Monomial Algebras

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CRC Press, Dec 5, 2000 - Mathematics - 474 pages
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An introduction to the methods used to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings, and toric varieties. It emphasizes square-free quadratics and effective computational methods, and applies a combinatorial description of the integral closure of the corresponding monomial subring to graph theory.
 

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Contents

Commutative Algebra
ix
12 Graded modules
10
13 CohenMacaulay modules
14
14 Normal rings
21
15 Koszul homology
26
Affine and Graded Algebras
29
22 CohenMacaulay graded algebras
33
23 Hilbert Nullstellensatz
43
Monomial Subrings
199
72 Integral closure of subrings
207
73 Integral closure of monomial ideals
231
74 Normality of some Rees algebras
237
75 Ideals of mixed products
246
76 Degree bounds for some integral closures
257
77 Degree bounds in the squarefree case
259
78 Some lexicographical Grobner bases
274

24 Grobner bases
46
25 Minimal resolutions
56
Rees Algebras and Normality
63
32 Rees algebras and syzygetic ideals
64
33 Complete and normal ideals
67
34 A criterion of Jurgen Herzog
80
35 Jacobian criterion
87
Hilbert Series
95
42 ainvariants and hvectors
104
43 Extremal algebras
108
44 Initial degrees of Gorenstein ideals
116
45 A symbolic study of Koszul homology
123
Monomial Ideals and StanleyReisner Rings
127
52 Simplicial complexes and homology
136
53 Face rings
140
54 Hilbert series of face rings
148
55 Upper bound conjectures
152
Edge Ideals
159
62 CohenMacaulay graphs
170
63 Trees
176
64 Bipartite graphs
180
65 Links of some edge ideals
183
66 First syzygy module of an edge ideal
186
67 Edge rings with linear resolutions
190
Monomial Subrings of Graphs
279
82 Rees algebras of edge ideals
284
83 Incidence matrix of a graph
292
84 Circuits of a graph and Grobner bases
294
85 Edge subrings of bipartite planar graphs
301
86 Normality of bipartite graphs
309
87 The integral closure of an edge subring
313
88 The equations of the edge cone
323
Semigroup Rings of Complete Graphs
333
92 Monomial subrings of complete graphs
348
93 Noether normalizations of edge subrings
357
Monomial Curves
365
102 Symmetric semigroups
375
103 Ideals generated by critical binomials
383
104 An algorithm for critical binomials
397
Affine Toric Varieties and Toric Ideals
401
112 Affine toric varieties
407
113 Curves in positive characteristic
412
Graph Diagrams
419
A2 Unmixed graphs
422
Bibliography
423
Notation
445
Index
447
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