On Homological Dimensions of <i>S</i> ‐Acts With Unique Zero

In this paper, we introduce homological dimensions of S ‐acts for a commutative monoid S with a zero element and analyze their properties. We define projective (flat) dimension by means of double‐arrow complexes and construct an injective resolution of an S ‐act. It is shown that the flat dimension of an S ‐act is a refinement of a projective dimension. For an S ‐act X and a multiplicatively closed subset T of S , we prove that the projective (flat) dimension of the localization T −1 X , as a T −1 S ‐act, is less than or equal to the projective (flat) dimension of X .