An elementary derivation of certain homological inequalities

Extract

Let R be a ring, not necessarily commutative, with an identity element, and let A be a left R-module. We shall describe this situation by writing (_RA). If

is an exact sequence of left. R-modules and R-homomorphisms in which each P_i (i ≥ 0) is R-projective, then the sequence

which we denote by P, is called an R-projective resolution of A. Suppose now that A is non-trivial; if P_i = 0 when i > n and if there are no R-projective resolutions of A containing fewer non-zero terms, then A is said to have left projective (or homological) dimension n, and we write 1.dim _RA = n. If no finite resolutions of this type exist, we write l.dim _RA = ∞. As a convention, we put l.dim _R0 = −1. If M denotes a variable left R-module, then is called the left global dimension of the ring R and is denoted by l.gl. dim R. It is well known that l.dim _RA < n if and only if for all left R-modules B and that l.gl.dim R < m if and only if regarded as a functor of left R-modules, takes only null values.