LOCAL STRUCTURE OF ALGEBRAIC MONOIDS

Transcription

1 LOCAL STRUCTURE OF ALEBRAIC MONOIDS MICHEL BRION Abstract. W dscrib th local structur of an irrducibl algbraic monoid M at an idmpotnt lmnt. Whn is minimal, w show that M is an inducd varity ovr th krnl M M (a homognous spac) with br th two-sidd stabilizr M (a connctd an monoid having a zro lmnt and a dns unit group). This yilds th irrducibility of stabilizrs and cntralizrs of idmpotnts whn M is normal, and critria for normality and smoothnss of an arbitrary monoid M. Also, w show that M is an inducd varity ovr an ablian varity, with br a connctd an monoid having a dns unit group. 0. Introduction An algbraic monoid is an algbraic varity quippd with an associativ product map, which is a morphism of varitis and admits an idntity lmnt. Algbraic monoids ar closly rlatd to algbraic groups: th group of invrtibl lmnts of any irrducibl algbraic monoid M is a connctd algbraic group, opn in M. Thus, M is an quivariant mbdding of its unit group with rspct to th action of via lft and right multiplication; this mbdding has a uniqu closd orbit, th krnl of th monoid. This rlationship taks a particularly prcis form in th cas of an (or, quivalntly, linar) monoids and groups. Indd, by work of Vinbrg and Rittator, th an irrducibl algbraic monoids ar xactly th an quivariant mbddings of connctd linar algbraic groups. Furthrmor, any irrducibl algbraic monoid having an an unit group is an (s [Vi95, Ri98, Ri06]). An irrducibl algbraic monoids hav bn intnsivly invstigatd, primarily by Putcha and Rnnr (s th books [Pu88, R05]). Th idmpotnts play a fundamntal rol in thir thory: for instanc, th krnl contains idmpotnts, and ths form a uniqu conjugacy class of th unit group. From th viwpoint of algbraic groups, th idmpotnts ar xactly th limit points of multiplicativ on-paramtr subgroups. It follows asily that vry irrducibl algbraic monoid having a rductiv unit group is unit rgular, that is, any lmnt is th product of a unit and an idmpotnt. Such rductiv monoids ar of spcial intrst (s th abov rfrncs); thir study has applications to compactications 1

2 2 MICHEL BRION of rductiv groups (s [Ti03]) and to dgnrations of varitis with group actions (s [AB04]). In contrast, littl was known about th non-an cas until th rcnt classication of normal algbraic monoids by Rittator and th author (s [BR07]). Loosly spaking, any such monoid is inducd from an ablian varity, with br a normal an monoid. This rsult xtnds, and builds on, Chvally's structur thorm for connctd algbraic groups (s [Ch60, Co02]); it holds in arbitrary charactristics, lik most of Putcha and Rnnr's rsults. Mor gnrally, any normal quivariant mbdding of a homognous varity undr an arbitrary algbraic group is inducd from an ablian varity, with br a normal quivariant mbdding of a homognous varity undr an an group; s [Br07], which also contains xampls showing that th normality assumption cannot b omittd. In th prsnt papr, w obtain a classication of all irrducibl algbraic monoids in th spirit of [BR07]: thy ar also inducd from ablian varitis, but brs ar allowd to b connctd an monoids having a dns unit group (Thorm 3.2.1). This answrs a longstanding qustion of Rnnr, s [R84]. Also, w charactriz th irrducibl algbraic monoids having a prscribd unit group, as thos quivariant mbddings X of such that th Albans morphism X is an (Corollary 3.3.3). Our approach dirs from thos of [BR07, Br07]; it rlis on a local structur thorm for an irrducibl algbraic monoid M at an idmpotnt (Thorm 2.2.1). Loosly spaking again, an opn nighbourhood of in M is an inducd varity ovr an opn subvarity of th product MM, with br th two-sidd stabilizr M = fx 2 M j x = x = g. Not that M is a closd submonoid of M with th sam idntity lmnt, and th zro lmnt ; w show that M is an and connctd, and its unit group is dns (Lmma 3.1.4). Whn lis in th krnl, our local structur thorm taks a global form: th whol varity M is inducd ovr th krnl M M, with br M (Corollary 2.3.2). As a dirct consqunc, th normality or smoothnss of M is quivalnt to that of M. This raiss th qustion of classifying all smooth monoids having a zro lmnt; such a monoid is isomorphic (as a varity) to an an spac, by Corollary Anothr opn problm arising from our local structur thorm is th classication of all connctd algbraic monoids having a dns unit group and a zro lmnt; to mak this problm tractabl, on may assum that th unit group is rductiv. Our rsults ar obtaind ovr an algbraically closd ld of charactristic zro. Thy may b adaptd to arbitrary charactristics, by considring group schms and monoid schms at appropriat placs.

