Summary: In [24] Matsumoto associated to each shift space (also called a subshift) an Abelian group which is now known as Matsumoto's $K_0$-group. It is defined as the cokernel of a certain map and resembles the first cohomology group of the dynamical system which has been studied in for example [2], [28], [13], [16] and [11] (where it is called the dimension group). In this paper, we will for shift spaces having a certain property $\coni$, show that the first cohomology group is a factor group of Matsumoto's $K_0$-group. We will also for shift spaces having an additional property $\conii$, describe Matsumoto's $K_0$-group in terms of the first cohomology group and some extra information determined by the left special elements of the shift space. We determine for a broad range of different classes of shift spaces if they have property $\coni$ and property $\conii$ and use this to show that Matsumoto's $K_0$-group and the first cohomology group are isomorphic for example for finite shift spaces and for Sturmian shift spaces. Furthermore, the ground is laid for a description of the Matsumoto $K_0$-group as an emphordered group in a forthcoming paper.

37B10, 54H20, 19K99

shift spaces, subshifts, symbolic dynamics, matsumoto's $K$-groups, dimension groups, cohomology, special elements