researches on the representation theory of algebra …iyama/yamagata65/miyachi.pdf · researches on...

TerminologiesTrivial extensions and representations

Tilting modules and several equivalencesGorenstein rings

Trivial extensions and stable equivalences

Researches on the Representation Theory ofAlgebra at University of Tsukuba

The view from the old seminar at Univesity of Tsukuba

Jun-ichi Miyachi

November 14, 2013, Nagoya

Jun-ichi Miyachi Researches on the Representation Theory of Algebra at University of Tsukuba The view from the old seminar at Univesity of Tsukuba

Let R be a ring, M a right R-module (default),Mod R (res., mod R, Proj R, proj R) : the category of right (resp., finitelypresented, projective, finitely generated projective) R-modules.modR : the stable category of mod R by projectives.AddB (resp., addB) : the category consisting of objects which are directsummands of direct (resp., finite direct) sums of objects of B.indB : the category of isomorphism classes of indecomposable objects ofB, ind R := ind(mod R). pdim M, idim M, fdim M: the projective,injective, flat dimension of M. Denote the minimal injective resoletion ofM by 0→ M → E0

R(M)→ E1R(M)→ · · · → En

R(M)→ · · · .X <⊕ Y means that X is a direct summand of Y .soc M (resp., rad M, top M) : the socle (resp., radical, M/ rad M) of M.For an artin algebra A over a commutative artinian ring C ,D := HomC (−,E0

C (top C )), τA = DTrA, τ−1A = TrAD are Auslanter-

Reiten translations.G(A) : the quiver of A, ΓA : the Auslander-Reiten quiver of A, sΓA : thestable Auslander-Reiten quiver of A.

1. Trivial extensions and representations

Definition

Let A be an art in algebra, M an A-bimodule, The trivial extensionA n M = A⊕M of A by M is a ring with identity (1A, 0) definedby the multiplication:

(a,m)(a′,m′) = (aa′, am′ + ma′)

for any (a,m), (a′,m′) ∈ A⊕M.We denote by A n DA by T(A).

Theorem (Tachikawa 80)

Let A be a hereditary artin algebra. Then | ind T(A)| = 2| ind A|.Especially T(A) is of finite representation type if and only if so is A.

Definition

Let A be an art in algebra, M an A-bimodule, The trivial extensionA n M = A⊕M of A by M is a ring with identity (1A, 0) definedby the multiplication:

(a,m)(a′,m′) = (aa′, am′ + ma′)

for any (a,m), (a′,m′) ∈ A⊕M.We denote by A n DA by T(A).

Theorem (Tachikawa 80)

Let A be a hereditary artin algebra. Then | ind T(A)| = 2| ind A|.Especially T(A) is of finite representation type if and only if so is A.

Theorem (Iwanaga-Wakamatsu 80)

Let A be an artin algebra with rad2 A = 0. T(A) is of finiterepresentation type if and only if G(A) is disjoint union of Dynkingraph s (An,Bn,Cn,Dn,E6,E7,E8,F4,G2).

Theorem (Yamagata 81)

Let A be an artin algebra. If T(A) is of finite representation type,then G(A) does not contain an oriented cycle.

Theorem (Iwanaga-Wakamatsu 80)

Let A be an artin algebra with rad2 A = 0. T(A) is of finiterepresentation type if and only if G(A) is disjoint union of Dynkingraph s (An,Bn,Cn,Dn,E6,E7,E8,F4,G2).

Let A be an artin algebra. If T(A) is of finite representation type,then G(A) does not contain an oriented cycle.

Let A be an Artinian ring with a QF-module Q, that is, anA-bimodule such that (1) QA, AQ are f.g., (2) soc(AQ) ' top(AA),soc(QA) ' top(AA). Then the following are equivalent

1 There is a ring extension 0→ Q → R → A→ 0 such thatΦ : ind A→ ind R \ ind A is bijective, where Φ(M) = ΩR(M)if M /∈ proj A or eR if M ' eA.

2 A ' A1 × A2 as rings, where A1 is a hereditary ring with aQF-module Q1, and where A2 is a serial ring with aQF-module Q2.

