Question

The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup over f.g. right modules of finite projective dimension.

The right global dimension of a ring is r.gl.dim$(R) =$ sup{proj.dim(M) | $M\in R$-mod}

It is clear that Findim$(R)\leq$ r.gl.dim$(R)$. According to T.Y. Lam's book Lectures on Modules and Rings, the right global dimension of $R$ is infinite iff there is some right $R$-module $M$ of infinite projective dimension. So if r.gl.dim$(R)<\infty$ then all $R$-modules have finite projective dimension and Findim$(R)=$r.gl.dim$(R)$. If r.gl.dim$(R)=\infty$ then there must be a module of infinite projective dimension but there could also be a chain of modules of finite but increasing projective dimension, i.e. big finitistic dimension can be finite or infinite. Clearly the only way to have Findim$(R)\neq$ r.gl.dim$(R)$ is to have infinite right global dim but finite big finitistic dimension.

What is an example of a ring with Findim$(R)\neq$ r.gl.dim$(R)$?

The classical example of a ring of infinite global dimension is $k[t]/(t^2)$ or really any ring with a nilpotent element. I suspect this ring has finite Findim, but I don't have a good enough sense of what $k[t]/(t^2)$-modules look like to easily pick out the ones of finite projective dimension.

If Findim$(R)=n$, can we find such an example for any $n$?

I have also seen the term "finitistic global dimension" running around. Here it is defined exactly as little finitistic dimension above.

Are these two terms always interchangeable?

Answer 1

A non-semisimple self-injective algebra like $k[t]/(t^2)$ has the property that its modules are either of infinite projective dimension or projective. So its Findim is actually zero, while its gldim is infinite.

For your second question, pick a ring $R$ with global dimension $n$ and consider the direct product ring $R\times k[t]/(t^2)$. Its global dimension is infinite, and its Findim is $n$.

Answer 2

Apologies for responding to a year-old question- I'm certainly not attempting any sort of necromancy- but I didn't see the point in making a new question when this is close to what I would like to know.

Anywho, an example of a ring $R$ with Findim $\neq$ r.gl.dim is $\mathcal{O}G$, with $\mathcal{O}$ a complete discrete valuation ring and $G$ a finite group. Since $\mathcal{O}$ is hereditary, the $\mathcal{O}G$-modules induced from $\mathcal{O}$-modules have projective dimension $\leq 1$, and the non-projective $\mathcal{O}G$-lattices (free as $\mathcal{O}$-modules) have infinite projective dimension, and it is not hard to see that no other projective dimensions are possible ($\mathcal{O}G$ is nearly self-injective).

My question is (and I strongly believe the answer to be yes): Are the only $kG$-modules (where $k$ is the residue field of $\mathcal{O}$, algebraically closed of charactistic $p$ dividing $|G|$) with finite projective dimension (as $\mathcal{O}G$-modules) the projective $kG$-modules? I know for sure that the simple modules have infinite projective dimension, and it feels like this is certainly known (if not well-known).

I should also point out that an $\mathcal{O}G$ module can have projective dimension 1, but not be induced from the trivial subgroup.

Thanks for any help!