The ladder construction of Prüfer modules.

Transcription

1 SS 26 Selected Topics CMR The ladder costructio of Prüfer modules Claus Michael Rigel Let R be ay rig We deal ith (left) R-modules Our aim is to cosider pairs of maps, : V ith a proper moomorphism Let M be a module If there exists a edomorphism φ of M hich is surjectie, locally ilpotet, ad ith o-zero kerel W of fiite legth, the M ill be said to be a Prüfer module (ith respect to φ, ad ith basis W) The basic costructio A pair of exact sequeces W ad K Q yields a module 2 ad a pair of exact sequeces 2 W ad K 2 Q by formig the iduced exact sequece of W usig the map : K K W 2 W Q Q 2 The ladder sig iductio, e obtai i this ay modules i ad pairs of exact sequeces i i i+ W ad K i i i+ Q

2 for all i We may combie the pushout diagrams costructed iductiely ad obtai the folloig ladder of commutatie squares: We form the iductie limit = i i (alog the maps i ) Sice all the squares commute, the maps i iduce a map hich e deote by : 2 2 i i = i i = We also may cosider the factor modules / ad / The map : maps ito, thus it iduces a map : / / Claim The map is a isomorphism Namely, the commutatie diagrams ca be reritte as i i i W i i i i i+ W i i i i / i i i i i i i+ i+ / i ith a isomorphism i : i / i i+ / i The map is a map from a filtered module ith factors i / i (here i ) to a filtered module ith factors i+ / i (agai ith i ), ad the maps i are just those iduced o the factors It follos: The compositio of maps / p / / ith p the projectio map is a epimorphism φ ith kerel / It is easy to see that φ is locally ilpotet 2

3 Summery The maps i yield a map : ith kerel K ad cokerel Q This map iduces a isomorphism : / / Composig the ierse of this isomorphism ith the caoical projectio p, e obtai a edomorphism φ / p / / ad / is a Prüfer module ith respect to φ, ith basis W (sig a termiology itroduced for strig algebras, e also ca say: is expadig, / is cotractig) If ecessary, e ill use the folloig otatio: i (; ) = i, for all i N ad also for i =, ad P(; ) = / for the Prüfer module (here, =, = ) Sice P(; ) is a Prüfer module ith basis the cokerel W of, e ill sometimes rite W[] = / Examples () The classical example: Let R = Z, ad also = = Z Maps Z Z are gie by the multiplicatio ith some iteger, thus e deote it just by Let = 2 ad = If is odd, the P(2; ) is the ordiary Prüfer group for the prime 2, ad (2; ) = Z[ 2 ] (the subrig of Q geerated by ) If is ee, the P(2; ) is a 2 elemetary abelia 2-group (2) Let R = K(2) be the Kroecker algebra oer some field k Let be simple projectie, idecomposable projectie of legth ad : a o-zero map ith cokerel W (oe of the idecomposable modules of legth 2) The module P( ; ) is the Prüfer module for W if ad oly if / k, otherise it is a direct sum of copies of W () Triial cases: First, let be a split moomorphism The the Prüfer module ith respect to ay map α: is just the coutable sum of copies of W Secod, let : be a arbitrary moomorphism, let β: be a edomorphism The P(; β) is the coutable sum of copies of W (4)] Assume there exists a split moomorphism α:, say = X ad α = : The [ X W is a Riedtma-Zara sequece, thus W is a degeeratio of X Accordig to Zara, there is such that W[ + ] W[] X for all