Jump to Serial rings.

Let A be a ring, GA a generator with B = EndA(G ), then thefollowing are equivalent.

1 B is serial Noetherian.

2 A is serial Noetherian, andG ∈ add(A⊕ E0(top Aa)⊕ E0(top Aa)/ soc E0(top Aa)).

Here Aa is the Artinian component of A [Warfield 75].

1 B is serial prime Noetherian.

2 A is serial prime Noetherian, and G is f.g. projective.

Riedtmann showed that in the case that A is a representation-finiteself-injective algebra A over an algebraically closed field K ,

sΓA ' Z∆/G , where ∆ is either Dynkin graphs An,Dn,E6,E7,E8,and G is an automorphism group of Z∆ (Riedtmann 80).

Theorem (Hoshino 82)

Let A be a connected self-injective artin algebra and assume thatthere exists a component of sΓA whose class is a Dynkin graph(An,Bn,Cn,Dn,E6,E7,E8,F4,G2).Then A is of finiterepresentation type.

The converse was proved in [Todorov 80].

Let A be a connected self-injective artin algebra and assume thatthere exists a component of sΓA whose class is a Dynkin graph(An,Bn,Cn,Dn,E6,E7,E8,F4,G2).

Then A is of finiterepresentation type.

Let A be a f.d. algebra over a field K . The following areequivalent.

1 T(A) is of finite representation type.

2 G(A) contains no oriented cycles, and for some duality D,there is a ring extension 0→ DA→ R → A→ 0 such that Ris of finite representation type.

3 For any duality D, any ring extension 0→ DA→ R → A→ 0,R is of finite representation type.

In the case that A is a f.d. algebra over a field, Tachikawaintroduced the following 2 conjectures related to NakayamaConjecture (Tachikawa 73):

1 If Exti (DA,A) = 0 (i > 0), then A is self-injective.

2 In the case that A is self-injective, if a finitely generatedA-module satisfies Ext i (M,M) = 0 (i > 0), then M isprojective.

Let A be a symmetric artin algebra with rad3 A = 0. Let X be aA-module with ExtiA(X ,X ) = 0 for all i > 0. Then X is projective.

Let Λ = T(A) with A hereditary. Let X be a Λ-module withExtiΛ(X ,X ) = 0 for all i > 0. Then X is projective.

1 If Exti (DA,A) = 0 (i > 0), then A is self-injective.2 In the case that A is self-injective, if a finitely generated

A-module satisfies Ext i (M,M) = 0 (i > 0), then M isprojective.

2. Tilting modules and several equivalences

Definition (Happel-Ringel 82)

A module T over a ring A is a tilting module of pdim ≤ 1 if

(a) An exact 0→ P1 → P0 → T → 0 exists with P0,P1 ∈ proj A.

(b) Ext1A(T ,T ) = 0.

(c) An exact 0→ A→ T0 → T1 → 0 exists with T0,T1 ∈ add T .

Definition

A pair (T ,F) of full subcategories of an abelian category A iscalled a torsion theory in A provided that

(i) HomA(T ,F) = 0;

(ii) for any object X in A, there exists an exact sequence0→ XT → X → XF → 0 in A with XT ∈ T and XF ∈ F .

Moreover, it is called splitting if any incecomposable object iseither in T or in F .

(b) Ext1A(T ,T ) = 0.

Definition

(i) HomA(T ,F) = 0;

(b) Ext1A(T ,T ) = 0.

Definition

(i) HomA(T ,F) = 0;

Theorem (Happel-Ringel 82)

Let T be a tilting module of pdim ≤ 1 with B = EndA(T ).T (TA) := M | Ext1

A(T ,M) = 0 F(TA) := N | HomA(T ,N) = 0X (T ) := X | X ⊗B T = 0 Y(T ) := Y | Tor1

B(Y ,T ) = 0 Then

1 (T (T ),F(T )) (resp., (X (T ),Y(T ))) is a torsion theory in mod A(resp., mod B).

2 HomA(T ,−) and −⊗B T induce T (T ) ' Y(T ), Ext1A(T ,−) and

TorB1 (−,T ) induce F(T ) ' X (T ).

