Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>𝔫</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(S,{n})} be a commutative noetherian local ring and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ω</m:mi> <m:mo>∈</m:mo> <m:mi>𝔫</m:mi> </m:mrow> </m:math> {ω∈{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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒫</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{Mon}(ω,P)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒢</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{Mon}(ω,G)} , are both Frobenius categories with the same projective objects. It is also proved that the stable category <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder accentunder="true"> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo>¯</m:mo> </m:munder> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒫</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{Mon}(ω,P)} is triangle equivalent to the category of D-branes of type B, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝖣𝖡</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {DB(ω)} , which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder accentunder="true"> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo>¯</m:mo> </m:munder> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒫</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{Mon}(ω,P)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder accentunder="true"> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo>¯</m:mo> </m:munder> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒢</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{Mon}(ω,G)} are closely related to the singularity category of the factor ring <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>R</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>S</m:mi> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {R=S/({ω)}} . Precisely, there is a fully faithful triangle functor from the stable category <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder accentunder="true"> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo>¯</m:mo> </m:munder> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒢</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{Mon}(ω,G)} to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>𝖣</m:mi> <m:mi>𝗌𝗀</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{{Dsg}}(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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder accentunder="true"> <m:mi>𝖬𝗈𝗇</m:mi> <m:mo>¯</m:mo> </m:munder> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒫</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{Mon}(ω,P)} , guarantees the regularity of the ring S .
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Bahlekeh et al. (2024) studied this question.