The stable monomorphism category of a Frobenius category

For a Frobenius abelian category A, we show that the category Mon(A) of monomorphisms in A is a Frobenius exact category; the associated stable category Mon(A) modulo projective objects is called the stable monomorphism category of A. We show that a tilting object in the stable category A of A modulo projective objects induces naturally a tilting object in Mon(A).We show that if 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 × 2 upper triangular matrix algebra T 2 (A).As an application, we give two characterizations to the stable category of Ringel-Schmidmeier.