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98 7. EXCEPTIONAL OBJECTSof modules over a hereditary algebra by Crawley-Boevey [20] and for the categoryof coherent sheaves over a weighted projective line by Meltzer [78]. Ringel [94]simplified and extended Crawley-Boevey’s result to arbitrary hereditary Artin algebras.This last result gave a hint that Meltzer’s result should also be true forarbitrary exceptional curves. On the other hand, the proof for an algebraicallyclosed field presented in [78] did not work over an arbitrary field. Moreover, theresults in [53, 56] on tubular curves have shown that it is often not predictablewhether new effects occur or not.We now briefly sketch the idea of the proof from [60]. The proof is by inductionon the rank n of the Grothendieck group K 0 (X). The following lemma is crucial.Lemma 7.1.2. Let X be a non-homogeneous exceptional curve. Then each completeexceptional sequence in H can be shifted by the braid group to an exceptionalsequence which contains a simple object.7.1.3 (Proof of Theorem 7.1.1). Relying on the lemma, the proof of Theorem7.1.1 is straight-forward by induction, like in the algebraically closed case: Forn =2,thatis,ifX is homogeneous, the proof is easy. Assume n>2. Then there is a“canonical” complete exceptional sequence C =(C 1 ,...,C n ) given by a certain tiltingbundle in H ([70]). Given any complete exceptional sequence E =(E 1 ,...,E n )one can assume by the lemma that E n = S is an exceptional simple object. Bythe special structure of C we can also assume C n = S. Then we consider the rightperpendicular category S ⊥ of S, which is an exceptional curve where the rank ofthe Grothendieck group is n − 1, and use the induction hypothesis.□The proof of Lemma 7.1.2 uses the following rank formula [60] which followsfrom [43]. Forming the right perpendicular category E ⊥ to an exceptional vectorbundle E, we switch to a module category E ⊥ ≃ mod(Λ), where Λ is hereditary(not necessarily connected) with n − 1 simple modules.Proposition 7.1.4. Let E be an exceptional object in H + . Denote byS 1 ,...,S n−1 a complete system of simple modules in E ⊥ and by P 1 ,...,P n−1 theirprojective covers. Thenrk(E) 2 n−1[End(E) :k] = ∑i=1rk(P i ) · rk(S i )[End(P i ):k] .7.1.5 (Proof of Lemma 7.1.2). It is sufficient to show that in each orbit Othere is a complete exceptional sequence E =(E 1 ,...,E n ) such that E i is of finitelength for some i. Assume that for the orbit O this is not the case. Then in Oappears an exceptional vector bundle E, such thatrk(E) 2[End(E):k]is minimal. We canassume, that E appears in E with E = E n .FormingE ⊥ , the exceptional sequence(E 1 ,...,E n−1 )iscompleteinE ⊥ . Using the transitivity of the braid group actionfor mod(Λ) proved by Ringel [94], the sequence E can be shifted to the exceptionalsequences (P 1 ,...,P n−1 ,E)and(S n−1 ,...,S 1 ,E), where the P i and the S i are asin the preceding proposition, suitably ordered with P 1 = S 1 . From the rank formulawe get the contradictionrk(P 1 ) 2[End(P 1 ):k]
7.2. EXCEPTIONAL OBJECTS AND GRADED FACTORIALITY 99Corollary 7.1.6. The list of endomorphism skew fields appearing in a completeexceptional sequence in H is invariant.□7.2. Exceptional objects and graded factorialityIf k is algebraically closed then there is a relationship between the concept ofgraded factoriality and the existence of exceptional objects, as illustrated by thefollowing results:7.2.1. ForasmoothprojectivecurveC over an algebraically closed field k, thefollowing are equivalent [69]:(1) C is of genus zero.(2) coh(C) admits an exceptional object.(3) coh(C) admits a tilting object.(4) There is a (commutative) Z-graded factorial k-algebra R, affineofKrulldimension two, such that coh(C) ≃ modZ mod Z(R).0Moreover, it follows from [53] that this is also true for k = R.7.2.2. A similar result which follows from [55, 67], see [68] is: LetH be a smallabelian connected category over an algebraically closed field k. Then the followingassertions are equivalent:(1) H is equivalent to the category of coherent sheaves over an exceptionalcurve.(2) H is of the form modH (R)for a (commutative) H-graded factorial affine k-mod H 0 (R)algebra R of Krull dimension two, where H is a finitely generated abeliangroup of rank one.The results of Chapters 1 and 6 indicate that the implication (1)⇒(2) (replacing“commutative” by “noncommutative”) remains valid for an arbitrary base field (upto the insertion of weights into non-central prime elements). But the converse andalso 7.2.1 is wrong in general, even in a commutative situation, as Lenzing pointedout in [69]:Example 7.2.3. Let k = F 2 and R be the commutative Z-graded algebraF 2 [X, Y, Z]/(X 6 + Y 3 + Z 2 + X 2 Y 2 + X 3 Z),where deg(X) =1,deg(Y ) = 2 and deg(Z) = 3. (Note that R is not generatedin degree zero and one.) Then R is Z-graded factorial and the quotient categorymod Z (R)mod Z 0is equivalent to the category coh(C) of coherent sheaves of a smooth projectivecurve C of genus one (and not zero).(R)□It would be interesting to characterize the class of (noncommutative) gradedfactorial algebras which are related to the exceptional curves.
