pure-injective modules over tubular algebras and string algebras

293 Pages·2011·1.09 MB·English

PURE-INJECTIVE MODULES OVER
TUBULAR ALGEBRAS AND STRING
ALGEBRAS
A thesis submitted to the University of Manchester
for the degree of Doctor of Philosophy
in the Faculty of Engineering and Physical Sciences
2011
Richard James Harland
School of Mathematics
Contents
Abstract 8
Declaration 9
Copyright Statement 10
Acknowledgements 11
1 Introduction 12
2 Background 18
2.1 Homological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Projective and injective modules . . . . . . . . . . . . . . . . . 20
2.1.2 Ext and Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Model theory of modules . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Pure-injectives . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 Pp-pairs and finitely presented functors . . . . . . . . . . . . . 27
2.2.3 Pure-injective hulls . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Ziegler spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 Cantor-Bendixson rank . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Bound quiver algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 The quiver of an algebra . . . . . . . . . . . . . . . . . . . . . 33
2.5 Auslander-Reiten quivers . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.1 Translation quivers . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.2 The Auslander-Reiten quiver . . . . . . . . . . . . . . . . . . . 35
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2.5.3 Projective covers in quiver algebras . . . . . . . . . . . . . . . 37
2.6 Torsion pairs and tilting . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6.1 Tilting functors . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Tubular Algebras 41
3.1 Tubular algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1 Integral quadratic forms . . . . . . . . . . . . . . . . . . . . . 42
3.1.2 Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.3 Generalised tubes . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.4 Krause’s canonical exact sequence . . . . . . . . . . . . . . . . 49
3.1.5 Tubular families . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.6 Hereditary algebras and concealed algebras . . . . . . . . . . . 52
3.1.7 Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.8 Tubular extensions . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.9 Basic properties of a tubular algebra . . . . . . . . . . . . . . 58
3.2 Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Modules in stable tubes . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Lattices and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 Modules arising from separating tubular families . . . . . . . . . . . . 70
3.5.1 Infinite dimensional modules . . . . . . . . . . . . . . . . . . . 70
3.5.2 The pure-injective modules in Supp(T ) . . . . . . . . . . . . . 74
3.5.3 The CB-rank of Supp(T ) . . . . . . . . . . . . . . . . . . . . . 75
3.6 Irrational cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.6.1 Pp-formulas at an irrational cut . . . . . . . . . . . . . . . . . 79
3.6.2 The lattice of pp-formulas at an irrational cut . . . . . . . . . 83
4 Modules of Irrational Slope 86
4.1 Modules in tubular families . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 The tubular algebras, C(4,λ), C(6), C(7) and C(8) . . . . . . . . . . 90
4.2.1 C(4,λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.2 C(6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
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4.2.3 C(7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.4 C(8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Indecomposables over C(4,λ), C(6), C(7) and C(8) . . . . . . . . . . 95
4.3.1 The dimension vectors of C-modules . . . . . . . . . . . . . . 96
4.3.2 Pp-pairs near an irrational cut . . . . . . . . . . . . . . . . . . 100
4.4 rad+(χ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5 The width of pp/ ∼ . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C r
4.6 Other classes of tubular algebras . . . . . . . . . . . . . . . . . . . . . 110
4.6.1 Canonical algebras . . . . . . . . . . . . . . . . . . . . . . . . 111
4.6.2 Bush algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.7 Shrinking functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.7.1 Inducing an embedding of pp-lattices . . . . . . . . . . . . . . 115
4.7.2 Shrinking functors between different tubular algebras . . . . . 118
5 String Algebras 120
5.1 String algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.1.1 Finite dimensional string modules . . . . . . . . . . . . . . . . 121
5.1.2 Finite dimensional band modules . . . . . . . . . . . . . . . . 123
5.2 Infinite words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2.1 Infinite dimensional string modules . . . . . . . . . . . . . . . 127
5.2.2 Some isomorphisms between string modules . . . . . . . . . . 128
5.2.3 Ringel’s List . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 Some pp-formulas over string algebras . . . . . . . . . . . . . . . . . 130
5.3.1 Partially ordering the set of words . . . . . . . . . . . . . . . . 130
5.3.2 pp-definable subsets obtained from words . . . . . . . . . . . . 134
5.3.3 Subword notation . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.3.4 Results about pp-formulas defined by words . . . . . . . . . . 136
5.3.5 Pre-Subwords and Post-Subwords . . . . . . . . . . . . . . . . 139
5.4 Comparing infinite and finite strings . . . . . . . . . . . . . . . . . . 140
5.4.1 Comparing pp-types . . . . . . . . . . . . . . . . . . . . . . . 144
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5.4.2 Comparing words with similar subwords . . . . . . . . . . . . 147
