1. Cone Length for DG Modules and Global Dimension of DG Algebras.
- Author
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Mao, X.-F. and Wu, Q.-S.
- Subjects
MODULES (Algebra) ,DIFFERENTIAL algebra ,COMMUTATIVE rings ,HOMOLOGY theory ,MATHEMATICAL analysis ,MATHEMATICS - 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., Aop-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 Dc(A) are not bigger than the DG free class of a minimal semifree resolution X of the DG Ae-module A. [ABSTRACT FROM AUTHOR]
- Published
- 2011
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