical Journal

Let ( R , 𝔪 ) be a standard graded K -algebra over a field K . Then R can be written as S / I , where I ( x 1 , ... , x n ) 2 is a graded ideal of a polynomial ring S = K [ x 1 , ... , x n ] . Assume that n 3 and I is a strongly stable monomial ideal. We study the symmetric algebra Sym R ( Syz 1 ( 𝔪 ) ) of the first syzygy module Syz 1 ( 𝔪 ) of 𝔪 . When the minimal generators of I are all of degree 2, the dimension of Sym R ( Syz 1 ( 𝔪 ) ) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.

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