Changchang Xi ( ) derived categories Beijing, China - PowerPoint PPT Presentation

Noncommutative Algebraic Geometry: Shanghai Workshop 2011, Shanghai, China, September 12-16, 2011 Overview and schedule Definitions and examples Happel’s Theorem for Infinitely Generated Tilting Two results Modules Recollements Main result Stratification of Changchang Xi ( ➝ ➝ ➝ ✚ ✚ ✚⑦ ⑦ ⑦ ) derived categories Beijing, China Email: xicc@bnu.edu.cn

Overview Overview and schedule Given an infinitely generated tilting module, Definitions and examples the derived category of its endomorphism ring Two results Recollements admits a recollement by derived categories of Main result rings Stratification of derived categories Jordan-H¨ older Theorem fails for stratifications of derived module categories by derived module categories. This talk reports a part of joint works with Hongxing Chen.

Schedule Overview and schedule Definitions and examples I. Definitions and examples Two results II. Two results on tilting modules Recollements Main result III. Recollements Stratification of derived categories IV. Main result V. Stratifications of derived categories

Notations R : ring with 1 Overview and schedule R -Mod: cat. of all left R -modules Definitions and examples R -mod: cat. of f. g. left R -modules Two results M : R -module Recollements M ( I ) : Main result direct sum of I copies of M Stratification of Add ( M ) : derived full subcat. of R -Mod, categories dir. summands of M ( I ) add ( M ) : full subcat. of R -mod, dir. summands of M ( I ) , I : finite pd ( M ) : proj. dim. of M

Notations R : ring with 1 Overview and schedule R -Mod: cat. of all left R -modules Definitions and examples R -mod: cat. of f. g. left R -modules Two results M : R -module Recollements M ( I ) : Main result direct sum of I copies of M Stratification of Add ( M ) : derived full subcat. of R -Mod, categories dir. summands of M ( I ) add ( M ) : full subcat. of R -mod, dir. summands of M ( I ) , I : finite pd ( M ) : proj. dim. of M

Notations R : ring with 1 Overview and schedule R -Mod: cat. of all left R -modules Definitions and examples R -mod: cat. of f. g. left R -modules Two results M : R -module Recollements M ( I ) : Main result direct sum of I copies of M Stratification of Add ( M ) : derived full subcat. of R -Mod, categories dir. summands of M ( I ) add ( M ) : full subcat. of R -mod, dir. summands of M ( I ) , I : finite pd ( M ) : proj. dim. of M

Notations R : ring with 1 Overview and schedule R -Mod: cat. of all left R -modules Definitions and examples R -mod: cat. of f. g. left R -modules Two results M : R -module Recollements M ( I ) : Main result direct sum of I copies of M Stratification of Add ( M ) : derived full subcat. of R -Mod, categories dir. summands of M ( I ) add ( M ) : full subcat. of R -mod, dir. summands of M ( I ) , I : finite pd ( M ) : proj. dim. of M

Notations R : ring with 1 Overview and schedule R -Mod: cat. of all left R -modules Definitions and examples R -mod: cat. of f. g. left R -modules Two results M : R -module Recollements M ( I ) : Main result direct sum of I copies of M Stratification of Add ( M ) : derived full subcat. of R -Mod, categories dir. summands of M ( I ) add ( M ) : full subcat. of R -mod, dir. summands of M ( I ) , I : finite pd ( M ) : proj. dim. of M

Notations R : ring with 1 Overview and schedule R -Mod: cat. of all left R -modules Definitions and examples R -mod: cat. of f. g. left R -modules Two results M : R -module Recollements M ( I ) : Main result direct sum of I copies of M Stratification of Add ( M ) : derived full subcat. of R -Mod, categories dir. summands of M ( I ) add ( M ) : full subcat. of R -mod, dir. summands of M ( I ) , I : finite pd ( M ) : proj. dim. of M

Notations R : ring with 1 Overview and schedule R -Mod: cat. of all left R -modules Definitions and examples R -mod: cat. of f. g. left R -modules Two results M : R -module Recollements M ( I ) : Main result direct sum of I copies of M Stratification of Add ( M ) : derived full subcat. of R -Mod, categories dir. summands of M ( I ) add ( M ) : full subcat. of R -mod, dir. summands of M ( I ) , I : finite pd ( M ) : proj. dim. of M

Notations R : ring with 1 Overview and schedule R -Mod: cat. of all left R -modules Definitions and examples R -mod: cat. of f. g. left R -modules Two results M : R -module Recollements M ( I ) : Main result direct sum of I copies of M Stratification of Add ( M ) : derived full subcat. of R -Mod, categories dir. summands of M ( I ) add ( M ) : full subcat. of R -mod, dir. summands of M ( I ) , I : finite pd ( M ) : proj. dim. of M