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M EMOIRSof theAmerican Mathematical
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M EMOIRSof theAmerican Mathematical
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Dedicated to the memory of my belov
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viiiCONTENTS5.1. The automorphism g
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IntroductionCurves of genus zero. I
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INTRODUCTION 3(although not under t
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INTRODUCTION 5ideal P y associated
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INTRODUCTION 7It is important to st
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INTRODUCTION 9of the commutativity
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12 0. BACKGROUNDFor each λ ∈ k l
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14 0. BACKGROUND0.3. Canonical alge
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16 0. BACKGROUND0.3.8 (Exceptional
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18 0. BACKGROUNDWe say that M is a
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20 0. BACKGROUNDThen there is a nat
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22 0. BACKGROUND0.4.8. Let x ∈ X
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0.6. RATIONAL POINTS 25By [89] ther
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CHAPTER 1Graded factorialityIn this
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1.1. EFFICIENT AUTOMORPHISMS 31I M
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1.2. PRIME IDEALS AND UNIVERSAL EXT
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1.3. PRIME IDEALS AS ANNIHILATORS 3
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1.3. PRIME IDEALS AS ANNIHILATORS 3
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40 1. GRADED FACTORIALITYProof. Let
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42 1. GRADED FACTORIALITYOne can su
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44 1. GRADED FACTORIALITYThe non-si
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46 1. GRADED FACTORIALITYProof. The
Page 61 and 62: CHAPTER 2Global and local structure
Page 63 and 64: 2.2. LOCALIZATION AT PRIME IDEALS 5
Page 69 and 70: 2.3. NONCOMMUTATIVITY AND THE MULTI
Page 71 and 72: 2.5. ZARISKI TOPOLOGY AND SHEAFIFIC
Page 73 and 74: CHAPTER 3Tubular shifts and prime e
Page 75 and 76: 3.2. NON-CENTRAL PRIME ELEMENTS AND
Page 77 and 78: 3.2. NON-CENTRAL PRIME ELEMENTS AND
Page 79: 3.2. NON-CENTRAL PRIME ELEMENTS AND
Page 82 and 83: 70 4. COMMUTATIVITY AND MULTIPLICIT
Page 88 and 89: 76 5. AUTOMORPHISM GROUPSan involut
Page 90 and 91: 78 5. AUTOMORPHISM GROUPSwhere Pic(
Page 92 and 93: 80 5. AUTOMORPHISM GROUPSN(a) =1. T
Page 94 and 95: 82 5. AUTOMORPHISM GROUPS5.5. The q
Page 96 and 97: 84 5. AUTOMORPHISM GROUPSzX 2 )(Y 2
Page 98 and 99: 86 5. AUTOMORPHISM GROUPSWe have to
Page 101 and 102: CHAPTER 6Insertion of weightsIn thi
Page 103 and 104: 6.2. INSERTION OF WEIGHTS INTO CENT
Page 105 and 106: 6.2. INSERTION OF WEIGHTS INTO CENT
Page 107 and 108: 6.3. AUTOMORPHISM GROUPS FOR WEIGHT
Page 109: CHAPTER 7Exceptional objectsIn this
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Page 118 and 119: 106 8. TUBULAR EXCEPTIONAL CURVES2T
Page 120 and 121: 108 8. TUBULAR EXCEPTIONAL CURVESwh
Page 122 and 123: 110 8. TUBULAR EXCEPTIONAL CURVES(4
Page 124 and 125: 112 8. TUBULAR EXCEPTIONAL CURVESPr
Page 126 and 127: 114 A. AUTOMORPHISM GROUPS OVER THE
Page 128 and 129: 116 A. AUTOMORPHISM GROUPS OVER THE
Page 131: APPENDIX BThe tubular symbols( ) (
Page 134 and 135: 122 BIBLIOGRAPHY22. V. Dlab, An int
Page 136 and 137: 124 BIBLIOGRAPHY69. , Twenty-one ch
Page 139 and 140: Indexp-cycle, 89q-symbol, 104Algebr
Page 141 and 142: Editorial InformationTo be publishe
Page 143: EditorsThis journal is designed par
Page 146 and 147: TITLES IN THIS SERIES918 Jonathan B
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