Monomial Algebras

✍ Scribed by Rafael Villarreal

• Year: 0
• Tongue: English
• Leaves: 474
• Series: Monographs and Textbooks in Pure and Applied Mathematics
• Edition: 1st
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents......Page 16
Preface......Page 12
1.1 Module theory......Page 20
1.2 Graded modules......Page 31
1.3 Cohen-Macaulay modules......Page 35
1.4 Normal rings......Page 42
1.5 Koszul homology......Page 47
2.1 Noether normalizations......Page 50
2.2 Cohen-Macaulay graded algebras......Page 54
2.3 Hilbert Nullstellensatz......Page 64
2.4 Gröbner bases......Page 67
2.5 Minimal resolutions......Page 77
3.1 Symmetric algebras......Page 84
3.2 Rees algebras and syzygetic ideals......Page 85
3.3 Complete and normal ideals......Page 88
3.4 A criterion of Jürgen Herzog......Page 101
3.5 Jacobian criterion......Page 108
4.1 Hilbert-Serre's Theorem......Page 116
4.2 a-invariants and h-vectors......Page 125
4.3 Extremal algebras......Page 129
4.4 Initial degrees of Gorenstein ideals......Page 137
4.5 A symbolic study of Koszul homology......Page 144
5.1 Primary decomposition......Page 148
5.2 Simplicial complexes and homology......Page 157
5.3 Face rings......Page 161
5.4 Hilbert series of face rings......Page 169
5.5 Upper bound conjectures......Page 173
6.1 Graphs and ideals......Page 180
6.2 Cohen-Macaulay graphs......Page 191
6.3 Trees......Page 197
6.4 Bipartite graphs......Page 201
6.5 Links of some edge ideals......Page 204
6.6 First syzygy module of an edge ideal......Page 207
6.7 Edge rings with linear resolutions......Page 211
7.1 Basic properties......Page 220
7.2 Integral closure of subrings......Page 228
7.3 Integral closure of monomial ideals......Page 252
7.4 Normality of some Rees algebras......Page 258
7.5 Ideals of mixed products......Page 267
7.6 Degree bounds for some integral closures......Page 278
7.7 Degree bounds in the square-free case......Page 280
7.8 Some lexicographical Gröbner bases......Page 295
8 Monomial Subrings of Graphs......Page 300
8.1 The subring associated to a graph......Page 301
8.2 Rees algebras of edge ideals......Page 305
8.3 Incidence matrix of a graph......Page 313
8.4 Circuits of a graph and Gröbner bases......Page 315
8.5 Edge subrings of bipartite planar graphs......Page 322
8.6 Normality of bipartite graphs......Page 330
8.7 The integral closure of an edge subring......Page 334
8.8 The equations of the edge cone......Page 344
9.1 Monomial subrings of bipartite graphs......Page 354
9.2 Monomial subrings of complete graphs......Page 369
9.3 Noether normalizations of edge subrings......Page 378
10.1 Defining equations of monomial curves......Page 386
10.2 Symmetric semigroups......Page 396
10.3 Ideals generated by critical binomials......Page 404
10.4 An algorithm for critical binomials......Page 418
11.1 Systems of binomials in toric ideals......Page 422
11.2 Affine toric varieties......Page 428
11.3 Curves in positive characteristic......Page 433
A.1 Cohen-Macaulay graphs......Page 440
A.2 Unmixed graphs......Page 443
Bibliography......Page 444
Notation......Page 466
C......Page 468
F......Page 469
I......Page 470
M......Page 471
P......Page 472
S......Page 473
Z......Page 474