known about the local theory and little about global problems. The present paper is an attempt to define and give some basic properties of the sheaves that are related to differential equations. Let M be a smooth domain in some euclidean space and let D be a linear partial differential operator whose coefficients are Coo on M. We define the sheaf A on M by requiring that, for each xEM, the stalk A. of A at x consists of all functionsf which are defined and CI in a neighborhood of x and satisfy Df = 0 in this neighborhood of x. We consider the cohomology groups Hi(A) of M with coefficients in A. We show that, if D has constant coefficients, or if D is elliptic or hyperbolic, then Hi(A) = 0 for j ? 2. Moreover, if 8 denotes the space of Co functions on M, then H'(A) -/D&, so H'(A) measures how many functions in 8 are not of the form Df for fE 8.