3 LOCAL STRUCTURE OF ALEBRAIC MONOIDS 3 For xampl, th stabilizr M should b undrstood as a closd submonoid schm of M; this subschm turns out to b rducd in charactristic zro (Rmark 2.2.2), but this fails in positiv charactristics,.g., for crtain non-normal an toric varitis. This may b a motivation for dvloping a thory of monoid schms; not that, unlik group schms, many monoid schms ovr a ld of charactristic zro ar not rducd. For xampl, viw th an plan A 2 as a monoid with product (x 1 ; y 1 ) (x 2 ; y 2 ) = (x 1 x 2 ; x 1 y 2 + x 2 y 1 ) and unit (1; 0). Thn th closd subschm with idal (x 2 ; xy) is a non-rducd submonoid, th an lin with a fat point at th origin. This papr is organizd as follows. W bgin by gathring som basic dnitions and rsults on algbraic varitis, algbraic groups and inducd varitis. In Sction 1, w study various stabilizrs and cntralizrs associatd with idmpotnts in an irrducibl algbraic monoids. This builds on work of Putcha (xposd in [Pu88, Chaptr 6]), but w hav modid som of his trminology in ordr to comply with standard convntions in algbraic gomtry and algbraic groups. Sction 2 is dvotd to th local structur of an irrducibl algbraic monoids, with applications to critria for normality or smoothnss, and to th irrducibility of stabilizrs and cntralizrs in normal monoids. In th nal Sction 3, w obtain our classication thorm and driv som consquncs,.g., all irrducibl algbraic monoids ar quasiprojctiv varitis. Notation and convntions. W considr algbraic varitis ovr an algbraically closd ld k of charactristic zro; morphisms ar undrstood to b k-morphisms. By a varity, w man a sparatd rducd schm of nit typ ovr k; in particular, varitis ar not ncssarily irrducibl. A point will always man a closd point. An algbraic group is a group schm of nit typ ovr k; thn is a smooth varity, as k has charactristic 0. Also rcall that is an if and only if it is linar, i.., isomorphic to a closd subgroup of som gnral linar group. ivn an arbitrary algbraic group and a closd subgroup H, thr xists a quotint morphism q :! =H, whr =H is a quasiprojctiv varity, and q is a principal H-bundl. Furthrmor, q is an if and only if H is an. Mor gnrally, givn a varity Y whr H acts algbraically, considr th product Y whr H acts via h (g; y) = (gh 1 ; hy). If Y is quasi-projctiv, thn thr xists a quotint morphism q Y : Y! ( Y )=H =: H Y; whr H Y is a quasi-projctiv varity, and q Y is a principal H- bundl. Th inducd varity H Y is quippd with a -action and a morphism : H Y! =H p Y

4 4 MICHEL BRION such that p Y and q Y ar -quivariant; th (schm-thortic) br of p Y at th bas point of =H is H-quivariantly isomorphic to Y. Morovr, p Y is an if and only if Y is an (for ths facts, s [S58a, Proposition 4] and [MFK94, Proposition 7.1]. In particular, H Y is an whnvr =H and Y ar both an. 1. Stabilizrs and cntralizrs 1.1. Stabilizrs. Throughout this sction, M dnots an irrducibl an algbraic monoid, 1 2 M th idntity lmnt, and = (M) th unit group. Thn acts on M via (x; y) z = xzy 1 (th twosidd action); w also hav th lft action of on M via x y = xy, and th right action via x y = yx 1. W x an idmpotnt 2 M and dnot by (1.1.1) ` : M! M; x 7! x th lft multiplication by. Clarly, ` is a rtraction of th varity M onto th closd subvarity M = fx 2 M j x = xg. Likwis, th right multiplication by, (1.1.2) r : M! M; x 7! x is a rtraction of M onto M = fx 2 M j x = xg. W also hav a rtraction of varitis (1.1.3) t : M! M; x 7! x: Morovr, M = fx 2 M j x = xg = M \M is a closd irrducibl submonoid of M with idntity lmnt. W put (1.1.4) M ` (th lft stabilizr of in M) and := fx 2 M j x = g (1.1.5) ` := \ M ` : Clarly, M ` is a closd submonoid of M with idntity lmnt 1 and unit group `. Morovr, ` is dns in M ` by [Pu88, Thorm 6.11]. Not that M ` is th st-thortic br of r at. In fact, M ` is also th schm-thortic br, as w shall s in Rmark (i). Likwis, th right stabilizr of in M, (1.1.6) M r := fx 2 M j x = g; is a closd submonoid of M with idntity lmnt 1, and dns unit group (1.1.7) r := \ M r : Th (two-sidd) stabilizr of in M, (1.1.8) M := fx 2 M j x = x = g = M ` \ M r ;

5 LOCAL STRUCTURE OF ALEBRAIC MONOIDS 5 is also a closd submonoid of M with idntity lmnt 1, zro lmnt, and unit group (1.1.9) := \ M : Morovr, is dns in M by [Pu88, Thorm 6.11] again. Our notation for M and dirs from that of Putcha in [Pu88]: his M and ar th irrducibl componnts of ours that contain 1. Also, not that M, ` M r and M ar gnrally rducibl; quivalntly, `, r and ar gnrally non-connctd. This happns for many non-normal an toric varitis, rgardd as commutativ monoids; s [Pu88, Exampl 6.12] for an xplicit xampl. Yt th stabilizrs in M ar always connctd, as follows from th xistnc of a multiplicativ on-paramtr subgroup of with limit point : Lmma (i) Thr xists a homomorphism of algbraic groups : m! which xtnds to a morphism of varitis : A 1! M such that (0) =. In particular, th closur of 0 in M contains. (ii) Th brs of ` and r ar connctd. (iii) M `, M r and M ar connctd. Proof. (i) By [Pu88, Corollary 6.10], thr xists a maximal torus T such that lis in T (th closur of T in M). Sinc T is a (possibly non-normal) toric varity, thr xist a on-paramtr subgroup : m! T and an lmnt t 2 T such that xtnds to a morphism : A 1! M such that (0) = t. Thn is a homomorphism of monoids, so that (0) is idmpotnt. Sinc t and commut, it follows that t =. Morovr, (x) = (x)(0) = (0) = = (x) for all x 2 k. (ii) Clarly, ` is invariant undr th lft action of `. In particular, ach br ` 1 (y), y 2 M, is stabl undr lft multiplication by 0. So, for any x 2 ` 1 (y), th closur of th orbit 0 x is an irrducibl subvarity of ` 1 (y) containing both points x and x = y. It follows that ` 1 (y) is connctd. (iii) Th connctdnss of M r (rsp. M ` ) follows from (ii). To show th connctdnss of M, not as abov that th closur of any orbit 0 x is an irrducibl subvarity of M containing both points x and x = Th cntralizr. Lt (1.2.1) C M () := fx 2 M j x = xg; this is th cntralizr of in M. Clarly, C M () is a closd submonoid of M with idntity lmnt 1 and unit group (1.2.2) C () := \ C M ():

