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Abstract Let (S, 𝔫) (S, {n) } be a commutative noetherian local ring and let ω ∈ 𝔫 {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 ω. It is shown that these categories, which will be denoted by 𝖬𝗈𝗇 (ω, 𝒫) {Mon (, P) } and 𝖬𝗈𝗇 (ω, 𝒢) {Mon (, G) }, are both Frobenius categories with the same projective objects. It is also proved that the stable category 𝖬𝗈𝗇 ¯ (ω, 𝒫) {Mon (, P) } is triangle equivalent to the category of D-branes of type B, 𝖣𝖡 (ω) DB (), which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories 𝖬𝗈𝗇 ¯ (ω, 𝒫) {Mon (, P) } and 𝖬𝗈𝗇 ¯ (ω, 𝒢) {Mon (, G) } are closely related to the singularity category of the factor ring R = S / (ω) R=S/ ({) }. Precisely, there is a fully faithful triangle functor from the stable category 𝖬𝗈𝗇 ¯ (ω, 𝒢) {Mon (, G) } to 𝖣 𝗌𝗀 (R) {Dₒ₆ (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 𝖬𝗈𝗇 ¯ (ω, 𝒫) {Mon (, P) }, guarantees the regularity of the ring S.
Bahlekeh et al. (Mon,) studied this question.