The stable category of monomorphisms between (Gorenstein) projective modules with applications

Abstract 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,{\mathfrak{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> {\omega\in{\mathfrak{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> {{\mathsf{Mon}}(\omega,\mathcal{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> {{\mathsf{Mon}}(\omega,\mathcal{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> {\underline{\mathsf{Mon}}(\omega,\mathcal{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> {\mathsf{DB}(\omega)} , 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> {\underline{\mathsf{Mon}}(\omega,\mathcal{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> {\underline{\mathsf{Mon}}(\omega,\mathcal{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> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0317_eq_013