6 6 MICHEL BRION Morovr, C () is connctd by [Pu88, Thorm 6.16]. But th xampl blow (a variant of [Pu88, Exampl 6.15]) shows that C M () is gnrally rducibl; in othr words, C () may not b dns in C M (). Exampl Lt V, W b vctor spacs of dimnsions m; n 2. Considr th multiplicativ monoids End(V ), End(W ) and th map ' : End(V ) End(W )! End(V W ); (A; B) 7! A B: Thn ' is a homomorphism of monoids, and is th invariant-thortical quotint by th m -action via t (x; y) = (tx; t 1 y). Thus, th imag of ' is a closd normal submonoid, M := End(V ) End(W ) End(V W ): Its unit group is th quotint of L(V ) L(W ) by m mbddd via t 7! (t id V ; t 1 id W ). ivn two idmpotnts 2 End(V ) and f 2 End(W ), th idmpotnt f 2 M satiss It follows asily that C M ( f) = fx y 2 M j x yf = x fyg: C ( f) = C L(V ) () C L(W ) (f); whil C M ( f) (1 ) End(W ): Thus, C M ( f) is rducibl whnvr ; f 6= 0; 1. Howvr, th cntralizrs in M ar always connctd, as shown by th following: Lmma (i) Th morphism (1.2.3) : C M ()! M; x 7! x = x = x is a rtraction of algbraic monoids. (ii) Th brs of ar connctd. In particular, C M () is connctd. (iii) W hav an xact squnc of algbraic groups (1.2.4) 1!! C () (iv) Th normalizr N ( ) quals C ().! (M)! 1: Proof. (i) is straightforward. (ii) Not that is invariant undr th (lft or right) action of C (). So th assrtion follows by arguing as in th proof of Lmma 1.1.1(ii). (iii) Clarly, rstrict to a homomorphism C ()! (M) with krnl. This homomorphism is surjctiv by [Pu88, Rmark 1.3(ii), Thorm 6.16]. (iv) C () normalizs by (1.2.4). Convrsly, if x 2 normalizs, thn it normalizs M (th closur of in M). Thus, x commuts with th zro lmnt of M.

7 LOCAL STRUCTURE OF ALEBRAIC MONOIDS 7 Nxt, w considr th action of on M by conjugation. Thn th isotropy group of is C (), so that th conjugacy class of is isomorphic to =C (). Lmma Th -conjugacy class of is closd in M. In particular, th varity =C () is an. Proof. W adapt a classical argumnt for th closdnss of smi-simpl conjugacy classs in an algbraic groups. Lt B b a Borl subgroup of. Sinc =B is complt, it sucs to chck that th B-conjugacy class of is closd in M. W may assum that B contains a maximal torus T such that 2 T, s th proof of Lmma Thn T cntralizs, so that th B-conjugacy class of is an orbit of th unipotnt radical of B; hnc this class is closd in th an varity M, by [Ro61] Lft and right cntralizrs. Lt (1.3.1) C ` M() := fx 2 M j x = xg; th lft cntralizr of in M. For any x; y 2 C ` (), w hav M (1.3.2) xy = xy = xy = xy: Thus, C ` () is a closd submonoid of M with idntity lmnt 1. Th M unit group of C ` () quals M (1.3.3) C() ` := \ CM() ` (indd, this is a closd submonoid of, and hnc a subgroup by [R05, Exrciss 1 and 2]). Morovr, C ` () is connctd by [Pu88, Thorm 6.16]. Howvr, C ` () is gnrally rducibl. For M instanc, with th notation of Exampl 1.2.1, w hav C ` ( f) = C ` () C ` ` (f) whil C ( f) End(V )(1 ) End(W ). M W now xtnd th statmnt of Lmma to lft cntralizrs: Lmma (i) C ` () is th primag of M undr th morphism M r of (1.1.2). Morovr, r rstricts to a rtraction of algbraic monoids (1.3.4) : C ` M()! M; x 7! x = x: (ii) Th brs of ar connctd. In particular, C ` () is connctd. M (iii) W hav an xact squnc (1.3.5) 1! `! C ` ()! (M)! 1 and th quality (1.3.6) C() ` = ` C (): (iv) Th normalizr N (` ) quals C ` ().

8 8 MICHEL BRION Proof. (i) is a dirct vrication. (ii) follows from (i) togthr with Lmma (iii) Clarly, yilds a homomorphism of algbraic groups C ` ()! (M) with krnl `. By 1.2.4, th rstriction of this homomorphism to C () is surjctiv. This implis both statmnts. (iv) C ` () normalizs ` by (1.3.5). For th convrs, if x 2 normalizs `, thn x 1 Mx ` = M ` as M ` is th closur of `. In particular, x 1 x 2 M, ` i.., x 1 x =, and x = x. Nxt, for latr us, w show that crtain homognous spacs ar an: Lmma (i) Th varity ` = is isomorphic to C ` ()=C (). (ii) Both varitis ` = and C ` ()= ar an. Proof. (i) W hav ` = = ` \ C =(` ()) = ` C ()=C () = C()=C ` (); whr th lattr quality follows from (1.3.6). (ii) C ` ()=C () is closd in =C (), and hnc is an by Lmma Morovr, C ` ()= is an inducd varity ovr th homognous spac C ` ()=C (), with br C ()=. Th lattr is an by Lmma 1.2.2; this implis th scond statmnt. Similar assrtions hold for th right cntralizr of in M rsp., (1.3.7) C r M() := fx 2 M j x = xg; C r () := \ C r M(): (Hr again, our notation dirs from that of Putcha: his C ` () is M our C r ().) In particular, th morphism M ` of (1.1.1) yilds an xact squnc of algbraic groups (1.3.8) 1! r! C r ()! (M)! 1 and th quality (1.3.9) C r () = r C (): This implis radily th following dscription of th stabilizr of in M M, (M M) := f(x; y) 2 M M j x = yg: and of its stabilizr in, ( ) = f(x; y) 2 j xy 1 = g: Not that (M M) is a closd submonoid of th product monoid M M, with idntity lmnt (1; 1) and unit group ( ). Lmma (i) (M M) = f(x; y) 2 CM() ` CM() r j (x) = (y)g:

9 LOCAL STRUCTURE OF ALEBRAIC MONOIDS 9 (ii) Th two projctions M M! M yild surjctiv morphisms (M M)! C ` (), (M M) M! C r () with connctd brs. In M particular, (M M) is connctd. (iii) Th two projctions! yild xact squncs 1! r! ( )! C() `! 1; 1! `! ( )! C r ()! 1: Rmark If is rductiv, thn C ` () and C r () ar opposit parabolic subgroups of, with common Lvi subgroup C (); morovr, th unipotnt radical R U (C ` ()) is containd in ` (s [R05, Thorm 4.5]). In viw of (1.2.4), it follows that and (M) ar rductiv. Morovr, ` is th smi-dirct product of R u (C ` ()) with C ()\`. Likwis, r is th smi-dirct product of R u (C r ()) with. Th stabilizr ( ) is dscribd in [AB04, Sction 3], in th mor gnral stting of stabl rductiv varitis. 2. Th local structur of affin irrducibl monoids 2.1. Local structur for th lft action. Throughout this sction, w maintain th notation and assumptions of Sction 1. W rst rcord th following consqunc of a rsult of Putcha: Lmma (i) Th product C r () is an opn an subvarity of M, isomorphic to C r ()=. (ii) Th product C r ()` is an opn an subvarity of, isomorphic to C r () ` whr acts on C r () ` via x (y; z) = (yx 1 ; xz). Proof. By [Pu88, Thorm 6.16], M is containd in C r (), th closur of C r () in M. Thus, M C r (), that is, C r () = M. So C r () is dns in M. But C r () is an orbit, and hnc is opn in M; th isotropy group of is C r () \ ` =. Togthr with Lmma 1.3.2, this implis (i). Not that C() r = C()= r = C()=(C r () r \ ` ) = C()` r =` is an opn an subvarity of = =`. Sinc th morphism r j :! is an (as its sourc is an), this implis (ii). Likwis, th product C ` () is an opn an subvarity of M, isomorphic to C ` ()=. Also, combining Lmmas and 2.1.1, w s that th product map r C () `! inducs an isomorphism = C r ()` = r C ` r C () ` () = C()C r (); ` and th right-hand sid is an opn an subvarity of. Nxt, w show that an an nighbourhood of in M is an inducd varity rlativ to th lft action of C r (): =

10 10 MICHEL BRION Proposition (i) Th subvarity (2.1.1) M r 0 := fx 2 M j x 2 C r ()g is opn in M, an, stabl undr th two-sidd action of C r () C` () on M, and contains M `. (ii) Th product map C r () M `! M inducs an isomorphism (2.1.2) f r : C r () M `! M r 0 ; quivariant undr th two-sidd action of th subgroup C r () ` C r () C ` (). (iii) Th schm-thortic intrsction M\M ` consists of th (rducd) point. Proof. (i) Not that M r 0 is th primag of C r () M undr th morphism r of (1.1.2). Sinc that morphism is an, and C r () is opn and an (by Lmma 2.1.1), M r 0 is opn and an as wll. Clarly, M r 0 contains M ` and is stabl undr C r (). To show th stability undr C ` (), considr x 2 M r 0 and g 2 C ` (). Thn xg = xg 2 C r ()C () = C r (); as g 2 C () by (1.3.9). (ii) Sinc r is quivariant undr C r (), th natural map C r () r 1 ()! r 1 (C r ()) = M r 0 is an isomorphism, whr r 1 () dnots th schm-thortic br. So it sucs to chck th quality (2.1.3) M ` = r 1 (): Clarly, M ` is containd in r 1 () as its maximal closd rducd subschm. Morovr, M ` is stabl undr th lft action of. So C r () M ` is a closd subschm of C r () r 1 (), and both hav th sam closd points. But C r () r 1 () is an opn subschm of M, and hnc is rducd; this implis (2.1.3). (iii) By (ii), f r rstricts to an isomorphism C() r (M \ M) ` = M \ M r 0 : Morovr, is th uniqu closd point of M \ M. ` Sinc M \ M r 0 is an irrducibl varity, it follows that M \ M ` = fg as schms, by arguing as in th proof of (ii). Rmarks (i) As shown in th abov proof, M ` is th schmthortic br of r at. Also, M ` may b rgardd as a slic at to th orbit C r (), or to its closur M. (ii) Th right action of C ` () on th opn subvarity (2.1.4) M ` 0 := fx 2 M j x 2 C ` ()g