5.4.3 A further comment on these subwords . . . . . . . . . . . . . 148
5.5 Simple String Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.6 Pure embeddings between string modules . . . . . . . . . . . . . . . . 152
5.6.1 Periodic and almost-periodic results . . . . . . . . . . . . . . . 152
5.6.2 Aperiodic and half-periodic results . . . . . . . . . . . . . . . 153
5.7 Pp-formulas obtained from finite words . . . . . . . . . . . . . . . . . 155
5.7.1 Links to simple string maps . . . . . . . . . . . . . . . . . . . 156
5.7.2 Homomorphisms between string modules . . . . . . . . . . . . 159
5.7.3 Simple string endomorphisms . . . . . . . . . . . . . . . . . . 161
5.8 1-Sided Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.8.1 1-Sided Modules over Aperiodic Words . . . . . . . . . . . . . 166
5.8.2 1-Sided Modules over Contracting Words . . . . . . . . . . . . 171
5.8.3 1-Sided Modules over Expanding Words . . . . . . . . . . . . 175
6 Indecomposable Pure-Injective Modules 181
6.1 Indecomposable Direct Product Modules . . . . . . . . . . . . . . . . 182
6.1.1 Words satisfying (IC) and the descending chain condition . . . 184
6.1.2 Words not satisfying the descending chain condition . . . . . . 187
6.1.3 Words not satisfying (IC) . . . . . . . . . . . . . . . . . . . . 191
6.2 Pure-injective direct sum string modules . . . . . . . . . . . . . . . . 200
6.3 Maps between string and band modules . . . . . . . . . . . . . . . . . 201
6.3.1 Tubes in the AR-quiver of a string algebra . . . . . . . . . . . 201
6.3.2 Maps between string and band modules . . . . . . . . . . . . . 202
6.3.3 Results about simple string maps . . . . . . . . . . . . . . . . 204
6.3.4 A variant of Ko¨nig’s Lemma . . . . . . . . . . . . . . . . . . . 205
6.4 Spanning sets and almost-invertible maps . . . . . . . . . . . . . . . . 207
6.4.1 The spanning set of a pp-definable subgroup . . . . . . . . . . 209
6.4.2 Almost-invertible morphisms . . . . . . . . . . . . . . . . . . . 210
6.4.3 Facts about almost-invertible maps . . . . . . . . . . . . . . . 212
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6.4.4 Inverses of almost-invertible morphisms . . . . . . . . . . . . . 215
6.5 Infinite almost-invertible chains . . . . . . . . . . . . . . . . . . . . . 216
6.6 Periodic string modules . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.6.1 Power series rings . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.6.2 Implications for M-sequences . . . . . . . . . . . . . . . . . . 222
6.7 M-sequences are eventually stationary . . . . . . . . . . . . . . . . . 227
6.7.1 Rearranging band modules . . . . . . . . . . . . . . . . . . . . 229
6.7.2 The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6.8 Finding spanning sequences . . . . . . . . . . . . . . . . . . . . . . . 235
6.8.1 Pp-formulas freely realised in string modules . . . . . . . . . . 238
6.8.2 Pp-formulas freely realised in band modules . . . . . . . . . . 239
6.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7 Two-Directed Modules 245
7.1 Two-directed modules . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.1.1 Left-words and right-words . . . . . . . . . . . . . . . . . . . . 246
7.1.2 Fundamental elements . . . . . . . . . . . . . . . . . . . . . . 247
7.1.3 Fundamental and homogeneous elements . . . . . . . . . . . . 252
7.1.4 The pp-type of a fundamental element . . . . . . . . . . . . . 257
7.1.5 Extending lemma 161 to almost periodic Z-words . . . . . . . 258
7.2 Pure-injective hulls of string modules . . . . . . . . . . . . . . . . . . 262
7.2.1 Periodic and almost periodic modules . . . . . . . . . . . . . . 262
7.2.2 Aperiodic and half periodic modules . . . . . . . . . . . . . . 264
7.3 Words with similar sets of finite subwords . . . . . . . . . . . . . . . 270
7.3.1 Supp(M(w(cid:48))) ⊆ Supp(M(w)) implies w(cid:48) (cid:52) w . . . . . . . . . 270
7.3.2 w(cid:48) (cid:52) w implies Supp(M(w(cid:48))) ⊆ Supp(M(w)) . . . . . . . . . 272
7.3.3 Words with the same set of finite subwords . . . . . . . . . . . 273
7.4 Algebras with a pp-lattice of defined width . . . . . . . . . . . . . . . 275
7.4.1 The link between fundamental elements and the width of the
pp-lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
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7.4.2 Domestic string algebras with w(pp) < ∞ . . . . . . . . . . . 280
7.5 The equivalence of M(w) and M(w) . . . . . . . . . . . . . . . . . . 281
7.6 Direct summands: an example . . . . . . . . . . . . . . . . . . . . . . 286
8 Conclusions 288
Bibliography 291
Word count 74,613
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The University of Manchester
Richard James Harland
Doctor of Philosophy
Pure-Injective Modules over Tubular Algebras and String Algebras
January 30, 2011
We show that, for any tubular algebra, the lattice of pp-definable subgroups of the
direct sum of all indecomposable pure-injective modules of slope r has m-dimension
2 if r is rational, and undefined breadth if r is irrational- and hence that there are no
superdecomposablepure-injectivesofrationalslope, buttherearesuperdecomposable
pure-injectives of irrational slope, if the underlying field is countable.
We determine the pure-injective hull of every direct sum string module over a
string algebra. If A is a domestic string algebra such that the width of the lattice
of pp-formulas has defined breadth, then classify “almost all” of the pure-injective
indecomposable A-modules.
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Declaration
No portion of the work referred to in this thesis has been
submittedinsupportofanapplicationforanotherdegree
or qualification of this or any other university or other
institute of learning.
9
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