Notations R : ring with 1 Overview and schedule R -Mod: cat. of all left R -modules Definitions and examples R -mod: cat. of f. g. left R -modules Two results M : R -module Recollements M ( I ) : Main result direct sum of I copies of M Stratification of Add ( M ) : derived full subcat. of R -Mod, categories dir. summands of M ( I ) add ( M ) : full subcat. of R -mod, dir. summands of M ( I ) , I : finite pd ( M ) : proj. dim. of M

Notations R : ring with 1 Overview and schedule R -Mod: cat. of all left R -modules Definitions and examples R -mod: cat. of f. g. left R -modules Two results M : R -module Recollements M ( I ) : Main result direct sum of I copies of M Stratification of Add ( M ) : derived full subcat. of R -Mod, categories dir. summands of M ( I ) add ( M ) : full subcat. of R -mod, dir. summands of M ( I ) , I : finite pd ( M ) : proj. dim. of M

Tilting modules Tilting modules (or tilting complexes, objects) Overview and schedule occur in Repr. Theory of Algebras. Definitions and examples Linked to: Two results Recollements Algebraic groups (Donkin’s works) Main result Stratification of Lie Theory (Irving, Cline-Parshall-Scott, derived categories Soegel) Algebraic geometry (Lenzing’s works) Modular representation theory of f. groups (Brou´ e’s conjecture)

Tilting modules Tilting modules (or tilting complexes, objects) Overview and schedule occur in Repr. Theory of Algebras. Definitions and examples Linked to: Two results Recollements Algebraic groups (Donkin’s works) Main result Stratification of Lie Theory (Irving, Cline-Parshall-Scott, derived categories Soegel) Algebraic geometry (Lenzing’s works) Modular representation theory of f. groups (Brou´ e’s conjecture)

Tilting modules Tilting modules (or tilting complexes, objects) Overview and schedule occur in Repr. Theory of Algebras. Definitions and examples Linked to: Two results Recollements Algebraic groups (Donkin’s works) Main result Stratification of Lie Theory (Irving, Cline-Parshall-Scott, derived categories Soegel) Algebraic geometry (Lenzing’s works) Modular representation theory of f. groups (Brou´ e’s conjecture)

Tilting modules Tilting modules (or tilting complexes, objects) Overview and schedule occur in Repr. Theory of Algebras. Definitions and examples Linked to: Two results Recollements Algebraic groups (Donkin’s works) Main result Stratification of Lie Theory (Irving, Cline-Parshall-Scott, derived categories Soegel) Algebraic geometry (Lenzing’s works) Modular representation theory of f. groups (Brou´ e’s conjecture)

Definitions of f. g. tilting modules R T ∈ R -mod is called a classical tilting module if Overview and schedule Definitions and (1) ∃ exact seq. in R -mod with P j proj. : examples Two results 0 → P n → ··· → P 0 → T → 0 . Recollements Main result Stratification of (2) Ext i R ( T , T ) = 0 for all i > 0 . derived categories (3) ∃ exact seq. 0 → R → T 0 → T 1 → ··· → T m → 0 , T i ∈ add ( T ) . Brenner-Butler (1979), Happel-Ringel (1982), Miyashita (1986).

General definition of tilting modules R T ∈ R -Mod is called a tilting module if Overview and schedule Definitions and examples (1) pd ( R T ) < ∞ , Two results (2) Ext i R ( T , T ( I ) ) = 0 for all sets I , i > 0 . Recollements Main result (3) ∃ exact seq. Stratification of derived categories 0 → R → T 0 → T 1 → ··· → T m → 0 , T i ∈ Add ( T ) . In 1995 by Colpi-Trlifaj, Bazzoni.

Good tilting modules Overview and schedule Definitions and examples T : tilting R -module is called good if the Two results Recollements T i ∈ add ( T ) in Main result Stratification of ( 3 ) : 0 → R → T 0 → T 1 → ··· → T m → 0 derived categories for all i .

Good tilting modules Overview and schedule Definitions and examples Relationship: Two results Recollements Classical tilting = ⇒ Good tilting = ⇒ Tilting Main result Stratification of ⇒ T ′ : = ⊕ n T : tilting = j = 0 T j is good. derived categories Note: T and T ′ have the same torsion theory in R -Mod.

Tilting modules of projective dimension one Overview and schedule From now on, in this talk, Definitions and examples By tilting modules we mean tilting modules of pd at Two results most 1 , that is, Recollements Main result Stratification of (1) pd ( R T ) ≤ 1 , derived categories R ( T , T ( I ) ) = 0 for all sets I , (2) Ext 1 (3) ∃ exact seq. 0 → R → T 0 → T 1 → 0 , T i ∈ Add ( T ) .