On the Finite F-representation type and F-signature of hypersurfaces

Abstract

Let S = K[x 1 , ..., x n ] or S = K[[x 1 , ..., x n ]] be either a polynomial or a formal power series ring in a finite number of variables over a field K of characteristic p > 0 with [K : K p ] < ∞. Let R be the hypersurface S/f S where f is a nonzero nonunit element of S. If e is a positive integer, F e * (R) denotes the R-algebra structure induced on R via the e-times iterated Frobenius map ( r → r p e ). We describe a matrix factorizations of f whose cokernel is isomorphic to F e * (R) as R-module. The presentation of F e * (R) as the cokernel of a matrix factorization of f enables us to find a characterization from which we can decide when the ring S [[u, v]]/(f + uv) has finite F-representation type (FFRT) where S = K[[x 1 , ..., x n ]]. This allows us to create a class of rings that have finite F-representation type but not finite CM type. For S = K[[x 1 , ..., x n ]], we use this presentation to show that the ring S[[y]]/(y p d + f ) has finite F-representation type for any f in S. Furthermore, we prove that S/I has finite F-representation type when I is a monomial ideal in eitherFinally, this presentation enables us to compute the F-signature of the rings S [[u, v]]/(f + uv) and S[[z]]/(f + z 2 ) where S = K[[x 1 , ..., x n ]] and f is a monomial in the ring S. When R is a Noetherian ring of prime characteristic that has FFRT, we prove that R[x 1 , ..., x n ] and R[[x 1 , ..., x n ]] have FFRT. We prove also that over local ring of prime characteristic a module has FFRT if and only it has FFRT by a FFRT system. This enables us to show that if M is a finitely generated module over Noetherian ring R of prime characteristic p, then the set of all prime ideals Q such that M Q has FFRT over R Q is an open set in the Zariski topology on Spec(R).when I was studying instead of playing with them. Thanks for putting up with me.A special thanks goes to Prof. Salah-Eddine Kabbaj, Prof. Abdeslam Mimouni and Prof. Jawad Abuihlail for the courses in commutative algebra I took with them at King Fahd University of Petroleum and Minerals of Saudi Arabia. Thanks for their encouragement for me to pursue my PhD. I would like to also thank Prof. Craig Huneke for the other courses in commutative algebra that I took with him at the University of Kansas while I was studying my master degree.I am very grateful to King Khalid University of Saudi Arabia for funding my studies abroad. Without their financial support, this would not have been accomplished.Last but not least, I am so grateful for my best friend Nabil Alhakamy for his constant support and encouragement throughout my journey. I would always remember his supportive friendship. I must also thank my classmate Mehmet Yesil for being a nice friend. I would like to thank my friends in Sheffield for the treasuring moments we shared together. I find myself lucky of having them during my stay in Sheffield.5 On the FFRT over hypersurfaces 5.1 The presentation of F e * (S/f S) as a cokernel of a Matrix Factorization of f .