# Different ways of having infinite global dimension

Is there any ring $R$ of infinite global dimension such that any $R$-module is a retract (i.e. direct summand) of some $\oplus_{i\in I}M_i$ where each $M_i$ has finite projective dimension?

I ask this because in the easy examples of rings of infinite global dimension I have in mind, there is always a simple $R$-module with infinite projective dimension. I wanted to know if this happens in general, or if pathological examples exist. In the situation I ask for above, the global dimension would be infinite, morally for asymptotic reasons.

• I seem to vaguely recall that by a result of Auslander the global dimension is the sup of the projective dimensions of all cyclic R-modules. This would imply what you want can't happen if I am recalling correctly – Benjamin Steinberg May 21 '15 at 17:59
• @BenjaminSteinberg yes, the projective dimension can be computed on cyclic modules, but I don't see why this answers negatively my question. What am I missing? – Fernando Muro May 21 '15 at 18:04
• If a cyclic R-modules is a retract of a direct sum then it is a retract of a finite direct sum because the splitting takes the generator into finitely many of the summands. – Benjamin Steinberg May 21 '15 at 18:09
• Since Ext commutes with direct sums you get that a summand in a finite direct sun of modules of finite projective dimension has finite projective dimension. – Benjamin Steinberg May 21 '15 at 18:10
• @BenjaminSteinberg thanks, I just didn't come up with the idea of using that f.g. modules are 'compact' w.r.t. direct sums. – Fernando Muro May 21 '15 at 18:36

In https://projecteuclid.org/download/pdf_1/euclid.nmj/1118799684 Auslander shows that the global dimension of $R$ is the supremum of the projective dimensions of cyclic modules.
If a cyclic module is a retract of a direct sum of modules of finite projective dimension, then it would be a retract of a finite direct sum of such modules and hence have finite projective dimension. So what you want can't happen: there is no such $R$.
Added for clarity: the projective dimension of a module $M$ is the largest $n$ such that $Ext^n(M,-)$ is non-zero. Since Ext commutes with direct sums a direct summand of a finite direct sum of modules of finite projective dimension has finite projective dimension.