Cone Length for DG Modules and Global Dimension of DG Algebras

Abstract

As the definition of free class of differential modules over a commutative ring in [1], we define DG free class for semifree DG modules over an Adams connected DG algebra A. For any DG A-modules M, we define its cone length as the least DG free classes of all semifree resolutions of M. The cone length of a DG A-module plays a similar role as projective dimension of a module over a ring does in homological ring theory. The left (resp., right) global dimension of an Adams connected DG algebra A is defined as the supremum of the set of cone lengths of all DG A-modules (resp., A ^op -modules). It is proved that the definition is a generalization of that of graded algebras. Some relations between the global dimension of H(A) and the left (resp. right) global dimension of A are discovered. When A is homologically smooth, we prove that the left (right) global dimension of A is finite and the dimension of D(A) and D ^c (A) are not bigger than the DG free class of a minimal semifree resolution X of the DG A ^e -module A.

Key Words:

• Adams connected
• Compact DG module
• Cone length
• Differential graded algebra
• DG free class
• Global dimension

2000 Mathematics Subject Classification:

• Primary 16E45
• 16E10

ACKNOWLEDGMENTS

The authors thank Peter Jørgensen and the referees for their valuable comments and suggestions. Q.-S. Wu is supported by the NSFC (key project 10731070) and by the Doctorate Foundation (No. 20060246003), Ministry of Education of China. X.-F. Mao is supported by the National Natural Science Foundation of China (No. 11001056), by the China Postdoctoral Science Foundation (No. 20090450066), by the China Postdoctoral Science Foundation (No. 20103244), and by Key Disciplines of Shanghai Municipality (S30104).

Notes

Communicated by D. Nakano.