Table Of ContentUniversitext
W. Frank Moore
Mark Rogers
Sean Sather-Wagstaff
Monomial
Ideals and Their
Decompositions
Universitext
Universitext
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San Francisco State University
Carles Casacuberta
Universitat de Barcelona
Angus MacIntyre
Queen Mary University of London
Kenneth Ribet
University of California, Berkeley
Claude Sabbah
École polytechnique, CNRS, Université Paris-Saclay, Palaiseau
Endre Süli
University of Oxford
Wojbor A. Woyczyński
Case Western Reserve University
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W. Frank Moore Mark Rogers
(cid:129)
Sean Sather-Wagstaff
Monomial Ideals and Their
Decompositions
123
W.Frank Moore SeanSather-Wagstaff
Department ofMathematics Schoolof Mathematical andStatistical
WakeForest University Sciences
Winston-Salem, NC,USA Clemson University
Clemson, SC,USA
Mark Rogers
Department ofMathematics
MissouriState University
Springfield,MO, USA
ISSN 0172-5939 ISSN 2191-6675 (electronic)
Universitext
ISBN978-3-319-96874-2 ISBN978-3-319-96876-6 (eBook)
https://doi.org/10.1007/978-3-319-96876-6
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Dedicated to the memory of Diana Taylor,
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Contents
Introduction... .... .... .... ..... .... .... .... .... .... ..... .... xi
Overview... .... .... .... ..... .... .... .... .... .... ..... .... xii
Audience... .... .... .... ..... .... .... .... .... .... ..... .... xiii
Summary of Contents. .... ..... .... .... .... .... .... ..... .... xiv
Notes for the Instructor/Independent Reader. .... .... .... ..... .... xvii
Possible Course Outlines... ..... .... .... .... .... .... ..... .... xviii
Acknowledgments.... .... ..... .... .... .... .... .... ..... .... xxiii
Part I Monomial Ideals
1 Fundamental Properties of Monomial Ideals . . . . . . . . . . . . . . . . . . 5
1.1 Monomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Integral Domains (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Generators of Monomial Ideals. . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Noetherian Rings (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Exploration: Counting Monomials. . . . . . . . . . . . . . . . . . . . . . . 28
1.6 Exploration: Numbers of Generators . . . . . . . . . . . . . . . . . . . . . 31
2 Operations on Monomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1 Intersections of Monomial Ideals . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Unique Factorization Domains (optional). . . . . . . . . . . . . . . . . . 40
2.3 Monomial Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Exploration: Reduced Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5 Colons of Monomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.6 Bracket Powers of Monomial Ideals . . . . . . . . . . . . . . . . . . . . . 64
2.7 Exploration: Saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.8 Exploration: Generalized Bracket Powers . . . . . . . . . . . . . . . . . 73
2.9 Exploration: Comparing Bracket Powers and Ordinary
Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
vii
viii Contents
3 M-Irreducible Ideals and Decompositions. . . . . . . . . . . . . . . . . . . . . 81
3.1 M-Irreducible Monomial Ideals. . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Irreducible Ideals (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.3 M-Irreducible Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.4 Irreducible Decompositions (optional). . . . . . . . . . . . . . . . . . . . 102
3.5 Exploration: Decompositions in Two Variables, part I . . . . . . . . 106
Part II Monomial Ideals and Other Areas
4 Connections with Combinatorics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1 Square-Free Monomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 Graphs and Edge Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.3 Decompositions of Edge Ideals. . . . . . . . . . . . . . . . . . . . . . . . . 125
4.4 Simplicial Complexes and Stanley-Reisner Ideals . . . . . . . . . . . 131
4.5 Decompositions of Stanley-Reisner Ideals . . . . . . . . . . . . . . . . . 139
4.6 Facet Ideals and Their Decompositions . . . . . . . . . . . . . . . . . . . 146
4.7 Exploration: Alexander Duality. . . . . . . . . . . . . . . . . . . . . . . . . 152
5 Connections with Other Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.1 Krull Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.2 Vertex Covers and PMU Placement . . . . . . . . . . . . . . . . . . . . . 165
5.3 Cohen-Macaulayness and the Upper Bound Theorem. . . . . . . . . 175
5.4 Hilbert Functions and Initial Ideals . . . . . . . . . . . . . . . . . . . . . . 188
5.5 Resolutions of Monomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . 199
Part III Decomposing Monomial Ideals
6 Parametric Decompositions of Monomial Ideals . . . . . . . . . . . . . . . . 221
6.1 Parameter Ideals and Parametric Decompositions. . . . . . . . . . . . 221
6.2 Corner Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
6.3 Finding Corner Elements in Two Variables. . . . . . . . . . . . . . . . 241
6.4 Finding Corner Elements in General . . . . . . . . . . . . . . . . . . . . . 246
6.5 Exploration: Decompositions in Two Variables, part II . . . . . . . 252
6.6 Exploration: Decompositions of Some Powers of Ideals. . . . . . . 253