11 LOCAL STRUCTURE OF ALEBRAIC MONOIDS 11 is dscribd in similar trms. (iii) By th argumnt of Proposition 2.1.2, th product of M inducs an opn immrsion è M `! M; this yilds a local structur rsult for th lft action of. Howvr, th orbit = =` is gnrally not an; this happns, for xampl, if M = End(V ) and 6= 0; 1. As a consqunc, th varity è M ` is gnrally not an ithr Local structur for th two-sidd action. W now show that an an nighbourhood of in M is an inducd varity rlativ to th two-sidd action of C r () `. Thorm (i) Th subvarity (2.2.1) M 0 := fx 2 M j x 2 C() r and x 2 C()g ` = M r 0 \ M ` 0 is opn in M, an, stabl undr th two-sidd action of C r () C` () on M, and contains M. (ii) Th product map C r () M `! M inducs an isomorphism (2.2.2) f : C r () M `! M 0 ; quivariant undr C r () `. (iii) Th schm-thortic intrsction MM \ M consists of th (rducd) point. Proof. (i) follows fom Proposition togthr with th fact that th intrsction of any two an opn subvaritis is an. (ii) By Proposition again, th natural map C r () (M ` \ M 0 )! M 0 is an isomorphism. Thus, it sucs to show that th natural map M `! M ` \ M 0 is an isomorphism. Lt x 2 M ` \ M 0, thn x = and x = g for som g 2 C ` (). Thus, = x = g = g, that is, g 2 `. Hnc M ` \ M 0 = fx 2 M ` j x 2 ` g: Sinc ` is opn and dns in M `, th product ` and dns in M. ` Thus, th natural map (M ` \ M r ) `! M ` \ M 0 = ` = is opn is an isomorphism, whr M ` \ M r dnots th schm-thortic intrsction. Th lattr intrsction quals M as a st, and hnc as a schm by th argumnt of Proposition This yilds th dsird isomorphism. (iii) By th argumnt of Proposition again, it sucs to chck that MM \ M = fg as sts. For this, rcall that M is isomorphic to a closd submonoid of th multiplicativ monoid End(V ), whr V

12 12 MICHEL BRION is a nit-dimnsional vctor spac; s [Pu88, Thorm 3.15]. Lt x 2 MM \ M. Thn rk(x) rk() and x = + y whr y 2 End(V ) satiss y = y = 0; thus, rk(x) = rk() + rk(y). It follows that y = 0, and x =. Rmarks (i) M may b rgardd as a slic at to th orbit C r ()`, or to its closur MM. Morovr, M (rgardd as a closd subschm of M) is rducd and quals th schm-thortic intrsction of M ` and M r. (ii) On may wondr whthr this local structur rsult xtnds to th two-sidd action of th whol group. Th answr is positiv for rductiv monoids and minimal idmpotnts, by a corollary of th Luna slic thorm (s [AB04, Lmma 4.3]). Howvr, th answr is gnrally ngativ: if M 0 is a -stabl nighbourhood of admitting an quivariant morphism f to th orbit = ( )=( ), thn M 0 contains th opn orbit = ( )= diag(). Thus, th isotropy group, diag(), is containd in a conjugat of ( ) in. But this dos not hold in gnral,.g., whn M = End(V ) and 6= 0; 1. Th lft and right actions do not play symmtric rols in th statmnt of Thorm W now rformulat this rsult in a symmtric way, and apply it to th local structur of th cntralizr of : Corollary (i) With th notation of Thorm 2.2.1, th product of M inducs isomorphisms and C () M = CM () \ M 0 r (C M () \ M 0 ) ` = M0 : (ii) C M () \ M 0 is irrducibl. In particular, C M () is irrducibl at. Proof. (i) Lt g 2 C r (), x 2 M C M (). Thn g = gxh = gxh = gxh = gh and h 2 ` b such that gxh 2 so that g = gh = g. Thus, g 2 C ` ()\C r () = C (). It follows that = h, that is, h 2 r \` =. Combind with Thorm 2.2.1, this implis th rst assrtion. Th scond assrtion is a consqunc of that thorm in viw of th isomorphism C r () = r C (); which follows in turn from (1.3.9). (ii) By (i), C ()M is an opn nighborhood of in C M (). Morovr, C () is dns in C ()M, sinc is dns in M. Thus, C ()M = C M () \ M 0 is irrducibl. Similar argumnts yild:

13 LOCAL STRUCTURE OF ALEBRAIC MONOIDS 13 Corollary (i) With th notation of Thorm 2.2.1, th product of M inducs isomorphisms and C r () M = C r M() \ M 0 ( ) (M M ) = (M 0 M 0 ) : (ii) C r () \ M M 0 is irrducibl. In particular, C r () is irrducibl at. M Anothr gomtric consqunc of Proposition and Thorm is th following normality critrion: Corollary If M is normal at, thn: (i) Th stabilizrs M `, M r and M ar irrducibl and normal. In particular, `, and r ar connctd. (ii) Th two-sidd stabilizr ( ) is connctd as wll. (iii) C M (), C ` (), C r () and (M M) M M ar normal at. Convrsly, if on of th varitis M `, M r, M, C M (), C ` (), M C r (), (M M) M is normal at, thn M is also normal at. Proof. (i) Dnot by : f M `! M ` th normalization map of M. ` Thn th lft action of on M ` lifts to an action on M f. ` This yilds a nit morphism ' : C() r M f`! C r () M ` which rstricts to an isomorphism ovr a dns opn subvarity. Sinc C r () M ` is irrducibl and normal (by th normality of M and Proposition 2.1.2), ' is an isomorphism. Thus, is an isomorphism, that is, M ` is normal; this varity is also connctd by Lmma 1.1.1, and hnc irrducibl. It follows that ` is irrducibl as wll. Th sam argumnt shows that M r is normal. Likwis, th normality of M follows from Lmma and Thorm (ii) follows from th connctdnss of in viw of Lmma (iii) is a consqunc of th normality of M togthr with Corollaris and Th convrs statmnt is provd similarly. Rmark Assum that is rductiv. Thn th natural map R u (C()) r (C () M ) R u (C()) `! M is an opn immrsion with imag M 0. This statmnt follows from Thorm combind with Rmark 1.3.4; altrnativly, this may b dducd from a local structur thorm for actions of rductiv groups, s [Ti03, Sction 6] or [AB04, Lmma 2.8]. Also, not that ach orbit of M for th (lft or right) -action contains an idmpotnt. Hnc th abov statmnt dscribs th local structur of M at an arbitrary point.

