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

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Noncommutative Algebraic Geometry: Shanghai Workshop 2011, Shanghai, China, September 12-16, 2011 Overview and schedule Definitions and examples Happels Theorem for Infinitely Generated Tilting Two results Modules Recollements Main

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Noncommutative Algebraic Geometry: Shanghai Workshop 2011, Shanghai, China, September 12-16, 2011

Happel’s Theorem for Infinitely Generated Tilting Modules Changchang Xi (➝

➝ ➝✚ ✚ ✚⑦ ⑦ ⑦)

Beijing, China

Email: xicc@bnu.edu.cn

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Overview Given an infinitely generated tilting module, the derived category of its endomorphism ring admits a recollement by derived categories of rings Jordan-H¨

  • lder Theorem fails for stratifications
  • f derived module categories by derived

module categories.

This talk reports a part of joint works with Hongxing Chen.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Schedule

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

Notations

R :

ring with 1

R-Mod:

  • cat. of all left R-modules

R-mod:

  • cat. of f. g. left R-modules

M: R-module M(I) :

direct sum of I copies of M

Add(M) :

full subcat. of R-Mod,

  • 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
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SLIDE 5

Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Notations

R :

ring with 1

R-Mod:

  • cat. of all left R-modules

R-mod:

  • cat. of f. g. left R-modules

M: R-module M(I) :

direct sum of I copies of M

Add(M) :

full subcat. of R-Mod,

  • 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
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SLIDE 6

Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Notations

R :

ring with 1

R-Mod:

  • cat. of all left R-modules

R-mod:

  • cat. of f. g. left R-modules

M: R-module M(I) :

direct sum of I copies of M

Add(M) :

full subcat. of R-Mod,

  • 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
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SLIDE 7

Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Notations

R :

ring with 1

R-Mod:

  • cat. of all left R-modules

R-mod:

  • cat. of f. g. left R-modules

M: R-module M(I) :

direct sum of I copies of M

Add(M) :

full subcat. of R-Mod,

  • 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
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SLIDE 8

Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Notations

R :

ring with 1

R-Mod:

  • cat. of all left R-modules

R-mod:

  • cat. of f. g. left R-modules

M: R-module M(I) :

direct sum of I copies of M

Add(M) :

full subcat. of R-Mod,

  • 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
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SLIDE 9

Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Notations

R :

ring with 1

R-Mod:

  • cat. of all left R-modules

R-mod:

  • cat. of f. g. left R-modules

M: R-module M(I) :

direct sum of I copies of M

Add(M) :

full subcat. of R-Mod,

  • 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
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SLIDE 10

Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Notations

R :

ring with 1

R-Mod:

  • cat. of all left R-modules

R-mod:

  • cat. of f. g. left R-modules

M: R-module M(I) :

direct sum of I copies of M

Add(M) :

full subcat. of R-Mod,

  • 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
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SLIDE 11

Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Notations

R :

ring with 1

R-Mod:

  • cat. of all left R-modules

R-mod:

  • cat. of f. g. left R-modules

M: R-module M(I) :

direct sum of I copies of M

Add(M) :

full subcat. of R-Mod,

  • 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
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SLIDE 12

Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Notations

R :

ring with 1

R-Mod:

  • cat. of all left R-modules

R-mod:

  • cat. of f. g. left R-modules

M: R-module M(I) :

direct sum of I copies of M

Add(M) :

full subcat. of R-Mod,

  • 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
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SLIDE 13

Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Notations

R :

ring with 1

R-Mod:

  • cat. of all left R-modules

R-mod:

  • cat. of f. g. left R-modules

M: R-module M(I) :

direct sum of I copies of M

Add(M) :

full subcat. of R-Mod,

  • 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
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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Tilting modules Tilting modules (or tilting complexes, objects)

  • ccur in Repr. Theory of Algebras.

Linked to: Algebraic groups (Donkin’s works) Lie Theory (Irving, Cline-Parshall-Scott, Soegel) Algebraic geometry (Lenzing’s works) Modular representation theory of f. groups (Brou´ e’s conjecture)

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Tilting modules Tilting modules (or tilting complexes, objects)

  • ccur in Repr. Theory of Algebras.

Linked to: Algebraic groups (Donkin’s works) Lie Theory (Irving, Cline-Parshall-Scott, Soegel) Algebraic geometry (Lenzing’s works) Modular representation theory of f. groups (Brou´ e’s conjecture)

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Tilting modules Tilting modules (or tilting complexes, objects)

  • ccur in Repr. Theory of Algebras.

