In this work we study the relation between tilting and standard strati cation. We recall that for each standardly strati ed algebra corresponds a tilting module. We show that the poset given by the dierent strati cations of one algebra is a subposet of the poset formed by the tilting modules. Also, we show several examples, in particular we see that in the orientedA n for n = 2; 3; 4 all tilting modules are given by strati cations.

Advíncula, Fidel Hernández

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Cadernos Espinosanos (E-Journal)

HeaderBodyFooterMarginNotesi8ﬀ -i7?6i1ﬀ -ﬀ-i3 i10ﬀ -ﬀ-i96?i11i2?66?i46?i56?i61 one inch + \hoffset 2 one inch + \voffset3 \oddsidemargin = 54pt 4 \topmargin = 22pt5 \headheight = 19pt 6 \headsep = 19pt7 \textheight = 475pt 8 \textwidth = 330pt9 \marginparsep = 11pt 10 \marginparwidth = 90pt11 \footskip = 28pt \marginparpush = 5pt (not shown)\hoffset = 0pt \voffset = 0pt\paperwidth = 614pt \paperheight = 794ptHeaderBodyFooterMarginNotesi8ﬀ -i7?6i1ﬀ -ﬀ-i3 i10ﬀ -ﬀ-i96?i11i2?66?i46?i56?i61 one inch + \hoffset 2 one inch + \voffset3 \evensidemargin = 54pt 4 \topmargin = 22pt5 \headheight = 19pt 6 \headsep = 19pt7 \textheight = 475pt 8 \textwidth = 330pt9 \marginparsep = 11pt 10 \marginparwidth = 90pt11 \footskip = 28pt \marginparpush = 5pt (not shown)\hoffset = 0pt \voffset = 0pt\paperwidth = 614pt \paperheight = 794ptSa˜o Paulo Journal of Mathematical Sciences 4, 2 (2010), 135–139RELATIONS BETWEEN TILTING ANDSTRATIFICATION.Fidel Herna´ndez Adv´ınculaDepartament of MathematicFaculty of Mathemathic and Computer SciencesUniversidad de La Habana, CubaE-mail address: fidel@matcom.uh.cuAbstract. In this work we study the relation between tiltingand standard stratification. We recall that for each standardlystratified algebra corresponds a tilting module. We show thatthe poset given by the different stratifications of one algebra isa subposet of the poset formed by the tilting modules. Also, weshow several examples, in particular we see that in the orientedAn for n = 2, 3, 4 all tilting modules are given by stratifications.PreliminarsIn this work all algebras are finite dimensional K - algebras, ba-sic and indecomposables, K is an algebraically closed field and it isknown that an algebra Λ with these properties is of the form Λ = KQIwhere Q is a finite quiver and I an admissible ideal.Let v1, ...vn be the vertices of Q in a fixed order and S1, ..., Sn thecorresponding order of simple modules, Pi the projective cover of Siand Qi the injective envelope of Si. The standard module ∆i is definedas the maximal factor of Pi with composition factors Sj , j ≤ i[R]. Indual way, it is defined the co-standard ∇i as the maximal submoduleof Qi with composition factors Sj , j ≤ i[R]Let ∆ = {∆1, ...,∆n}, consider F (∆), the full subcategory ofmod Λ, consisting by M ∈ mod Λ such that M has a filtration with135136 Fidel Herna´ndez Advinculafactors in ∆, this is, 0 = M0 ⊂ M1 ⊂ ... ⊂ Mt = M con MiMi−1 ' ∆k.Dually, it is defined F (∇).There are the following subcategories of mod Λ:• Y (∆) = {Y ∈ modΛ/Ext1(F (∆), Y ) = 0}• F (∆) ∩ Y (∆)• W (∇) = {W ∈ modΛ/Ext1(W,F (∇)) = 0}• W (∇) ∩ F (∇)The algebra Λ is called standardly stratified if Λ ∈ F (∆).If also, the endomorphisms ring of each standard module is simple,Λ is called quasi - hereditary (see for instance [R] and [X]).1. A tilting module associated to the standard stratifica-tionAn A - module T is called tilting (generalized) if:(1) pdT <∞.(2) Exti (T, T ) = 0,∀i > 0(3) There is an exact sequence 0→ A→ T0 → T1 → ....→ Ts →0, with Ti ∈ addT, ∀i.If the algebra Λ is standardly stratified, we have that F (∆) is aresolving category ([X]), i. e. is closed under extensions, kernel ofsurjections and contains the projectives.Let $(∆) be the interseccion of the subcategories F (∆) and Y (∆)There is the following fact, proved in [X], Theor. 4.3:Proposition 1. If Λ is standardly stratified. Then there is a tiltingmodule T , unique except for the multiplicity of the indecomposabledirect summands such that add (T ) = $ (∆) .2. A Poset given by the standard stratificationsFor an Artin algebra Λ, consider the set TΛ of all tilting moduleswith direct summands of multiplicity one.Sa˜o Paulo J.Math.Sci. 4, 2 (2010), 135–139RELATIONS BETWEEN TILTING AND STRATIFICATION. 