Almost Commutative Manifolds and Their Modular Classes

An almost commutative algebra, or a $\rho$-commutative algebra, is an algebra which is graded by an abelian group and whose commutativity is controlled by a function called a commutation factor. The same way as a formulation of a supermanifold as a ringed space, we introduce concepts of the $\rho$-commutative versions of manifolds, Q-manifolds, Berezin volume forms, and the modular classes. They are generalizations of the ones in supergeometry. We give examples including a $\rho$-commutative version of the Schouten bracket and a noncommutative torus.