4 The chessboard Assume o that both maps, : are moomorphisms The e get the folloig arragemet of commutatie squares: We see both horizotally as ell as ertically ladders: the horizotal ladders yield ( ; ) (ad its edomorphism ); the ertical ladders yield ( ; ) (ad its edomorphism ) Let Λ be a arti algebra First applicatio: Degeeratios Propositio Let, V be modules, ad let W ad W be cokerels of moomorphisms V If Ext (W, W) =, the there exists a module X ad a exact sequece X X W W Note that the existece of a exact sequece of the form X X W W may be iterpreted as assertig that W is a degeeratio of W, accordig to Riedtma ad Zara [Z] Corollary Let, V be modules, ad let W ad W be cokerels of moomorphisms V Assume that both Ext (W, W) = ad Ext (W, W ) = The the modules W ad W are isomorphic Both assertios are ell-ko i case k is a algebraically closed field: i this case, the coclusio of propositio just asserts that W is a degeeratio of W i the sese of algebraic geometry The mai poit here is to deal ith the geeral case he Λ is a arbitrary arti algebra Our iterest i this questio as raised by a series of lectures by Serre Smalø at the Mar del Plata coferece, March 26 The corollary stated aboe (uder the additioal assumptios that V is projectie ad that (), () are cotaied 4

5 i the radical of V ) is due to Bautista ad Perrez [BP] ad this result as preseted by Smalø ith a e proof [S] at Mar del Plata Lemma Let W be a module ith Ext (W, W) = Let 2 be a sequece of iclusios of modules ith i / i = W for all i The there is a atural umber such that + is a split moomorphism for all Lemma is ell-ko, it is based o the fact that Ext (W, ) he cosidered as a k-module is of fiite legth A proof ill be gie belo Let us use it i order to fiish the proof of propositio We apply Lemma to the chai of iclusios 2 2 ad see that there is such that : + splits This shos that + is isomorphic to W But e also hae the exact sequece + W Replacig + by W, e see that e get a exact sequece of the form (a Riedtma-Zara sequece), as asserted W W Proof of Corollary It is ell-ko that the existece of exact sequeces X X W W ad Y Y W W implies that the modules W- ad W are isomorphic But i our case e just hae to chage oe lie i the proof of propositio i order to get the required isomorphism Thus, assume that both Ext (W, W) = ad Ext (W, W ) = Choose such that both the iclusio maps : + ad : + split The + is isomorphic both to W ad to W, thus it follos from the Krull-Remak-Schmidt theorem that W ad W are isomorphic Remark Assume that, :, V are moomorphisms ith cokerels W ad W, respectiely, ad that Ext (W, W) = ad Ext (W, W ) = The splits if ad oly if splits Proof: Accordig to the corollary, e ca assume W = W Assume that splits, thus V is isomorphic to W Look at the exact sequece V W If it does ot split, the dimed(v ) < dim Ed( W), but V is isomorphic to W Proof of Lemma A assertio equialet to Lemma as used for example by Roiter i his proof of the first Brauer-Thrall cojecture, a correspodig proof ca be foud i [R] We iclude here a slightly differet proof 5

6 i Applyig the fuctor Hom(W, ) to the short exact sequece i i W, e obtai the exact sequece Ext (W, i ) Ext (W, i ) Ext (W, W) Sice the latter term is zero, e see that e hae a sequece of surjectie maps Ext (W, ) Ext (W, ) Ext (W, i ), beig iduced by the iclusio maps i The maps betee the Ext-groups are k-liear Sice Ext (W, ) is a k-module of fiite legth, the sequece of surjectie maps must stabilize: there is some such that the iclusio + iduces a isomorphism Ext (W, ) Ext (W, + ) for all No e cosider also some Hom-terms: the exactess of Hom(W, + ) Hom(W, W) Ext (W, ) Ext (W, + ) shos that the coectig homomorphism is zero, ad thus that the map Hom(W, + ) Hom(W, W) (iduced by the projectio map p: + W) is surjectie But this meas that there is a map h Hom(W, + ) ith ph = W, thus p: + W is a split epimorphism ad therefore the iclusio map + is a split moomorphism Remark I geeral, there is o actual boud o the umber Hoeer, i case of dealig ith the chai of iclusios 2 such a boud exists, amely the legth of Ext (W, ) as a k-module, or, ee better, the legth of Ext (W, ) as a E-module, here E = Ed(W) Proof: Look at the surjectie maps Ext (W, ) Ext (W, ) Ext (W, i ), beig iduced by the maps + (ad these maps are ot oly k-liear, but ee E-liear) Assume that Ext (W, ) Ext (W, + ) is bijectie, for some As e hae see aboe, this implies that the sequece ( ) + W splits No the map + is obtaied from ( ) as the iduced exact sequece usig the map With ( ) also ay iduced exact sequece ill split Thus + is a split moomorphism (ad Ext (W, + ) Ext (W, +2 ) ill be bijectie, agai) Thus, as soo as e get a bijectio Ext (W, ) Ext (W, + ) for some, the also all the folloig maps Ext (W, m ) Ext (W, m+ ) ith m > are bijectie 6

