Abstract. Let X be a resolving subcategory of an abelian category. In this paper we investigate the singularity category Dsg(X ) = D b (mod X )/K b (proj(mod X )) of the stable category X of X . We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type (A 1 ). We also generalize several results of Yoshino on totally reflexive modules.
Let X be a resolving subcategory of an abelian category. In this paper we investigate the singularity category Dsg(X ) = D b (mod X )/K b (proj(mod X )) of the stable category X of X . We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type (A 1 ). We also generalize several results of Yoshino on totally reflexive modules.
Let R be a commutative noetherian ring. Denote by D -(R) the derived category of cochain complexes X of finitely generated R-modules with H i (X) = 0 for i ≫ 0. Then D -(R) has the structure of a tensor triangulated category with tensor product − ⊗ L R − and unit object R. In this paper, we study thick tensor ideals of D -(R), i.e., thick subcategories closed under the tensor action by each object in D -(R), and investigate the Balmer spectrum Spc D -(R) of D -(R), i.e., the set of prime thick tensor ideals of D -(R). First, we give a complete classification of the thick tensor ideals of D -(R) generated by bounded complexes, establishing a generalized version of the Hopkins-Neeman smash nilpotence theorem. Then, we define a pair of maps between the Balmer spectrum Spc D -(R) and the Zariski spectrum Spec R, and study their topological properties. After that, we compare several classes of thick tensor ideals of D -(R), relating them to specialization-closed subsets of Spec R and Thomason subsets of Spc D -(R), and construct a counterexample to a conjecture of Balmer. Finally, we explore thick tensor ideals of D -(R) in the case where R is a discrete valuation ring.
Two left noetherian rings R and S are said to be singularly equivalent if their singularity categories are equivalent as triangulated categories. The aim of this paper is to give a necessary condition for two commutative noetherian rings to be singularly equivalent. To do this, we develop the support theory for triangulated categories without tensor structure.
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