The stable monomorphism category of a Frobenius category

Abstract

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 ([33]).

Recollements of Singularity Categories and Monomorphism Categories

“…( [31] and see [9] for n = 2 case). In particular, if is a self-injective algebra, then Theorem 3.6.…”

“…We refer the following direct consequence to Chen [9]. With T n we denote the generalized lower n × n triangular matrix over the algebra .…”

On derived equivalences of lines, rectangles and triangles

“…The relevance of these algebras to other branches of representation theory is demonstrated by the result of Kussin, Lenzing and Meltzer [18], who have recently shown a relation between the 'lines' and the categories of coherent sheaves on weighted projective lines, through the notion of the stable category of vector bundles [20]. In addition, as shown in the same work, and also recently in [7], the stable categories of submodules of nilpotent linear maps studied by Ringel and Schmidmeier [26] are equivalent to bounded derived categories of certain 'rectangles'.…”

“…Quivers with relations of four derived equivalent algebras. Starting at the upper left and going clockwise: the triangle Aus(kA7) with linear orientation on A7; Aus(kA7) with a symmetric orientation on A 7; kA7 ⊗ k kA3 with a symmetric orientation (on A7); and the rectangle kA 7 ⊗ k kA3 (with linear orientation).…”

Auslander–Reiten translations in monomorphism categories

“…Recently, after the deep and systematic work of C. M. Ringel and M. Schmidmeier ( [13][14][15]), the monomorphism category receives more attention. X. W. Chen [5] shows that H. Lenzing, and H. Meltzer [9] establish a surprising link between the stable submodule category with the singularity theory via weighted projective lines of type .2; 3; p/. In [19], S n .X/ is studied for any full subcategory X of A-mod, and it is proved that for a cotilting A-module T , there is a cotilting T n .A/-module m.T / such that S n .…”