for a noetherian scheme, we introduce its unbounded stable derived category. this leads to recollement which reflects the passage from bounded category of coherent sheaves quotient modulo subcategory perfect complexes. some applications are included, instance an analogue maximal cohen–macaulay approximations, construction tate cohomology, and extension classical grothendieck duality. in addition, relevance modular representation theory is indicated.

Nous proposons une façon nouvelle de définir notion support pour les objets d'une catégorie avec petits coproduits, engendrée par des compacts. Cette approche est basée sur construction foncteurs cohomologie locale catégories triangulées relativement à un anneau central d'opérateurs. Comme cas particuliers, on retrouve la théorie anneaux noethériens Foxby et Neeman, d'Avramov Buchweitz locaux d'intersection complète, ou variétés représentations groupes finis selon Benson, Carlson Rickard....

We classify localising subcategories of the stable module category a finite group that are closed under tensor product with simple (or, equivalently all) modules.One application is proof telescope conjecture in this context.Others include new proofs theorem and classification thick finitely generated modules which avoid use cyclic shifted subgroups.Along way we establish similar classifications for differential graded over polynomial rings, exterior algebras.

A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action a graded commutative noetherian ring R. The utility this demonstrated by establishing diverse consequences which follow when stratified Among them are classification the localizing subcategories in terms subsets set prime ideals R; thick subcategory compact objects T; and results concerning support R-module homomorphisms Hom_T^*(C,D) leading to analogue tensor product theorem...

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules equivalent, as triangulated category, to injective modules. Restricted compact objects, this statement reinterpretation Grothendieck's duality theorem. Using equivalence it (Verdier) quotient acyclic complexes projectives by its subcategory totally and corresponding consisting are equivalent. A new characterization provided in Auslander categories Bass such rings.

Introduction The functor category Definable subcategories Left approximations duality Ideals in the of finitely presented modules Endofinite Krull-Gabriel dimension infinite radical Functors between module categories Tame algebras Rings definable scalars Reflective Sheaves hereditary Coherent rings Appendix A. Locally coherent Grothendieck B. Dimensions C. Finitely functors and ideals Bibliography.

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules equivalent, as triangulated category, to injective modules. Restricted compact objects, this statement reinterpretation Grothendieck's duality theorem. Using equivalence it (Verdier) quotient acyclic complexes projectives by its subcategory totally and corresponding consisting are equivalent. A new characterization provided in Auslander categories Bass such rings.

For an artin algebra Λ, cotorsion pairs are studied for the category Mod Λ of finitely presented Λ-modules and all Λ-modules. It is shown that every pair induces a Λ. This has some interesting applications, even modules. Another theme paper interplay between torsion pairs. leads to conjecture which analogue telescope in stable homotopy theory.

The Hom closed colocalizing subcategories of the stable module category a finite group are classified. Along way, homotopy injectives over an exterior algebra, and derived formal commutative differential graded To this end, with eye towards future applications, notion local homology cosupport for triangulated categories is developed, building on earlier work authors cohomology support.

Auslander's formula shows that any abelian category \mathsf C is equivalent to the of coherent functors on modulo Serre subcategory all effaceable functors. We establish a derived version this equivalence. This amounts showing homotopy injective objects some appropriate Grothendieck (the ind-objects ) compactly generated and full compact bounded . The same approach for an arbitrary its are well-generated triangulated categories. For sufficiently large cardinals \alpha we identify their...

Given a tilting object of the derived category an abelian finite global dimension, we give (under suitable finiteness conditions) bound for dimension its endomorphism ring.