6.7 Exploration: Macaulay Inverse Systems. . . . . . . . . . . . . . . . . . . 256
7 Computing M-Irreducible Decompositions . . . . . . . . . . . . . . . . . . . . 261
7.1 M-Irreducible Decompositions of Monomial Radicals . . . . . . . . 261
7.2 M-Irreducible Decompositions of Bracket Powers . . . . . . . . . . . 264
7.3 M-Irreducible Decompositions of Sums. . . . . . . . . . . . . . . . . . . 267
7.4 M-Irreducible Decompositions of Colon Ideals . . . . . . . . . . . . . 271
7.5 Computing General M-Irreducible Decompositions . . . . . . . . . . 276
7.6 Exploration: Edge, Stanley-Reisner, and Facet Ideals
Revisited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
7.7 Exploration: Decompositions of Saturations. . . . . . . . . . . . . . . . 285
Contents ix
7.8 Exploration: Decompositions of Generalized Bracket
Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
7.9 Exploration: Decompositions of Products of Monomial
Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Part IV Commutative Algebra and Macaulay2
Appendix A: Foundational Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
A.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
A.2 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
A.3 Ideals and Generators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
A.4 Sums of Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
A.5 Products and Powers of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . 312
A.6 Colon Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
A.7 Radicals of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
A.8 Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
A.9 Partial Orders and Monomial Orders. . . . . . . . . . . . . . . . . . . . . 323
A.10 Exploration: Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . 327
Appendix B: Introduction to Macaulay2 . . . . . . . . . . . . . . . . . . . . . . . . 331
B.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
B.2 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
B.3 Ideals and Generators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
B.4 Sums of Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
B.5 Products and Powers of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . 338
B.6 Colon Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
B.7 Radicals of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
B.8 Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
B.9 Monomial Orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Further Reading ... .... .... ..... .... .... .... .... .... ..... .... 349
References.... .... .... .... ..... .... .... .... .... .... ..... .... 351
Index of Macaulay2 Commands, by Command ... .... .... ..... .... 355
Index of Macaulay2 Commands, by Description .. .... .... ..... .... 361
Index of Names.... .... .... ..... .... .... .... .... .... ..... .... 371
Index of Symbols... .... .... ..... .... .... .... .... .... ..... .... 373
Index of Terminology... .... ..... .... .... .... .... .... ..... .... 377
Introduction
TheFundamentalTheoremofArithmeticstatesthateveryintegern>2factorsintoa
productofprimenumbersinanessentiallyuniqueway.Inalgebraclass,onelearns
a similar factorization result for polynomials in one variable with real number
coefficients: every nonconstant polynomial factors into a product of linear poly-
nomialsandirreduciblequadratic polynomials inanessentially uniqueway. These
examples share some obvious common ideas.
First, in each case we have a set of objects (in the first example, the set of
integers; in the second example, the set of polynomials with real number coeffi-
cients) that can be added, subtracted, and multiplied in pairs so that the resulting
sums, differences, and products are in the same set. (We say that the sets are
“closed” under these operations.) Furthermore, addition and multiplication satisfy
certain rules (or axioms) that make them “nice”: they are commutative and asso-
ciative, they have identities and additive inverses, and they interact coherently
together via the Distributive Law. In other words, each of these sets is a commu-
tative ring with identity. We do not consider division in this setting because, e.g.,
the quotient of two nonzero integers need not be an integer. Commutative rings
with identity arise in many areas of mathematics, like combinatorics, geometry,
graph theory, and number theory.
Second, each example deals with factorization of certain elements into finite
products of “irreducible” elements, that is, elements that cannot themselves be
factored in a nontrivial manner. In general, given a commutative ring R with
identity, the fact that elements can be multiplied implies that elements can be
factored, even if only trivially. One way to study R is to investigate how well its
factorizations behave. For instance, one can ask whether the elements of R can be
factoredintoafiniteproductofirreducibleelements.(Therearenontrivialexamples
where this fails.) Assuming that the elements of R can be factored into a finite
product of irreducibleelements, one can askwhetherthefactorizations areunique.
pffiffiffiffiffiffiffi
ThefirstexampleonemightseewherethisfailsistheringZ½ (cid:2)5(cid:3)consistingofall
pffiffiffiffiffiffiffi
complex numbers aþb (cid:2)5 such that a and b are integers. This ring admits two
xi
