Frobenius Algebras And Constructions Of Equivalent Quotient Categories

In noncommutative projective algebraic geometry,we characterize the quotient category of a noetherian graded algebra by that of other algebras.In the theory of non-commutative resolutions for a noncommutative singularity,we characterize the quo-tient category of Gorenstein algebra by that of the endomorphism algebra of certain reflexive module.In chapter 2,if there exists a noetherian graded algebra A,we can construct an new algebra B by using matrices.Some equivalences between quotient categories of the noetherian graded algebra A and those of the new algebra are established.In non-commutative projective algebraic geometry,we sometimes use non-connected Koszul algebras.We prove that under some mild conditions,if A is a connected Koszul algebra,then B is a non-connected Koszul algebra.Frobenius algebras are related to different branches of mathematics.Smash prod-uct is applied to different branches of mathematics as a mathematical tool.The study of Frobenius properties of smash products will provide more examples of Frobenius algebras.In chapter 3,we make some detailed computations of lower dimensional Frobenius algebra by using the parastrophic determinant.Assume that A,B is Frobe-nius,R is a map.We analyze when the triple(A,B,R)is an R-smash product.We give some sufficient conditions on R so that A#_RB is Frobenius.