14 14 MICHEL BRION 2.3. Th cas of a minimal idmpotnt. In this subsction, w assum that th idmpotnt is minimal, that is, is th uniqu idmpotnt of M; quivalntly, lis in th krnl kr(m), th uniqu closd orbit of in M. Hnc (2.3.1) kr(m) = = MM: Morovr, M is an algbraic group with idntity lmnt ; th - conjugats of ar xactly th minimal idmpotnts of M (for ths rsults, s [Pu88, Chaptr 6] and [Hu05, Sction 1]). Combind with (1.2.4), it follows that (2.3.2) M = = C () = C (): Furthrmor, (2.3.3) = C()C r () ` by [Pu88, Thorm 6.30 and Corollary 6.34]. In viw of Lmma (iii), this implis in turn: (2.3.4) = C()` r = r C(): ` W now show that th opn subvaritis that occur in Proposition and Thorm ar all qual to M: Lmma (i) M = C r (); quivalntly, M = C r ()M `. Likwis, M = C ` () and M = M r C ` (). (ii) M r = r ; quivalntly, M r = r M. Likwis, M ` = ` and M ` = M `. (iii) M = C r ()M ` = C ` ()M r and kr(m) = C r ()` = C `. ()r (iv) M r 0 = M ` 0 = M 0 = M. Proof. (i) By (2.3.1), (2.3.2) and (2.3.4), M = MM = = C () = = C r ()` = C r (). (ii) Lt x 2 M r, thn x 2 C r () = C () r. Writ accordingly x = gh, thn = x = x = gh = gh = g: Thus, g 2 and x 2 r. (iii) follows from (i) and (ii) togthr with (2.3.1); likwis, (iv) follows from (i) and (iii). Togthr with Thorm and Corollaris and 2.2.4, this lmma implis th following global structur rsult: Corollary For any minimal idmpotnt, th product of M inducs isomorphisms C() r M ` = M; M ` = M ` ; C r () M = C r M(); C () M = CM ();

15 LOCAL STRUCTURE OF ALEBRAIC MONOIDS 15 and ( ) (M M ) = (M M) : Also, M is normal if and only if it is normal at som minimal idmpotnt, sinc th normal locus is stabl undr th two-sidd -action. Togthr with Corollaris and 2.3.2, this implis in turn: Corollary Lt b a minimal idmpotnt of an irrducibl algbraic monoid M. Thn th following assrtions ar quivalnt: (i) M is normal. (ii) All th varitis M, M `, M r, C M(), C ` (), C r () and (M M) M M ar irrducibl and normal. (iii) At last on of ths varitis is normal at. 3. Th structur of irrducibl monoids 3.1. Local structur. In this subsction, w xtnd most rsults of th prvious sctions to an arbitrary (possibly non-an) irrducibl algbraic monoid M with unit group. As in [BR07] which trats th cas whr M is normal, our main tool is a thorm of Chvally: thr xists a uniqu xact squnc of connctd algbraic groups (3.1.1) 1! a!! A()! 0; whr a is an and A() is an ablian varity (s [Ch60], and [Co02] for a modrn proof). It follows that a is th maximal closd connctd an subgroup of, whil th quotint morphism :! A() is th Albans morphism of th varity (th univrsal morphism to an ablian varity, s [S58b]). Dnot by M a th closur of a in M. Clarly, M a is a submonoid of M with idntity lmnt 1 and unit group a. In fact, M a is an by [Ri06, Thorm 2]; as a consqunc, M a is th maximal closd irrducibl an submonoid of M. Morovr, th natural map (3.1.2) : a M a! M; (g; x) a 7! gx is birational (sinc rstricts to an isomorphism a a! ) and propr (sinc = a = A() is complt). It follows that is surjctiv, that is, (3.1.3) M = M a : Lt C dnot th cntr of ; thn = C a (s.g. [S58a, Lmm 2]). As a consqunc, (3.1.4) = C 0 a and M = C 0 M a ; whr C 0 dnots th nutral componnt of C. In particular, (3.1.5) C 0 =(C 0 \ a ) = = a = A();

16 16 MICHEL BRION and th natural map (3.1.6) : C 0 C 0 \ a M a! M is propr and birational. [BR07, Corollary 2.4]: This yilds th following gnralization of Lmma Any idmpotnt of M is containd in M a. Proof. ivn x 2 M, th (st-thortical) br of at x may b idntid with th subvarity fz(c 0 \ a ) j z 2 C 0 ; z 1 x 2 M a g C 0 =(C 0 \ a ) = A(): If x is idmpotnt, thn th abov subvarity is a closd subsmigroup of A(), and hnc is a group by [R05, Exrciss 1 and 2]. It follows that x 2 M a. W now choos an idmpotnt 2 M a and dn th stabilizrs M; ` M r ; M M and ` ; r ; as in Sction 1.1. Thn again, M ` is a submonoid of M with idntity lmnt 1 and unit group `, and likwis for M r, M. Lmma Th stabilizrs `, r and ar an. Proof. Rcall that ` is th isotropy group of th point 2 M for th lft -action. Sinc this action is faithful, ` is an by [Ma63, Lmma p. 54]. So r and = ` \ r ar an as wll. Rmarks (i) Th two-sidd stabilizr ( ) is not ncssarily an, as it contains C 0 mbddd diagonally in. (ii) In gnral, th stabilizrs ar not containd in a, as shown by [BR07, Exampl 2.7]. Spcically, lt A b a non-trivial ablian varity, F A a non-trivial nit subgroup, and M th commutativ monoid obtaind from th product monoid A A 1 by idntifying th points (x; 0) and (x + f; 0), for all x 2 A and f 2 F. Thn = A m, a = m, and th imag of (0; 0) in M is an idmpotnt with stabilizr which strictly contains a. = ` = r = F m ; Similary, w may dn th cntralizrs C ` (); C r (); C M M M() M and C ` (); C r (); C () as in Sction 1.2. Thn C ` M() = C 0 C ` M a (); C r M() = C 0 C r M a (); C M () = C 0 C Ma () and likwis C ` () = C 0 C ` a (); C r () = C 0 C r a (); C () = C 0 C a (): In particular, ths closd subvaritis ar all connctd, and C r () = r C (); C ` () = ` C ():