Theorem (Happel-Ringel 82)

Let T be a tilting module of pdim ≤ 1 with B = EndA(T ).T (TA) := M | Ext1

A(T ,M) = 0 F(TA) := N | HomA(T ,N) = 0X (T ) := X | X ⊗B T = 0 Y(T ) := Y | Tor1

B(Y ,T ) = 0 Then

1 (T (T ),F(T )) (resp., (X (T ),Y(T ))) is a torsion theory in mod A(resp., mod B).

2 HomA(T ,−) and −⊗B T induce T (T ) ' Y(T ), Ext1A(T ,−) and

TorB1 (−,T ) induce F(T ) ' X (T ).

Definition

If a simple A-module S = eA/e rad A satisfiesHomA(DA,S) = Ext1

A(S , S) = 0, then τ−1S ⊕ (1− e)A is called aBB-tilting module (Brenner-Butler). Moreover, it is calledAPR-tilting if S is projective (Auslander-Platzeck-Reiten).

B is an iterated tilted algebra of type A if there is there are algebrasA = A0,A1, · · · ,An = B and Ai tilting modulesT0,T1, · · · ,Tn−1

with EndAi(Ti ) ' Ai+1 such that (T (Ti ),F(Ti )) is splitting.

or equivalenty if Db(mod B)4' Db(mod A) (Happel, Rickard

Schofield 88).

Theorem (Tachikawa-Wakamatsu 86)

Let A be artin algebra. T(A) is representation-finite iff A is aniterated APR-tilted algebra of Dynkin type.

Definition

A(S , S) = 0, then τ−1S ⊕ (1− e)A is called aBB-tilting module (Brenner-Butler). Moreover, it is calledAPR-tilting if S is projective (Auslander-Platzeck-Reiten).B is an iterated tilted algebra of type A if there is there are algebrasA = A0,A1, · · · ,An = B and Ai tilting modulesT0,T1, · · · ,Tn−1

Schofield 88).

Definition

Schofield 88).

Definition

Schofield 88).

The following assertions are equivalent for a torsion theory (T ,F)in mod A.

1 A torsion theory (T ,F) in mod A is splitting.

2 Ext1A(F , T ) = 0.

3 idim Y ≤ 1 for any Y ∈ F .

Let A be an artin algebra, (T ,F) be a torsion theory in mod Awith DA ∈ F . Suppose either | ind T | or | indF| <∞.LetT =

⊕X∈ind T ,Ext1

A(X ,T )=0 X , then T is a tilting module.

2 Ext1A(F , T ) = 0.

Let A be an artin algebra, (T ,F) be a torsion theory in mod Awith DA ∈ F . Suppose either | ind T | or | indF| <∞.

LetT =

⊕X∈ind T ,Ext1

2 Ext1A(F , T ) = 0.

Let A be an artin algebra, (T ,F) be a torsion theory in mod Awith DA ∈ F . Suppose either | ind T | or | indF| <∞.LetT =

⊕X∈ind T ,Ext1

Theorem (Kato-Hoshino-M 01)

Let (T ,F) be a torsion theory in mod A such that T contains amodule X generating T with Ext1

A(X , T ) = 0, F contains amodule Y cogenerating F with Ext1

A(F ,Y ) = 0 , andT ⊗ADA ⊂ T . Let M ˝

X be a minimal projective presentation of Xand N ˝

Y a minimal injective presentation of Y .

P ˝ = M ˝X ⊕ Hom˝

A(DA,N ˝Y )[1]

is a tilting complex such that T = T (P ˝) and F = F(P ˝).

Jump to triangulated categories.

Y a minimal injective presentation of Y .Then

P ˝ = M ˝X ⊕ Hom˝

A(DA,N ˝Y )[1]

Y a minimal injective presentation of Y .Then

P ˝ = M ˝X ⊕ Hom˝

A(DA,N ˝Y )[1]

Definition (M 91)

Let D be a triangulated category. A pair (U ,V) of fullsubcategories of D is called a stable t-structure in D provided:

(a)U = ΣU and V = ΣV,(b) HomD(U ,V) = 0,(c) For every X ∈ D,there exists a triangle U → X → V → ΣU with U ∈ U and V ∈ V.

