The theory of homological dimensions of complexes

Summary

The object of this article is to comment on the theory of homological dimensions of bounded complexes of modules over a Noetherian commutative ring. When a module is thought of as a complex concentrated in degree zero, this theory extends parts of the theory of homological dimensions of modules; thus the idea is just to replace “modules” by “complexes of modules” whenever possible. This idea is not new. For example, it is essential, and used extensively, in the seminar notes “Residues and duality” [Ha] and “Théorie des intersections et théorème de Riemann-Roch” [SGA6].

Now let X be a bounded complex of modules over a ring A. Thus

X = O → X^r → X^r+1 → … → X^s → O.

The homological dimensions mentioned in the title are the following:

pd X, the projective dimension of X;

fd X, the flat dimension of X;

id X, the injective dimension of X;

depth X, the depth of X.

(In order that we can define the depth the ring must be local.) In addition to these there is another important dimension, namely

dim X, the Krull dimension of X.

Let me point out some reasons why one would (or could) want to study these concepts.

(1) It is possible

As an example let me define id_AX. For an integer n we say that id_AX ≤ n if there exists a quasi-isomorphism ϕ: X → I where I is a bounded complex of injective modules such that I^l = O for all l > n.