Finding modules to check for finite global dimension

Question

Given a finite dimensional algebra $A$ and a generator-cogenerator $M$ and let $B:=End_A(M)$. $B$ has finite global dimension iff every $A$-module has finite $add(M)$-resolution dimension, which is usually hard to check.

But $B$ also has finite global dimension iff every simple module $S_i$ has finite projective dimension iff $\Omega^2(S_i)$ has finite projective dimension.

Now $\Omega^2(S)$ is in the image of the functor $F:=Hom_A(M,-)$.

Can one somehow find $A$-modules $N_i$ such that $F(N_i)=\Omega^2(S_i)$? Here I mean a more or less explicit description in terms of $M$. Having those $N_i$, $B$ has finite global dimesnion iff all $N_i$ have finite $add(M)$-resolution dimension.

Answer 1

This is a kind of Auslander–Reiten theory; for each indecomposable summand $M_i$ of $M$, you want to find a right-almost split morphism

$\varphi\colon\widehat{M}_i\to M_i$

in $\operatorname{add}{M}$. In other words, this should not be a split epimorphism, but any non-isomorphism $M_j\to M_i$ for any indecomposable summand $M_j$ of $M$ should factor through it.

Then $F\varphi$ is a projective presentation of $S_i$ (the simple top of the projective $FM_i$), and so setting $N_i=\operatorname{ker}(\varphi)$ and using left-exactness of $F$ shows that $FN_i=\Omega^2(S_i)$.

This is typically the approach used to show the equivalence of finite global dimension of $B$ and finite $\operatorname{add}(M)$-resolution dimension in $\operatorname{mod}(A)$ (and to explicitly bound $\operatorname{gldim}(B)$ by $2$ more than this resolution dimension), and it is also used by Iyama to bound the global dimension of endomorphism algebras of cluster-tilting objects in certain cases. Unfortunately, I don't know a situation where one can bound the $\operatorname{add}(M)$-resolution dimension just of the $N_i$s without just bounding it for all $A$-modules, as in these examples, since the construction doesn't make it very easy to understand these modules.

I should add that this is just a variant on the general argument that shows any finite-dimensional $B$-module (the crucial property being that it is finitely-presented) $X$ has $\Omega^2(X)=FN$ for some $N$; under Yoneda, a projective presentation of $X$ is of the form

$$FM_1\xrightarrow{F\varphi}FM_0\to X\to 0$$

for some $\varphi$, and $N=\operatorname{ker}(\varphi)$ has $FN=\Omega^2(X)$. On the other hand, in the case that $X$ is simple, it is possible to be a little bit more concrete about the properties of $\varphi$, which might be exploitable.