17 LOCAL STRUCTURE OF ALEBRAIC MONOIDS 17 Morovr, M = C 0 a contains C 0 C r a () = C r () as a dns opn subvarity, by Lmma Thus, all th statmnts of Proposition and Thorm hold in this stting, xcpt for th annss of M r, M ` and M 0 0 0; th proofs ar xactly th sam. Corollaris 2.2.3, and hold as wll, sinc thir proofs do not us any assumption of annss. Combind with th following rsult, this rducs th local structur of irrducibl algbraic monoids to that of connctd an monoids having a dns unit group. Lmma Th stabilizrs M `, M r and M ar an and connctd. Thir unit groups `,, r ar dns. Proof. With th notation of Proposition 2.1.2, th primag (f r ) 1 ( \ M r 0 ) = C() r ( \ M) ` = C() r ` is dns in C r () M, ` as is dns in M. It follows that ` is dns in M. ` Sinc ` is an, this implis th annss of M ` in viw of [Ri06, Thorm 3]. Th connctdnss of M ` is obtaind by arguing as in th proof of Lmma Likwis, th dsird proprtis of M follow from th statmnt of Thorm Ths considrations yild th following smoothnss critrion: Corollary Lt b an idmpotnt of an irrducibl algbraic monoid M. Thn is a smooth point of M if and only if th varity M is an an spac. Proof. By Thorm 2.2.1, M is smooth at if and only if M is smooth at. So th assrtion follows from th xistnc of an attractiv m - action on M with xd point. Spcically, lt : m! b as in Lmma Thn th m - action on via t x = (t)x xtnds to an action of th multiplicativ monoid A 1 on M, such that 0 x = x = for all x 2 M. This yilds a positiv grading of th algbra of rgular functions on th an varity M. Now th gradd vrsion of Nakayama's lmma implis our assrtion. Rmark Th abov smoothnss critrion raiss th qustion of classifying algbraic monoid structurs on a givn an n-spac, having th origin as thir zro lmnt. Whn th unit group is rductiv, such structurs corrspond bijctivly to dcompositions of n into a sum of squars of positiv intgrs, as th corrsponding monoids ar just products of matrix monoids. Indd, if M is a smooth monoid with rductiv unit group and zro lmnt 0, thn th varity M is quivariantly isomorphic to th -modul T 0 M, as follows from th Luna slic thorm. This

18 18 MICHEL BRION yilds a -quivariant isomorphism ' : M! Q m End(V i=1 i), whr V 1 ; : : : ; V m ar simpl -moduls; as a consqunc, ' is an isomorphism of monoids. Thus, is idntid to an opn subgroup of th product Q m L(V i=1 i), and hnc to th whol product. (This is also provd in [Ti03, Sction 11], via a rprsntation-thortic argumnt.) In th cas that is a minimal idmpotnt of M a, th subvarity a a is th uniqu closd orbit of a a in M a. As is propr, it follows that = C 0 a a is th uniqu closd -orbit in M. In othr words, is th krnl of M. Thn all th statmnts of Sction 2.3 hold, with xactly th sam proofs. Also, not that th minimal idmpotnts of M ar xactly thos of M a (by Lmma 3.1.1); thy form a uniqu conjugacy class of a or, quivalntly, of by (3.1.4). Sinc th smooth locus of M is stabl undr th two-sidd action of, w s that M is smooth if and only if M is an an spac for som minimal idmpotnt lobal structur. By th main rsult of [BR07], th map of (3.1.2) is an isomorphism whnvr M is normal, and thn M a is normal as wll. In othr words, any normal monoid is an inducd varity ovr an ablian varity, with br a normal an monoid. This statmnt dos not xtnd to arbitrary irrducibl monoids, in viw of [BR07, Exampl 2.7]. Yt w show that any such monoid is an inducd varity ovr an ablian varity, with br a connctd an monoid having a dns unit group: Thorm Lt M b an irrducibl algbraic monoid, and its unit group. Thn thr xists a closd submonoid N M satisfying th following proprtis: (i) N is an, connctd, and contains 1. (ii) Th unit group H := (N) is dns in N, and contains a as a subgroup of nit indx. In particular, M a is th irrducibl componnt of N containing 1. (iii) Th canonical map (3.2.1) ' : H N! M; (g; n)h 7! gn is an isomorphism of varitis. Morovr, th projction p : H N! =H is idntid with th Albans morphism of th varity M. In particular, H and N ar uniquly dtrmind by M. Proof. W bgin with th proof of th nal assrtion: w assum that M = H N whr H and N satisfy (i){(iii), and show that p quals th Albans morphism M : M! A(M). Th lattr morphism is uniquly dtrmind up to a translation in A(M); w normaliz it by imposing that M (1) = 0 (th origin of th ablian varity A(M)).

