Abstract: Differential geometry play a fundamental role in discussing partial differential equations（PDEs） in mathematical physics. Recently discrete differential geometry is an active mathematical terrain, which aims at the development and application of discrete equivalents of the geometric notions and methods of differential geometry. In this paper, a discrete theory of exterior differential calculus and the analogue of the Poincar′e lemma for differential-difference complex are proposed. They provide an intrinsic idea for developing the theory to discuss the integrability of difference equations. 

Key words: noncommutative differential calculus, differential-difference complex, exact