Mathematics is a subject that has always been considered to be a challenging discipline, particularly at the graduate level. With the advancements in technology, there are now a plethora of resources available for students to learn and master the subject. One such resource is the book “Graduate Texts in Mathematics” written by Jürgen Herzog and Takayuki Hibi. This book is considered to be an essential resource for students pursuing a degree in mathematics at the graduate level.

Jürgen Herzog is a renowned mathematician, researcher and professor at the University of Duisburg-Essen, Germany. He has extensive experience in the field of algebra and has contributed significantly to the development of the subject. Takayuki Hibi, on the other hand, is a professor at the Graduate School of Information Science and Technology, Osaka University, Japan. He is a renowned mathematician and researcher and has made significant contributions to the field of combinatorial mathematics.

**TABLE OF CONTENTS**

**Monomial Ideals**

- Basic properties of monomial ideals
- The K-basis of a monomial ideal
- Monomial generators
- The Zn-grading
- Algebraic operations on monomial ideals
- Standard algebraic operations
- Saturation and radical
- Primary decomposition and associated prime ideals
- Irreducible monomial ideals
- Primary decompositions
- Integral closure of ideals
- Integral closure of monomial ideals
- Normally torsionfree squarefree monomial ideals
- Squarefree monomial ideals and simplicial complexes
- Simplicial complexes
- Stanley–Reisner ideals and facet ideals
- The Alexander dual
- Polarization

**2 A short introduction to Gr ̈obner bases**

- Dickson’s lemma and Hilbert’s basis theorem
- Dickson’s lemma
- Monomial orders
- Gr ̈obner bases
- Hilbert’s basis theorem
- The division algorithm
- The division algorithm
- Reduced Gr ̈obner bases
- Buchberger’s criterion
- S-polynomials
- Buchberger’s criterion
- Buchberger’s algorithm.

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**3 Monomial orders and weights**

- Initial terms with respect to weights
- Gradings defined by weights
- Initial ideals given by weights
- The initial ideal as the special fibre of a flat family
- Homogenization
- A one parameter flat family
- Comparison of I and in(I)

**4 Generic initial ideals**

- Existence
- Stability properties of generic initial ideals
- The theorem of Galligo and Bayer–Stillman
- Borel-fixed monomial ideals
- Extremal Betti numbers
- Almost regular sequences and generic annihilator numbers
- Annihilator numbers and Betti numbers

**5 The exterior algebra**

- Graded modules over the exterior algebra
- Basic concepts
- The exterior face ring of a simplicial complex
- Duality
- Simplicial homology
- Gr ̈obner bases
- Monomial orders and initial ideals
- Buchberger’s criterion
- Generic initial ideals and generic annihilator numbers in the exterior algebra

**6 Hilbert functions and the theorems of Macaulay and Kruskal–Katona**

- Hilbert functions, Hilbert series and Hilbert polynomials
- The Hilbert function of a graded R-module
- Hilbert functions and initial ideals
- Hilbert functions and resolutions
- The h-vector of a simplicial complex
- Lexsegment ideals and Macaulay’s theorem
- Squarefree lexsegment ideals and the Kruskal–Katona Theorem

**7 Resolutions of monomial ideals and the Eliahou–Kervaire formula**

- The Taylor complex
- Betti numbers of stable monomial ideals
- Modules with maximal Betti numbers
- Stable monomial ideals
- The Bigatti–Hulett theorem
- Betti numbers of squarefree stable ideals
- Comparison of Betti numbers over the symmetric and exterior algebra

**8 Alexander duality and resolutions**

- The Eagon–Reiner theorem
- Hochster’s formula
- Reisner’s criterion and the Eagon–Reiner theorem
- Componentwise linear ideals
- Ideals with linear quotients
- Monomial ideals with linear quotients and shellable simplicial complexes
- Componentwise linear ideals
- Ideals with linear quotients and componentwise linear ideals
- Squarefree componentwise linear ideals
- Sequentially Cohen–Macaulay complexes
- Ideals with stable Betti numbers

**9 Alexander duality and finite graphs**

- Edge ideals of finite graphs
- Basic definitions
- Finite partially ordered sets
- Cohen–Macaulay bipartite graphs
- Unmixed bipartite graphs
- Sequentially Cohen–Macaulay bipartite graphs
- Dirac’s theorem on chordal graphs
- Edge ideals with linear resolution
- The Hilbert–Burch theorem for monomial ideals
- Chordal graphs and quasi-forests
- Dirac’s theorem on chordal graphs
- Edge ideals of chordal graphs
- Cohen–Macaulay chordal graphs
- Chordal graphs are shellable

**10 Powers of monomial ideals**

- Toric ideals and Rees algebras
- Toric ideals
- Rees algebras and the x-condition
- Powers of monomial ideals with linear resolution
- Monomial ideals with 2-linear resolution
- Powers of monomial ideals with 2-linear resolution
- Powers of vertex cover ideals of Cohen–Macaulay bipartite graphs
- Powers of vertex cover ideals of Cohen–Macaulay chordal graphs
- Depth and normality of powers of monomial ideals
- The limit depth of a graded ideal
- The depth of powers of certain classes of monomial ideals
- Normally torsionfree squarefree monomial ideals and Mengerian simplicial complexes
- Classes of Mengerian simplicial complexes

**11 Shifting theory**

- Combinatorial shifting
- Shifting operations
- Combinatorial shifting
- Exterior and symmetric shifting
- Exterior algebraic shifting
- Symmetric algebraic shifting
- Comparison of Betti numbers
- Graded Betti numbers of IΔ and IΔs
- Graded Betti numbers of IΔe and IΔc
- Graded Betti numbers of IΔ and IΔc
- Extremal Betti numbers and algebraic shifting
- Superextremal Betti numbers

**12 Discrete Polymatroids**

- Classical polyhedral theory on polymatroids
- Matroids and discrete polymatroids
- Integral polymatroids and discrete polymatroids
- The symmetric exchange theorem
- The base ring of a discrete polymatroid
- Polymatroidal ideals
- Weakly polymatroidal ideals.

**Download Now Book in PDF**

The book “Graduate Texts in Mathematics” covers a wide range of topics that are considered to be essential for students pursuing a degree in mathematics at the graduate level. The authors have used clear and concise language to explain complex mathematical concepts, making the book an accessible resource for students. The book also includes numerous examples and exercises to help students understand and apply the concepts they learn.

One of the key features of the book is the comprehensive coverage of topics. The authors have included a wide range of topics, starting from the basics of algebra and progressing to more advanced topics such as homological algebra. The book is well-structured, making it easy for students to follow and understand the concepts.

In conclusion, the book “Graduate Texts in Mathematics” by Jürgen Herzog and Takayuki Hibi is an essential resource for students pursuing a degree in mathematics at the graduate level. The authors have done an excellent job of covering a wide range of topics in a clear and concise manner, making the book accessible to students of all levels. We are pleased to inform students that this book is available in PDF format, making it easy for students to download and access.