19 LOCAL STRUCTURE OF ALEBRAIC MONOIDS 19 Considr a morphism (of varitis) : H N! A whr A is an ablian varity. Th rstriction of to th nutral componnt H 0 N is a morphism from a connctd an algbraic group to an ablian varity, and hnc is constant (as follows.g. from [Mi86, Corollary 3.9]. Thus, is constant on vry irrducibl componnt of N. Sinc N is connctd, maps N to a point; likwis, it maps ach br of p (that is, ach translat gn in H N) to a point. Togthr with Zariski's Main Thorm, this implis that factors as p followd by a morphism =H! A. This provs th dsird quality. In particular, N is idntid with th br of M at 0. W now show that this br satiss th proprtis (i){(iii). By rigidity, th rstriction M j is a homomorphism of algbraic groups (s.g. [Mi86, Corollary 3.6]). Thus, M is a homomorphism of algbraic monoids. In particular, N is a closd submonoid of M containing 1. Morovr, M j factors through a uniqu homomorphism A()! A(M), which is surjctiv as is dns in M. Sinc A() = = a, w may idntify A(M) with th homognous spac =H, whr H is a closd subgroup of containing a. This idntis M with H N, quivariantly for th right -action on M. Sinc M is connctd, it follows that H acts transitivly on th connctd componnts of N. Lt N 0 N b th connctd componnt containing 1, and H 0 H its stabilizr. Thn th canonical map H H0 N 0! N is an isomorphism, as H=H 0 is idntid with th st of connctd componnts of N. Thus, th analogous map H0 N 0! M is an isomorphism as wll. Morovr, sinc H 0 has nit indx in H, and =H is complt, it follows that =H 0 is complt as wll. Thus, th composit map M = H0 N 0! =H 0 factors through a - quivariant morphism =H! =H 0. This implis that H = H 0 and N = N 0, i.., N is connctd. Likwis, sinc is dns in M, it follows that H is dns in N. To complt th proof, it sucs to show that th quotint H= a is nit. Indd, this implis that H is an and, in turn, that N is an in viw of [Ri06, Thorm 3]. Th nitnss of H= a is quivalnt to th assrtion that th canonical homomorphism = a = A()! A(M) = =H has a nit krnl, and hnc to th xistnc of a -quivariant morphism : M! A()=F; whr F A() is a nit subgroup.

20 20 MICHEL BRION To construct such a morphism, choos a minimal idmpotnt 2 M and rcall that M = = =` (s Lmma 2.3.1). This yilds a -quivariant morphism : M! =`. Now lt : M! A(=` ) b th composition of with th Albans morphism of =`. Thn is -quivariant. Morovr, A(=` ) is th quotint of A() by th imag of th subgroup `, and th lattr imag is a nit group (as ` is an by Lmma 3.1.2). Rmark W may dn a natural structur of algbraic monoid on H N so that th map ' of (3.2.1) is an isomorphism of algbraic monoids. Indd, th canonical map C 0 C 0 \H N! H N is an isomorphism, as = C 0 H = C 0 C 0 \H H. Morovr, C 0 C 0 \H N is th quotint of th product monoid C 0 N by th cntral subgroup C 0 \ H, mbddd via x 7! (x; x 1 ). Altrnativly, on may obsrv that th H-action on N by conjugation xtnds uniquly to a -action, whr C 0 acts trivially (sinc C 0 \ H, a cntral subgroup of H, acts trivially on N by conjugation). Thus, on may form th smi-dirct product of monoids N: its product is givn by (x; a) (y; b) = (xy; a y 1 b) whr a z dnots th conjugat of a 2 N by z 2 (s [R05, Exampl 3.7]). Thn on chcks that this product inducs a uniqu product on H N such that th quotint map N! H N is a homomorphism of monoids. Th abov construction is an analogu for algbraic monoids of th induction of varitis with group actions Som applications. W bgin by stating two dirct consquncs of Thorm 3.2.1, rst obtaind in [BR07] for normal monoids: Corollary Any irrducibl algbraic monoid is quasi-projctiv. Corollary Th catgory of irrducibl algbraic monoids is quivalnt to th catgory having as objcts th tripls (; H; N), whr is a connctd algbraic group, H is a closd subgroup containing a as a subgroup of nit indx, and N is a connctd an algbraic monoid with unit group H, dns in N. Th morphisms from such a tripl (; H; N) to a tripl ( 0 ; H 0 ; N 0 ) ar th pairs ('; ), whr ' :! 0 is a homomorphism of algbraic groups such that '(H) H 0, and : N! N 0 is a homomorphism of algbraic monoids such that 'j H = j H. Anothr consqunc is a charactrization of monoids among (possibly non-normal) quivariant mbddings of algbraic groups:

21 LOCAL STRUCTURE OF ALEBRAIC MONOIDS 21 Corollary Lt b a connctd algbraic group and lt X b a -quivariant mbdding of. Thn X admits a (uniqu) structur of algbraic monoid if and only if X is an. Proof. If X is an irrducibl algbraic monoid, thn its Albans morphism is an by Thorm For th convrs, arguing as in th proof of that thorm, on shows that A(X) = =H, whr H is a closd subgroup containing a ; morovr, X is -quivariant. Thus, X = H Y, whr Y is an quivariant mbdding of th (possibly non-connctd) algbraic group H. Morovr, Y is an by assumption, and hnc is an algbraic monoid. In particular, its unit group H is an, so that H= a is nit. As in Rmark 3.2.2, th inducd varity X is thn an algbraic monoid. Nxt, w show how to rcovr th main rsult of [BR07] (Thorm 4.1 and its proof): Corollary For any irrducibl algbraic monoid M, th morphism : a M a! M of (3.1.2) is nit. In particular, M is normal if and only if th associatd tripl satis- s: H = a and N = M a is normal. Proof. Sinc is propr and -quivariant, and M = N, th nitnss of is quivalnt to th nitnss of its rstriction to 1 (N). But 1 (N) = H a M a by Thorm Furthrmor, j 1 (N) factors as th closd mbdding H a M a! H a N (corrsponding to th inclusion of M a into N), followd by th isomorphism H a N = (H= a ) N (sinc N is H-stabl), followd in turn by th projction (H= a ) N! N; a nit morphism. Sinc is birational and nit, it is an isomorphism whnvr M is normal, by Zariski's Main Thorm; it thn follows that M a is normal as wll. Morovr, H = a and N = M a by th uniqunss statmnt in Thorm Th convrs is obvious. Finally, on may show as in [BR07, Thorm 5.3] that any irrducibl algbraic monoid M has a faithful rprsntation by ndomorphisms of a homognous vctor bundl ovr an ablian varity (th Albans varity of M.)

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