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Abstract Let (S, π«) (S, {n) } be a commutative noetherian local ring and let Ο β π« {n} be non-zerodivisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective S -modules, such that their cokernels are annihilated by Ο. It is shown that these categories, which will be denoted by π¬ππ β’ (Ο, π«) {Mon (, P) } and π¬ππ β’ (Ο, π’) {Mon (, G) }, are both Frobenius categories with the same projective objects. It is also proved that the stable category π¬ππ Β― β’ (Ο, π«) {Mon (, P) } is triangle equivalent to the category of D-branes of type B, π£π‘ β’ (Ο) DB (), which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories π¬ππ Β― β’ (Ο, π«) {Mon (, P) } and π¬ππ Β― β’ (Ο, π’) {Mon (, G) } are closely related to the singularity category of the factor ring R = S / (Ο) R=S/ ({) }. Precisely, there is a fully faithful triangle functor from the stable category π¬ππ Β― β’ (Ο, π’) {Mon (, G) } to π£ ππ β‘ (R) {Dββ (R) }, which is dense if and only if R (and so S) are Gorenstein rings. Particularly, it is proved that the density of the restriction of this functor to π¬ππ Β― β’ (Ο, π«) {Mon (, P) }, guarantees the regularity of the ring S.
Bahlekeh et al. (Mon,) studied this question.
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