Theorem (M 91)

Thee following are equivalent.

1 (U ,V) is a stable t-structure in D.

2 There are triangle functors Ui∗ //D

i !bbs_

j∗//___

Vj∗bbsuch that

i∗ a i !, j∗ a j∗, and i !, j∗ induce D/V ' U and D/U ' V,where i∗ : U → D, j∗ : V → D are the canonical embeddings.

This goes to Polygons of Recollements [Iyama-Kato-M 2011]

Definition (M 91)

Let D be a triangulated category. A pair (U ,V) of fullsubcategories of D is called a stable t-structure in D provided:(a)U = ΣU and V = ΣV,

(b) HomD(U ,V) = 0,(c) For every X ∈ D,there exists a triangle U → X → V → ΣU with U ∈ U and V ∈ V.

Theorem (M 91)

i !bbs_

j∗//___

Vj∗bbsuch that

Definition (M 91)

Let D be a triangulated category. A pair (U ,V) of fullsubcategories of D is called a stable t-structure in D provided:(a)U = ΣU and V = ΣV,(b) HomD(U ,V) = 0,

(c) For every X ∈ D,there exists a triangle U → X → V → ΣU with U ∈ U and V ∈ V.

Theorem (M 91)

i !bbs_

j∗//___

Vj∗bbsuch that

Definition (M 91)

Let D be a triangulated category. A pair (U ,V) of fullsubcategories of D is called a stable t-structure in D provided:(a)U = ΣU and V = ΣV,(b) HomD(U ,V) = 0,(c) For every X ∈ D,there exists a triangle U → X → V → ΣU with U ∈ U and V ∈ V.

Theorem (M 91)

i !bbs_

j∗//___

Vj∗bbsuch that

Definition (M 91)

Theorem (M 91)

i !bbs_

j∗//___

Vj∗bbsuch that

Definition (M 91)

Theorem (M 91)

i !bbs_

j∗//___

Vj∗bbsuch that

Definition (M 91)

Theorem (M 91)

i !bbs_

j∗//___

Vj∗bbsuch that

This goes to Polygons of Recollements [Iyama-Kato-M 2011]Jun-ichi Miyachi Researches on the Representation Theory of Algebra at University of Tsukuba The view from the old seminar at Univesity of Tsukuba

Definition (Miyashita 86)

A right A-module T over a ring A is a tilting module of pdim ≤ rprovided that

(a) there is an exact sequence of right R-modules

0→ Pr → · · · → P1 → P0 → T → 0

where P0, · · ·Pr are finitely generated projective,

(b) ExtiA(T ,T ) = 0 for any 1 ≤ i ≤ r ,

(c) there is an exact sequence of right R-modules

0→ A→ T0 → T1 → · · · → Ts → 0

where T0, · · · ,Ts ∈ add(TA).

Theorem (Miyashita 86)

Let TA be a tilting module of pdim ≤ r with B = EndA(T )op.

1 BT a tilting module of pdim ≤ r with A = EndB(T ).

2 ExteA(T ,−),TorBe (−,T ) induce an equivalence betweenX ∈ Mod A | ExtiA(T ,X ) = 0(i 6= e) andY ∈ Mod Bop | TorBi (Y ,T ) = 0(i 6= e).

For an A-module M, let · · · → Pr → · · · → P1 → P0 → M → 0 bea minimal projective resolution of M. We denoteP∗0 → · · · → P∗r → TrR r−1M → 0, where ()∗ := HomA(−,A).

Let A be a basic artin algebra, S ′ = Ae/ rad Ae a simple leftA-module. Assume that

(a) ExtiA(S ′,A) = 0 for 0 ≤ i < r ,

(b) ExtiA(S ′, S ′) = 0 for 0 < i ≤ r .

Then (1− e)A⊕ TrR r−1S ′ is a tilting module of pdim ≤ r .

