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Let $A$ be an Artin algebra and $\mathscr{P}(A)$ the category of projective $A$-modules.

I don't know how to show the following facts:

  • All objects in $\mathscr{P}(A)$ are injective as objects of $\mathscr{P}(A)$ iff for each simple right $A$-module $S$ we have that $A$ contains a right $A$-submodule isomorphic to $S$.

  • mod-$A$ has no module of projective dimension 1 iff all objects of $\mathscr{P}(A)$ are injective as objects of $\mathscr{P}(A)$.

Thanks for the help.

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up vote 2 down vote accepted

If a module $M$ has projective dimension $1$, there is a non-split short exact sequence $0\to P\xrightarrow{i} Q\to M$ with $P$ and $Q$ projective. This, in particular, means there is no map $f:Q\to P$ such that $f\circ i=1_P$, so that $P$ is not injective in $\mathcal P(A)$.

Conversely, if there is a $P$ in $\mathcal P(A)$ which is not injective there, there is an injection $i:R\to S$ and a map $j:R\to P$ such that there is no map $g:S\to P$ such that $g\circ i=j$. This implies that the short exact sequence $0\to R\to S\to S/R\to 0$ is not-split, and therefore the projective dimension of $S/R$ is $1$.

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