Linked to: Algebraic groups (Donkin’s works) Lie Theory (Irving, Cline-Parshall-Scott, Soegel) Algebraic geometry (Lenzing’s works) Modular representation theory of f. groups (Brou´ e’s conjecture)

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Tilting modules Tilting modules (or tilting complexes, objects)

  • ccur in Repr. Theory of Algebras.

Linked to: Algebraic groups (Donkin’s works) Lie Theory (Irving, Cline-Parshall-Scott, Soegel) Algebraic geometry (Lenzing’s works) Modular representation theory of f. groups (Brou´ e’s conjecture)

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Definitions of f. g. tilting modules

RT ∈ R-mod is called a classical tilting module if

(1) ∃ exact seq. in R-mod with Pj proj. :

0 → Pn → ··· → P0 → T → 0.

(2) Exti

R(T,T) = 0 for all i > 0.

(3) ∃ exact seq.

0 → R → T0 → T1 → ··· → Tm → 0, Ti ∈ add(T).

Brenner-Butler (1979), Happel-Ringel (1982), Miyashita (1986).

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

General definition of tilting modules

RT ∈ R-Mod is called a tilting module if

(1) pd(RT) < ∞, (2) Exti

R(T,T(I)) = 0 for all sets I, i > 0.

(3) ∃ exact seq.

0 → R → T0 → T1 → ··· → Tm → 0, Ti ∈ Add(T).

In 1995 by Colpi-Trlifaj, Bazzoni.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Good tilting modules

T: tilting R-module is called good if the Ti ∈ add(T) in (3) : 0 → R → T0 → T1 → ··· → Tm → 0

for all i.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Good tilting modules Relationship: Classical tilting =

⇒ Good tilting = ⇒ Tilting T: tilting = ⇒ T′ := ⊕n

j=0Tj is good.

Note: T and T′ have the same torsion theory in R-Mod.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Tilting modules of projective dimension one From now on, in this talk, By tilting modules we mean tilting modules of pd at most 1, that is, (1) pd(RT) ≤ 1, (2) Ext1

R(T,T(I)) = 0 for all sets I,

(3) ∃ exact seq. 0 → R → T0 → T1 → 0,

Ti ∈ Add(T).

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Examples All tilting modules in the sense of Brenner-Butler (1979), Happel-Ringel (1982).

T := Z⊕ Q/Z: tiling Z-module.

(Angeleri-H¨ ugel + Sanchez):

R → S: injective ring epi, pd(RS) ≤ 1, = ⇒ T := S⊕ S/R is tilting R-module.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Examples All tilting modules in the sense of Brenner-Butler (1979), Happel-Ringel (1982).

T := Z⊕ Q/Z: tiling Z-module.

(Angeleri-H¨ ugel + Sanchez):

R → S: injective ring epi, pd(RS) ≤ 1, = ⇒ T := S⊕ S/R is tilting R-module.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Examples All tilting modules in the sense of Brenner-Butler (1979), Happel-Ringel (1982).

T := Z⊕ Q/Z: tiling Z-module.

(Angeleri-H¨ ugel + Sanchez):

R → S: injective ring epi, pd(RS) ≤ 1, = ⇒ T := S⊕ S/R is tilting R-module.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Happel’s Theorem or Happel-Cline-Parshall-Scott Theorem Two of many beautiful results in tilting theory.

Theorem

T :

  • f. g. tilting R-module ( equiv.ly, classical)

S : = EndR(T), = ⇒ D(R) ∼ D(S) (as triang. cat.s). D(R) : the unbounded derived cat. of R-Mod

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Happel’s Theorem or Happel-Cline-Parshall-Scott Theorem Two of many beautiful results in tilting theory.

Theorem

T :

  • f. g. tilting R-module ( equiv.ly, classical)

S : = EndR(T), = ⇒ D(R) ∼ D(S) (as triang. cat.s). D(R) : the unbounded derived cat. of R-Mod

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Comments on f. g. tilting modules Positive aspect: Invariants of der. equivalences. Negative aspect: f. g. tilting modules will NOT provide us new der. categories.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Bazzoni’s Theorem

Theorem

T :

good tilting R-module,

S : = EndR(T), j! : = T ⊗L

S −, the left total der. functor.

= ⇒ Ker(j!)

D(S)

j!

  • D(R)
  • .