Theorem (Fujita 92)

let A be a basic semiperfect Noetherian ring, S ′ = Ae/ rad Ae asimple left A-module. Assume that (a) ExtiA(S ′,A) = 0 for0 ≤ i < r and (b) Ext1

A(S ,′ S ′) = 0.Then for each i (1 ≤ i ≤ r),there exists a right A-module Yi with a local endomorphism ringEndA(Yi ) such that (1− e)A⊕ Yi is a tilting module of pdim i .

(a) ExtiA(S ′,A) = 0 for 0 ≤ i < r ,

(b) ExtiA(S ′, S ′) = 0 for 0 < i ≤ r .

Theorem (Fujita 92)

(a) ExtiA(S ′,A) = 0 for 0 ≤ i < r ,

(b) ExtiA(S ′, S ′) = 0 for 0 < i ≤ r .

Theorem (Fujita 92)

A(S ,′ S ′) = 0.

Then for each i (1 ≤ i ≤ r),there exists a right A-module Yi with a local endomorphism ringEndA(Yi ) such that (1− e)A⊕ Yi is a tilting module of pdim i .

(a) ExtiA(S ′,A) = 0 for 0 ≤ i < r ,

(b) ExtiA(S ′, S ′) = 0 for 0 < i ≤ r .

Theorem (Fujita 92)

Definition

Let A be a right coherent ring and B a lefty coherent ring. Abimodule BUA is a cotilting B-A-bimodule if it satisfies thefollowing conditions.

(a) The canonical morphimsms A∼→ End(BU) and

B∼→ End(UA)op are isomorphisms.

(b) idimB U ≤ r and idim UA ≤ r (r ≥ 0).

(c) ExtiA(U,U) = ExtiB(U,U) = 0 for all i > 0.

(d) Both BU and UA are finitely presented.

In case of A = B, U is called a dualizing A-bimodule.

Let A be a right Noetherian ring and B a Noetherian ring, and U acotilting B-A-bimodule. Then the following hold for e ≥ 0.

1 ExteA(−,U) and ExteB(−,U) induce a duality betweenX ∈ mod A | ExtiA(X ,U) = 0(i 6= e) andY ∈ mod Bop | ExtiB(Y ,U) = 0(i 6= e).

2 ⊕i≥0E iA(U) is an injective cogenerator for ModA.

Theorem (M 96)

Under the same assumption, the following hold.

1 R HomA(−,U) and R HomB(−,U) induce a duality betweenDb(mod A) and Db(mod Bop).

2 Every injective indecomposable A-module <⊕ ⊕i≥0E iA(U).

Theorem (M 96)

4. Gorenstein rings

Definition

A commutative local Noetherian ring (R,M, k) is a Gorenstein ringif idim R = d <∞. In this case,

Ei (R) '⊕

P∈Spec Rht P=i

E0(R/P), Ed(R) ' E0(R/M),

fdim E0(R/P) = ht P fdim E0(R/M) = d

⊕di=0 Ei (R) '

⊕P∈Spec R E0(R/P)

A Noetherian ring R is calledan Iwanaga-Gorenstein ring if idimR R, idim RR <∞. Moreover,Iwanaga-Gorenstein ring R is calledan Auslander-Goenstein ring if fdim E i

R(RR) ≤ i for any i ≥ 0(or equivalently if fdim E i

R(RR) ≤ i for any i ≥ 0[Auslander]).

Definition

Ei (R) '⊕

P∈Spec Rht P=i

E0(R/P), Ed(R) ' E0(R/M),

fdim E0(R/P) = ht P fdim E0(R/M) = d⊕di=0 Ei (R) '

Definition

Ei (R) '⊕

P∈Spec Rht P=i

E0(R/P), Ed(R) ' E0(R/M),

A Noetherian ring R is calledan Iwanaga-Gorenstein ring if idimR R, idim RR <∞. Moreover,Iwanaga-Gorenstein ring R is called

an Auslander-Goenstein ring if fdim E iR(RR) ≤ i for any i ≥ 0

(or equivalently if fdim E iR(RR) ≤ i for any i ≥ 0[Auslander]).

