Direct limit closure of Serre subcategories

Question

Let $C$ be a Grothendieck category and $T$ a Serre subcategory of $C$. Let $\tilde{T}$ be the full subcategory of $C$ consisting of all direct limits of objects in $T$. Is $\tilde{T}$ a Serre subcategory of $C$?

Any comment is welcome.

Answer 1

Not necessarily.

For example, let $k$ be a field, and $R$ the $k$-algebra $k\oplus V$, where $V$ is an infinite dimensional square-zero ideal. Let $C$ be the category of $R$-modules and $T$ the Serre subcategory of finite dimensional $R$-modules.

Then there is a short exact sequence $$0\to V\to R\to k\to0$$ where $V$ and $k$ are in $\tilde{T}$, but $R$ is not, since every map from a finite dimensional module to $R$ maps into $V$, so $R$ is not a direct limit of finite dimensional modules.