global dimension

For any ring $R$ , the left global dimension of $R$ is defined to be the supremum of projective dimensions of left modules of $R$ :

l.\operatorname{Gd}(R):=\operatorname{sup}\{\operatorname{pd}_{R}(M)\mid M% \mbox{ is a left R-module }\}.

Similarly, the right global dimension of $R$ is:

r.\operatorname{Gd}(R):=\operatorname{sup}\{\operatorname{pd}_{R}(M)\mid M% \mbox{ is a right R-module }\}.

If $R$ is commutative, then $l.\operatorname{Gd}(R)=r.\operatorname{Gd}(R)$ and we may drop $l$ and $r$ and simply use $\operatorname{Gd}(R)$ to mean the global dimension of $R$ .

Remarks.

1.

For a ring $R$ , $l.\operatorname{Gd}(R)=0$ iff $r.\operatorname{Gd}(R)=0$ (see the first example below). However, in general, $l.\operatorname{Gd}(R)$ is not necessarily the same as $r.\operatorname{Gd}(R)$ .
2.

The left (right) global dimension of a ring can also be defined in terms of injective dimensions. For example, for right global dimension of $R$ , we have: $r.\operatorname{Gd}(R)=\operatorname{sup}\{\operatorname{id}_{R}(M)\mid M\mbox% { is a right R-module }\}$ . This definition turns out to be equivalent to the one using projective dimensions.

Examples.

1.

$l.\operatorname{Gd}(R)=0$ iff $R$ is a semisimple ring iff $r.\operatorname{Gd}(R)=0$ .
2.

$r.\operatorname{Gd}(R)=1$ iff $R$ is a right hereditary ring that is not semisimple.

Title	global dimension
Canonical name	GlobalDimension
Date of creation	2013-03-22 14:51:51
Last modified on	2013-03-22 14:51:51
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	5
Author	CWoo (3771)
Entry type	Definition
Classification	msc 13D05
Classification	msc 16E10
Classification	msc 18G20
Synonym	homological dimension