In <https://www.sciencedirect.com/science/article/pii/0001870891900378> section 6 a cotilting module T over an algebra A is said to be strong in case $\hat{add(T)}$ coincides with the subcategory of modules having finite injective dimension.


Here $\hat{add(T)}$ is just the full subcategory of all module $M$ such that there is an exact sequence $0 \rightarrow T\_n \rightarrow ... \rightarrow T\_0 \rightarrow M \rightarrow 0$ with $T\_i \in add(T)$.


Let $\Gamma=End\_A(T)$.
Proposition 6.5. in this article states that T is a strong cotilting module iff every simple $\Gamma$-module is contained in T as a $\Gamma$-module.


Question: What means "every simple $\Gamma$-module is contained in T as a $\Gamma$-module"?


Im a bit confused here. For example let $A$ be an algebra of finite positive global dimension and $T=A$ (then $\Gamma=A$) with dominant dimension at least one (for example the quiver algebra of the quiver of Dynkin type $A\_2$).


Then $T$ is a strong cotilting module but soc(T)=soc(A) does not contain every simple module as a right or left module.


So I think that I have a thinking error or I use the wrong definition of "contained".



Question on strong cotilting modules
I believe I had this exact same question some time ago. Auslander and Reiten attribute this result to a paper of Auslander and Green. I still have the latter paper, but can no longer find where this result is in that paper. From my old notes, it looks like I concluded that this proposition is misstated in Auslander and Reiten's paper. Most likely, they may have meant to write "strong *tilting* module", rather than strong cotilting module. In your example of $A=kA\_2$, the strong tilting module is $DA$, and this contains all simple $A$ modules (as submodules) on both sides. Alternatively, for strong cotilting modules, I believe the proper statement is the dual that every simple $\Gamma^{op}$-module is a *quotient* of $T\_{\Gamma}$.


If you want to see a proof of the statement for strong tilting modules, see Prop 7.1 in the paper <https://arxiv.org/abs/1407.2690>