Ker(j!) = 0 ⇐ ⇒ T : classical.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Definition of recollements

Beilinson, Bernstein and Deligne (1981):

D′,D,D′′ : triang. cat.s. D : recollement of D′ and D′′ if ∃ 6 triangle

functors:

D′′ i∗=i! D

j!=j∗

  • i!
  • i∗
  • D′

j∗

  • j!
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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Definition of reollements 4 adjoint pairs, 3 fully faithful functors, 3 zero-compositions, 2 extension properties: for C ∈ D, ∃ triangles in D:

i!i!(C) − → C − → j∗j∗(C) − → i!i!(C)[1] j!j!(C) − → C − → i∗i∗(C) − → j!j!(C)[1].

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Back to Bazzoni’s Theorem and questions

T : good tilting R-module, S := EndR(T). Then: ∃ Recollement Ker(j!)

D(S)

j!

  • D(R)
  • .

Question: (1) What is Ker(j!) ? (2) Can it be D(R′) for some ring R′ ?

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Back to Bazzoni’s Theorem and questions

T : good tilting R-module, S := EndR(T). Then: ∃ Recollement Ker(j!)

D(S)

j!

  • D(R)
  • .

Question: (1) What is Ker(j!) ? (2) Can it be D(R′) for some ring R′ ?

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Universal localizations

R,S : rings with 1. S: universal localization of R if

(1) ∃ Σ = {f : P1 −

→ P0 | Pi f.g. proj. R-mod.s},

(2) ∃ λ : R → S: ring hom. s. t. S⊗R f is iso. for

f ∈ Σ, and

(3) λ is universal with (2).

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Main result

Theorem

T :

good tilting R-module

S : = End(RT), j! := T ⊗L

S −

= ⇒ ∃ ring epi S → U, recollement: D(U)

D(S)

j!

  • D(R)
  • .

Note: U is universal localization of S.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Corollary of the main result

Corollary

R → S : inj. ring epi., pd(RS) ≤ 1,

TorR

1(S,S) = 0,

T := R⊕ S/R, B := EndR(T). = ⇒ ∃ recollement: D(S⊔R S′)

D(B)

  • D(R)
  • .

S′ := EndR(S/R), S⊔R S′: coproduct of S and S′ over R.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Definition of stratifications For groups: Exact sequences ⇒ simple groups, composition series For derived categories: Recollements ⇒ derived simple categories, stratifications

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Definition of der. simple categories

D(R): derived simple if there is no non-trivial

recollement of the form

D(R1)

D(R)

  • D(R2)
  • ,

Ri: rings.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Definition of stratifications of der. module categories A stratification of D(R) is a series of reollements:

D(R1)

D(R)

  • D(R2)
  • ,

D(R11)

D(R1)

  • D(R12)
  • ,

D(R21

D(R2)

  • D(R22)
  • ,

and so on, s.t. all Ri,Rij,··· , are der. simple.

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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Question Jordan-H¨

  • lder Theorem: For a finite group, all

composition series have the same length and the same composition factors. Question: Is this Theorem true for stratifications of D(R)

  • f a ring R ? (up to der. equiv.)

Note: This is a question by Angeleri-H¨ ugel, K¨

  • nig and Liu.
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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Question Jordan-H¨

  • lder Theorem: For a finite group, all

composition series have the same length and the same composition factors. Question: Is this Theorem true for stratifications of D(R)

  • f a ring R ? (up to der. equiv.)

Note: This is a question by Angeleri-H¨ ugel, K¨

  • nig and Liu.
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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Question Jordan-H¨

  • lder Theorem: For a finite group, all

composition series have the same length and the same composition factors. Question: Is this Theorem true for stratifications of D(R)

  • f a ring R ? (up to der. equiv.)

Note: This is a question by Angeleri-H¨ ugel, K¨

  • nig and Liu.
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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

Answers

Corollary

∃ ring with two

stratifications of length 2 and 3, and different composition factors.

Jordan-H¨

  • lder Theorem fails for D(R), in general.
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Overview and schedule Definitions and examples Two results Recollements Main result Stratification of derived categories

References

Preprint is available at: http://math.bnu.edu.cn/∼ccxi/

  • H. X. CHEN and C. C. XI, Good tilting modules and

recollements of derived module categories. Preprint, arXiv:1012.2176v1, 2010.

  • H. X. CHEN and C. C. XI, Stratifications of derived categories

from tilting modules over tame hereditary algebras. Preprint, arXiv:1107.0444, 2011. ********