Definition

Ei (R) '⊕

P∈Spec Rht P=i

E0(R/P), Ed(R) ' E0(R/M),

Theorem (Iwanaga 79)

Let R be an Iwanaga-Gorenstein ring. Then ⊕i≥0 Ei (R) is aninjective cogenerator for ModR.

Let R be an Iwanaga-Gorenstein ring of idim R = d. ForM ∈ Mod R, the following are equivalent.

(1) pdim M <∞ (2) pdim M ≤ d(3) idim M <∞ (4) idim M ≤ d

Let R be an Iwanaga-Gorenstein ring of idim R = 2. Then everyinjective indecomposable R-module <⊕

⊕1i=0 E i (R), and E1(R)

has a non-zero socle.

⊕1i=0 E i (R), and E1(R)

Theorem

Let R be an Iwanaga-Gorenstein ring of idim R = d.

1 soc(Ed(R)) is essential in Ed(R) [Iwanaga-Sato 96].

2 Every injective indecomposable R-module <⊕ ⊕di=0 Ei (R) [M

3 For a R-module M, if idim M = d, then Ed(M) ∈ Add Ed(R).As a consequence pdim Ed(M) = fdim Ed(M) = d[Iwanaga-M 98].

4 For any injective indecomposable R-module I of fdim = d,I <⊕ Ed(R) and I ' E0(S) for some simple R-module S.Thus for an R-module M with idim M = d, Ed(M) has anessential socle [Iwanaga-M 98].

Theorem

1990ICRA5

3. Trivial extensions and stable equivalences

1990ICRA5

Let A be a f.d. algebra over a field K , T a tilting right A-moduleof pdim ≤ 1 with B = EndA(T )op.

Then we have an equivalencemod T(A) ' mod T(B).

Theorem (Wakamatsu 92)

Let A be a f.d. algebra over a field K , T a tilting right A-moduleof pdim <∞ with B = EndA(T )op. Then we have an equivalencemod T(A) ' mod T(B).

Moreover, Wakamatsu introduced the notion of generalized tiltingmodules, and studied the condition which induces the aboveequivalence.

1990ICRA5

Let A be a f.d. algebra over a field K , T a tilting right A-moduleof pdim ≤ 1 with B = EndA(T )op.Then we have an equivalencemod T(A) ' mod T(B).

1990ICRA5

Let A be a basic algebra over a field K whose ordinary quivercontains no oriented cycles. Let 0→ DA→ R → A→ 0 be a ringextension. Let modp R = mod R \ proj R, then

modp R ' modp T(A), R/ soc R ' A n (DA/ soc DA)

ΓR ' ΓT(A)

Theorem (Ohnuki 02)

Let Λ1 and Λ2 be selfinjective algebras such that Λ and Λ′ are socleequivalent, that is, p : Λ1/ soc Λ1 → Λ2/ soc Λ2 is an isomorphism.Assume there is an isomorphism ϕ : Λi

∼→ DΛi (i = 1, 2) such thatϕ1(1Λ1)(ab) = ϕ1(1Λ1)(a′b′) for any a1, b1 ∈ rad Λ1, a2, b2 ∈ rad Λ2

with a2 = p(a1), b2 = p(b1).Then modΛ1 ' modΛ2.

1990ICRA5

ΓR ' ΓT(A)

Theorem (Ohnuki 02)

1990ICRA5

ΓR ' ΓT(A)

Theorem (Ohnuki 02)

with a2 = p(a1), b2 = p(b1).

Then modΛ1 ' modΛ2.

1990ICRA5

ΓR ' ΓT(A)

Theorem (Ohnuki 02)

1990ICRA5

Theorem (Skowronski-Yamagata 98)

Let A be a self-injective algebra over a field K . The followingconditions are equivalent:

1 A is stably equivalent to a selfinjective algebra B/(ϕνB),where B is a tilted artin K -algebra not of Dynkin type and ϕis a positive automorphism of B.

2 A is socle equivalent to a selfinjective algebra B/(ϕνB), whereB is a tilted artin K -algebra not of Dynkin type and ϕ is apositive automorphism of B.

1990ICRA5

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