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Noncommutative Algebraic Geometry: Shanghai Workshop 2011, Shanghai, China, September 12-16, 2011 Happel s Theorem for Infinitely Generated Tilting Modules Changchang Xi ( ~) Beijing, China Email: xicc@bnu.edu.cn

Overview Given an infinitely generated tilting module, the category of its endomorphism ring admits a recollement by of rings Jordan-Hölder Theorem fails for stratifications of module by module. This talk reports a part of joint works with Hongxing Chen.

Schedule I. II. on tilting modules III. IV. V. Stratifications of

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

Tilting modules Tilting modules (or tilting complexes, objects) occur 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 (Broué s conjecture)

Definitions of f. g. tilting modules RT R-mod is called a classical tilting module if (1) exact seq. in R-mod with P j proj. : 0 P n P 0 T 0. (2) Ext i R (T,T) = 0 for all i > 0. (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 RT R-Mod is called a tilting module if (1) pd( R T) <, (2) Ext i R (T,T(I) ) = 0 for all sets I, i > 0. (3) exact seq. 0 R T 0 T 1 T m 0, T i Add(T). In 1995 by Colpi-Trlifaj, Bazzoni.

Good tilting modules T: tilting R-module is called good if the T i add(t) in for all i. (3) : 0 R T 0 T 1 T m 0

Good tilting modules Relationship: Classical tilting = Good tilting = Tilting T: tilting = T := n j=0 T j is good. Note: T and T have the same torsion theory in R-Mod.

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( R T) 1, (2) Ext 1 R (T,T(I) ) = 0 for all sets I, (3) exact seq. 0 R T 0 T 1 0, T i Add(T).

Examples All tilting modules in the sense of Brenner-Butler (1979), Happel-Ringel (1982). T := Z Q/Z: tiling Z-module. (Angeleri-Hügel + Sanchez): R S: injective ring epi, pd( R S) 1, = T := S S/R is tilting R-module.

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 : = End R (T), = D(R) D(S) (as triang. cat.s). D(R) : the unbounded cat. of R-Mod

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

Bazzoni s Theorem Theorem T : S : j! : = good tilting R-module, = End R (T), = T L S, the left total der. functor. Ker(j! ) D(S) j! D(R). Ker(j! ) = 0 T : classical.

Definition of recollements Beilinson, Bernstein and Deligne (1981): D,D,D : triang. cat.s. D : recollement of D and D if 6 triangle functors: i j! D i =i! D j! =j i! j D

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].

Back to Bazzoni s Theorem and questions T : good tilting R-module, S := End R (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?

Universal localizations R,S : rings with 1. S: universal localization of R if (1) Σ = {f : P 1 P 0 P i 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).

Theorem T : good tilting R-module S : = End( R T), j! := T L S = ring epi S U, recollement: D(U) D(S) j! D(R). Note: U is universal localization of S.

Corollary of the main result Corollary R S : inj. ring epi., pd( R S) 1, Tor R 1 (S,S) = 0, T := R S/R, B := End R (T). = recollement: D(S R S ) D(B) D(R). S := End R (S/R), S R S : coproduct of S and S over R.

Definition of stratifications For groups: Exact sequences simple groups, composition series For : simple, stratifications

Definition of der. simple D(R): simple if there is no non-trivial recollement of the form D(R 1 ) D(R) D(R 2 ), R i : rings.

Definition of stratifications of der. module A stratification of D(R) is a series of reollements: D(R 1 ) D(R) D(R 2 ), D(R 11 ) D(R 1 ) D(R 12 ), D(R 21 D(R 2 ) D(R 22 ), and so on, s.t. all R i,r ij,, are der. simple.

Question Jordan-Hölder 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) of a ring R? (up to der. equiv.) Note: This is a question by Angeleri-Hügel, König and Liu.

Answers Corollary ring with two stratifications of length 2 and 3, and different composition factors. Jordan-Hölder Theorem fails for D(R), in general.

References Preprint is available at: http://math.bnu.edu.cn/ ccxi/ H. X. CHEN and C. C. XI, Good tilting modules and recollements of module. Preprint, arxiv:1012.2176v1, 2010. H. X. CHEN and C. C. XI, Stratifications of from tilting modules over tame hereditary algebras. Preprint, arxiv:1107.0444, 2011. ********