they are objects of $\CC_w$. On the other hand the $\L$-module $$\def\objectstyle{\scriptstyle}\def\labelstyle{\scriptstyle}
X=
\xymatrix@-1.5pc{
2\ar@{.>}[rd]&  \\
 & 1
}$$ cannot be obtained as a factor module of (a direct sum of copies) of $V_\i$, hence it does not belong to $\CC_w$.

Frobenius categories
--------------------

Let us recall some definitions from homological algebra. Let $\CC$ be a subcategory of the category of modules over an algebra $A$, which is closed under extensions. Clearly, we have $$\Ext_{\CC}^1(X,Y) = \Ext_A^1(X,Y)$$ for all modules $X$ and $Y$ in $\CC$. An $A$-module $C$ in $\CC$ is called $\CC$-[*projective*]{} (resp. $\CC$-[*injective*]{}) if $\Ext_A^1(C,X) = 0$ (resp. $\Ext_A^1(X,C) = 0$) for all $X \in \CC$. If $C$ is $\CC$-projective and $\CC$-injective, then $C$ is also called $\CC$-[*projective-injective*]{}. We say that $\CC$ has [*enough projectives*]{} (resp. [*enough injectives*]{}) if for each $X \in \CC$ there exists a short exact sequence $0 \to Y \to C \to X \to 0$ (resp. $0 \to X \to C \to Y \to 0$) where $C$ is $\CC$-projective (resp. $\CC$-injective) and $Y \in \CC$. If $\CC$ has enough projectives and enough injectives, and if these coincide (i.e. an object is $\CC$-projective if and only if it is $\CC$-injective), then $\CC$ is called a 