ll relieson the existenceof a specialbasisfor minimal
MultiplicativeBases. Bautista'sproofof Brauer-Thrall
representation-infinite
algebras:a multiplicativeCartan basis.The existenceproof for such a basis is the
main aim of the second paper which we have to mentionhere: the joint work of Bautista,Gabriel,Roiter
bases,which has appearedin
algebrasand multiplicative
and Salmer'onwith the title Representation-finite
the journal lnventionesMathematicae[B-S].We assumeagain that A is a k-algebrawith k an algebraically
closedfield.ln addition,we requirethatA is basic:this meansthat the factoralgebraA of A moduloits radical
is a productof copiesof k (this additionalrequirementcan alwaysbe achievedby replacingA by a Morita
equivalentalgebraAo;note that for Moritaequivalentalgebras,the modulecategoriesare equivalent).We
say that a k-basisof A is a multiplicative Gartan basis providedthe followingthree conditionsare satisfied:
(1) ff bl, b, belongto B, then b,b, is eitherzeroor belongsto B,
(2) B containsa completeset B' of orthogonalprimitiveidempotents.
(3) The non-idempotent
elementsof B generatethe radicalrad A of A.
elementsof B. Then it is easy
lf B is a multiplicative
Cartan basis, let B" be the set of all non-idempotent
to see that B is the disjointunion of B' and B", that the non-zeroelementsin (8";t form a basis of (rad A)
t,fort=l,2,,..,andthatforanyelementbeB,thereareidempotentsel,e2eBsuchthatb=e,ber.The
latterconditionsjust meansthat the basis B consistsof homogeneouselementswith respectto the Cartan
A =I,,t€,Ae, (this is the reasonwhy we preferto call such a basisa Cartanbasis,in contrast
decomposition
property
basis).The multiplicativity
to the terminologyof the paperwhich speaksof a "normed"multiplicative
(1) assertsthat one dealswith a combinatorially
definedalgebra;howeverin applicationsalso the remaining
propertiesturn out to be of great importance:afterall, any groupalgebrahas,for trivialreasons,a basiswith
property('t), but usuallywill not have a multiplicative
Cartanbasis.
the multiplicative
The main resultof [B-S]assertsthat any representation-finite,
as well as any minimalrepresentation-infinite
algebra,has a multiplicativebasis.The paper is quite long, howeverit still is very condensed:the main
method used is the socalledcleavingprocedure,a sort of partial coveringtheory,and in the numerous
applicationsof this methodusuallyonly the main ingredientsare provided,whereasthe actualverifications
are left to the reader.
Reviews,lwrote in 1987:The resulthas a ratherlonghistory...The generalresultwas
In the Mathematical
announcedby Roiter in 1981, howeverhis proof was incompleteand partly incorrect.The first complete
'1983,but was not
seemsto havebeengivenby Bautistain his lecturesat U.N.A.M.(Mexico)in the springof
published.SinceBautista'sproofwas basedon the ideasof Roiter,the resultmay be referredto as a theorem
of Roiterand Bautista.Gabriel
aboutthis commentand insistedthat he also had a complete
proofat the same time,thus I wrote a corresponding
addendumfor the Reviews.
I remembervery well the spring1983:lwas visitingU.N.A.M.at that time and listenedto the lecturesof
'
work,of course(as mentionedbefore)based
Raymundo What he presentedwas clearlyhis , ,
O n t h e O l d d f a f t O f R O i t e f . , r - : : : : - . , : :: : : . : , . : , ; i . . ; .; 1 . , 1;:-., ; . 1
dumbbelland diamond,
as being nasty: there are three essentiallydifferentkinds, called penny-farthing,
and the essentialobservationassertsthat such factor algebrashave only minor overlappings.We should
in sections8 and 10, dealingwith the simplicialcomplexof a ray
also mentionthe topologicalconsiderations
categoryand showingthe vanishingof a secondcohomologygroup.
One would
The structuretheory presentedin [B-S] has obviouslyscared away all other mathematicians:
Butthis is notthe case!
expectto find a bigvarietyof paperswhichare basedon this marvelleousinvestigation.
This is reallya pity,for severalreasons:one shouldtry to squeezethe argumentsin orderto obtaina more
version;one needscorrespondingresultsin the speciescase (thusworkingwith k-algebras,
comprehensive
wherek is notalgebraically
closed);
andoneshouldtryto understand
in case
themoduletheoretic
behaviour
onedealswithslightlylargef overlappings
of criticalparts:whendo we stillhavetameness?
whendo we get
justas islandswhichcanstillbe separated
wildclusters,
fromthe surrounding.
aI should add that Bautista himself sfresses the parallel streamlinesand the progressive interrelation.He wrote to me: I
cannot claim priority on this work. lt is true that I had a preliminary version of a proof, but this was on lhe bases of joint
work with Salmer'on, on discussionsin Kiev in May 1982 with Roiterand Ovsienko...This version was never published,
may-be in the future I will look again at this old manuscript.