# 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. 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. 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. 1.

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

2. 2.

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

Title global dimension GlobalDimension 2013-03-22 14:51:51 2013-03-22 14:51:51 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 13D05 msc 16E10 msc 18G20 homological dimension