The Stable Monomorphism Category of a Frobenius category

For a Frobenius abelian category $\mathcal{A}$, we show that the category $\mathrm{M}\mathrm{o}\mathrm{n}(\mathcal{A})$ of monomorphisms in $\mathcal{A}$ is a Frobenius exact category; the associated stable category $\underset{‾}{\mathrm{M}\mathrm{o}\mathrm{n}}(\mathcal{A})$ modulo projective objects is called the stable monomorphism category of $\mathcal{A}$. We show that a tilting object in the stable category $\underset{‾}{\mathcal{A}}$ of $\mathcal{A}$ modulo projective objects induces naturally a tilting object in $\underset{‾}{\mathrm{M}\mathrm{o}\mathrm{n}}(\mathcal{A})$. We show that if $\mathcal{A}$ is the category of (graded) modules over a (graded) self-injective algebra $A$, then the stable monomorphism category is triangle equivalent to the (graded) singularity category of the (graded) $2\times 2$ upper triangular matrix algebra $T_2(A)$. As an application, we give two characterizations to the stable category of Ringel-Schmidmeier (\cite{RS3}).