Differential Graded Frobenius Algebars

In the study of non-commutative resolutions of singularities,it is necessary to investigate the differential structure of graded Frobenius algebras.Then,Koszul duality theory is used to transform the problem of non-commutative singularities into the problem of studying the categories of differential graded module over Frobenius algebras.Therefore,the concept of differential graded Frobenius algebras is introduced in this paper.The first chapter of this paper is the introduction,which mainly explains the research background,relevant preparatory knowledge,and summarizes the main results of this paper.In the second chapter,we introduce the definition of differential graded Frobenius algebra and provide three equivalent conditions for a differential graded algebra to be Frobenius based on the four equivalent conditions of Frobenius algebra given in the paper[19].In chapter 3,we study how to construct the differential graded algebra structures on the third-order exterior algebra,and further obtain the necessary and sufficient conditions for the third-order exterior algebra to be a differential graded Frobenius algebra.An isomorphic classification of differential graded Frobenius structures on third-order exterior algebra is given.In chapter 4,we generalize the differential graded Frobenius algebra structures on the third-order exterior algebra,and obtain the conditions that for a graded algebra generated from three dimensional linear space to be a differential graded Frobenius algebra.In chapter 5,we proved that the coalgcbra generated by a superpotential which is a coalgebra is a differential graded subcoalgebra of the corresponding differential tensor coalgebra,and the dual spaces of which can form differential graded Frobenius algebras.The dual coalgebrad of a differential graded Frobenius algebra generated by a finite dimensional space is a differential graded coalgebra.