Connected Differential Graded Algebra Cohomology Nature

Differential graded(DG for short)algebras occur naturally in commutative al-gebra,algebraic topology and algebraic geometry.As an important algebraic tool, DG algebra turns out to be very important.This makes it urgent to develop system-atically a theory of DG homological algebra.More and more research works appear in literature in recent years.In this dissertation,we concentrate on the homological properties of DG mod-ules over connected DG algebras and study various homological invariants.The theory of homological dimensions is one of the central parts of homological algebra.Some algebraists have considered to generalize it to differential graded case. Inspired by the recent research on the homological dimensions for DG algebras,we study homological invariants of DG modules over a connected DG algebra follow-ing Frankild-Jorgenson.Some homological identities,such as Cochain Auslander-Buchsbaum formula and Cochain Bass formula,are proved for compact DG modules over a connected DG algebra.The amplitude of a complex is defined to be the length of its cohomology.For compact DG modules over a connected DG algebra,we prove an equality,which is similar to the Amplitude inequality in[Jo5].By this equality,we prove that the amplitude of a compact DG module M over a cohomologically bounded DG algebra A is just the sum of the amplitude of A and the projective dimension of M.This yields the unboundedness of the cohomology of non-trivial(A(?)k)regular DG algebras.Many results in the rich theory of commutative Gorenstein rings have been generalized to DG algebras.In this dissertation,we define the Gorenstein condi-tions for connected DG algebras as in[JF3].We generalize some results on simply connected Gorenstein DG algebras in[FIJ]to connected cases.For cohomologically finite dimensional connected DG algebra A,we investigate the relations between the Gorenstein conditions of A and the existence of Auslander-Reiten triangles in D^c(A).When A is a non-trivial,cohomologically finite dimensional regular DG al-gebra,H(A)is not bounded above.We prove that D^c(A)doesn't admit Auslander-Reiten triangles.For an arbitrary regular DG algebra A,we turn to investigate the existence of Auslander-Reiten triangles in D_lf^b(A).We find that it relates closely with the Gorenstein conditions of A.For example,any Koszul regular DG algebra is Gorenstein if and only if its derived category of DG modules with finite dimensional cohomology admits Auslander-Reiten triangles.Many people have interests on a reasonable definition of global dimension for DG algebras.In this paper,we define the cone length for DG modules.The left (respectively,right)global dimension of a DG algebra A is defined to be the supre-mum of cone lengthes of all DG A-modules(respectively,DG A^op-modules).The global dimension of a connected graded algebra equals to the global dimension of it viewed as a DG algebra with zero differential.So our definition is a kind of DG generalization of the global dimension of connected graded algebras.In some cases, we find that the left(right)global dimension of a connected DG algebra A has a close relation with the global dimension of H(A).When the connected DG algebra A is regular,we prove that A has finite left and right global dimension.Finally,we prove that the left global dimension of a DG algebra A is larger than or equal to the dimension of the triangulated categories D(A)and D^c(A).Hence D(A)and D^c(A) has finite dimension when A is regular.