137For each tilting module T ∈ TΛ consider the right perpendicularcategory T⊥ = {X ∈ modΛ/Exti(T,X) = 0, ∀i}In [HU], it is defined a partial order in the class of all tilting modulesfor an Artin algebra by the following relation T1 6 T2 ⇔ T⊥1 ⊆ T⊥2 .For this relation T is minimal if and only if P<∞ is contravariantlyfinite ([HU]).Using the results of [AR], we see se that Y (∆) = T⊥.Theorem 2. The order among the different forms in that an algebracan be standardly stratified , given by inclusion between the respec-tive subcategories F (∆), induces an inverse order between the tiltingmodules corresponding to these stratifications.Proof. If we have two orders of simple modules such that Λ is stan-dardly stratified in these orders and F1 (∆) ⊂ F2 (∆) ⇒ Y2 (∆) ⊂Y1 (∆)(If Y ∈ Y2 (∆) ⇒ Ext1(X,Y ) = 0, X ∈ F2 (∆), as F1 (∆) ⊂F2 (∆)⇒ Ext1(X,Y ) = 0, X ∈ F1 (∆)⇒ Y ∈ Y1 (∆)).Then we have Y2 (∆) ⊂ Y1 (∆), and as Yi (∆) = T⊥i then T⊥2 ⊆T⊥1 We know that Proj ⊂ F (∆) ⊂ mod A, also F (∆) ⊂ P<∞.If F (∆) = P<∞, that is to say F (∆) is maximal then P<∞ iscontravariantly finite, well F (∆) it is, then T is minimal.If F (∆) = Proj, that is to say F (∆) is minimal then Y (∆) ={Y/Ext1(X,Y ) = 0, X ∈ F (∆)} = modA, then F (∆) ∩ Y (∆) =Proj, therefore T = P1 ⊕ ... ⊕ Pn = A, then T⊥ = A⊥ = modA andwe conclude that T is maximal.If F (∆) is maximal (minimal) not necessarily F (∆) = P<∞(Proj)Example 3. Let Am be the algebraKQI where Q is the quiverβ1 β2 βm−11•  2•  3• ... m−1•  m•α1 α2 αm−1Sa˜o Paulo J.Math.Sci. 4, 2 (2010), 135–139138 Fidel Herna´ndez Advinculaand I the ideal generated by αi+1αi, βiβi+1, αiβi − βi+1αi+1,1 ≤ i ≤ m− 2, αm−1βm−1.We can see that this algebra is quasi hereditary, only in this orderof simple modules, then F (∆) is maximal and minimal because theposet has only one element and F (∆) 6= P<∞ and F (∆) 6= Proj.3. Remarks and ExamplesRemark 4. We have several cases in that the maximal and the mini-mal are reached for the Poset given by the standard stratifications[HM1](1) The Hereditary algebras(2) The quasi hereditary algebras without oriented cycles, exceptloops(3) The algebras which are standardly stratified in all ordersFor the algebras with radical square zero, if it quasi triangular it isreached the minimal and the maximal.[HM2]Remark 5. For the hereditary algebras given by the quiver An for n =2, 3, 4, we can check that all tilting modules are given by stratifications,but for the hereditary algebra given by the quiver1•!!BBBBBBBB•32•==||||||||the tilting module T = P1⊕P2⊕I3 is not associated to stratification.In the Kronecker algebra, that is to say the hereditary algebra givenby the quiver 1•⇒ •2 we only have two stratifications: the one givenby the projectives and the other given by the injectives and we haveinfinite tilting modules.The algebra given by the quiver 1• •2 with radical square zero isnot standardly stratified in any orden and we have an unique tiltingmodule which is the trivial given by the sum of the projectives.Sa˜o Paulo J.Math.Sci. 4, 2 (2010), 135–139RELATIONS BETWEEN TILTING AND STRATIFICATION. 139References[AR] M. Auslander, I. Reiten, Aplications of Contravariantly Finite Subcate-gories, Advances in Mathematics, 86, 111-152, 1991.[HU] D. Happel and L. Unger, On a partial order of tilting modules, Algebr.Represent. Theory, 8(2), 147 -156, 2005.[HM1] F. Herna´ndez Adv´ıncula, E. N. Marcos Algebras standardly stratitifiedin all orders, Colloquium Mathematicum, vol. 107, pag. 81-87, 2007.[HM2] F. Herna´ndez Adv´ıncula, E. N. Marcos Stratifications of algebras withradical square zero, Annales des Sciences Mathe´matiques du Que´bec, vol.20, no. 2. pag. 135 - 142, 2006.[R] C. M. Ringel The category of modules with good filtrations over a quasi-hereditary algebras has almost split sequences, Mathematische Zeitschrift,Vol. 208, pag. 209-223, 1991.[S] C. Sa´enz, On modules with Weyl filtration for Schur algebras of finite type,Journal of Algebra, 237, pag. 49 - 63, 2001.[X] Changchang Xi, Standardly Stratified Algebras and Cellular Algebras,Mathematical Proceedings of the Cambridge Philosophical Society, 133,pag. 37-53, Cambridge University Press, 2002.Sa˜o Paulo J.Math.Sci. 4, 2 (2010), 135–139

RELATIONS BETWEEN TILTING AND STRATIFICATION

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RELATIONS BETWEEN TILTING AND STRATIFICATION

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