7 Example Cosider the D 4 -quier ith subspace orietatio: a b c d ad let Λ be its path algebra oer some field k We deote the idecomposable Λ-modules by the correspodig dimesio ectors Let =, = 2, W =, W = Note that a map : ith cokerel W exists oly i case the base-field k has at least elemets; of course, there is alays a map : ith cokerel W We hae dimext (W, ) = 2, ad it turs out that the module 2 is the folloig: 2 = The pushout diagram iolig the modules, (tice) ad 2 is costructed as follos: deote by µ a, µ b, µ c moomorphisms hich factor through the idecomposable projectie modules P(a), P(b), P(c), respectiely We ca assume that µ c = µ a µ b, so that a mesh relatio is satisfied Deote the summads of 2 by M a, M b, M c, ith o-zero maps ν a : M a, ν b : M b, ν c : M c, such that ν a µ a =, ν b µ b =, ν c µ c = There is the folloig commutatie square, for ay q k, e are iterested he q / {, }: =µ a +qµ b ] =µ a =[ ν b ν c = [ ν a ν b ( q)ν c ] 2 (the oly calculatio hich has to be doe cocers the third etries: ν c (µ a + qµ b ) = ( q)ν c µ a ) Note that (as ell as ) does ot split But o e deal ith a module 2 such that Ext (W, 2 ) = This implies that is isomorphic to 2 W Thus the ext pushout costructio yields a exact sequece of the form 2 2 W W 7

8 4 Secod applicatio: No-degeeratio Propositio 2 Let, : V be moomorphisms ith cokerel W, W, respectiely Assume Ed(W) is a brick, W, W are o-isomorphic, ad dim Ed(W) = dim Ed(W ) The Λ is ot of fiite represetatio type Proof: Let F = F(W) be the full category of modules ith a filtratio ith factors isomorphic to W This is a abelia category ith a uique simple object It is sufficiet to sho that F has ifiitely may isomorphism clases of idecomposable objects If ot, the F is a serial category, say ith l idecomposable objects It follos that the F-legth of ay object i F is bouded by l times its socle legth We cosider the chai of iclusios 2 correspodig to (thus, ith all factors isomorphic to W) Claim: oe of the iclusios has to split! Note that i / is a object of F-legth i Deote by s(i) the F-socle legth of i / We see = s() s(2) ith i l s(i), thus s(i) i/l I particular, this is a ubouded sequece Let i be the submodule of i cotaiig such that i / is the F-socle of i / The chai 2 is a sequece of extesios of by direct sums of copies of W, thus after a hile all the iclusios split Let be a idex such that + is a proper iclusio hich splits The + + = + ad the splittig of the iclusio + implies the splittig of + as e ated to sho But the splittig of implies that W is a degeeratio of W Sice dim Ed(W) = dim Ed(W ), it follos that W ad W are isomorphic, a cotradictio Refereces [BP] Raymudo Bautista ad Efre Perez: O modules ad complexes ithout self-extesios Commuicatios i Algebra (to appear) [R] Rigel: The Gabriel-Roiter measure Bull Sci math 29 (25) [S] Smalø: Homological cojectures ad degeeratios of modules Lectures Mar del Plata 26 [Z] Zara G Zara: A degeeratio-like order for modules Arch Math 7 (998),