# Singular

Preface (ps-file)

1 Rings, Ideals and Standard Bases
1.1 Rings, Polynomials and Ring Maps
1.2 Monomial Orderings
1.3 Ideals and Quotient Rings
1.4 Local Rings and Localization
1.5 Rings Associated to Monomial Orderings
1.6 Normal Forms and Standard Bases
1.7 The Standard Basis Algorithm
1.8 Operations on Ideals and their Computation
1.8.1 Ideal Membership
1.8.2 Intersection with Subrings
1.8.3 Zariski Closure of the Image
1.8.4 Solvability of Polynomial Equations
1.8.5 Solving Polynomial Equations
1.8.7 Intersection of Ideals
1.8.8 Quotient of Ideals
1.8.9 Saturation
1.8.10 Kernel of a Ring Map
1.8.11 Algebraic Dependence and Subalgebra Membership
2. Modules
2.1 Modules, Submodules and Homomorphisms
2.3 Standard Bases for Modules
2.4 Exact Sequences and free Resolutions
2.5 Computing Resolutions and the Syzygy Theorem
2.6 Modules over Principal Ideal Domains
2.7 Tensor Product
2.8 Operations on Modules and their Computation
2.8.1 Module Membership Problem
2.8.2 Intersection with Free Submodules
2.8.3 Intersection of Submodules
2.8.4 Quotients of Submodules
2.8.5 Radical and Zerodivisors of Modules
2.8.6 Annihilator and Support
2.8.7 Kernel of a Module Homomorphism
2.8.8 Solving Systems of Linear Equations
3. Noether Normalization and Applications
3.1 Finite and Integral Extensions
3.2 The Integral Closure
3.3 Dimension
3.4 Noether Normalization
3.5 Applications
3.6 An Algorithm to Compute the Normalization
3.7 Procedures
4. Primary Decomposition and Related Topics
4.1 The Theory of Primary Decomposition
4.2 Zero-dimensional Primary Decomposition
4.3 Higher Dimensional Primary Decomposition
4.4 The Equidimensional Part of an Ideal
4.6 Procedures
5. Hilbert Function and Dimension
5.1 The Hilbert Function and the Hilbert Polynomial
5.2 Computation of the Hilbert-Poincare Series
5.3 Properties of the Hilbert Polynomial
5.4 Filtrations and the Lemma of Artin-Rees
5.5 The Hilbert-Samuel Function
5.6 Characterization of the Dimension of Local Rings
5.7 Singular Locus
6. Complete Local Rings
6.1 Formal Power Series Rings
6.2 Weierstrass Preparation Theorem
6.3 Completions
6.4 Standard bases
7. Homological Algebra
7.1 Tor and Exactness
7.2 Fitting Ideals
7.3 Flatness
7.4 Local Criteria for Flatness
7.5 Flatness and Standard Bases
7.6 Koszul Complex and Depth
7.7 Cohen-Macaulay Rings
7.8 Further Characterization of Cohen-Macaulayness
A. Geometric Background
A.1 Introduction by Pictures (ps-file)
A.2 Affine Algebraic Varieties
A.3 Spectrum and Affine Schemes
A.4 Projective Varieties
A.5 Projective Schemes and Varieties
A.6 Morphisms between Varieties
A.7 Projective Morphisms and Elimination
A.8 Local versus Global Properties
A.9 Singularities
B. SINGULAR - A Short Introduction (ps-file)
B.2 Getting Started
B.3 Procedures and Libraries
B.4 Data Types
B.5 Functions
B.6 Control Structures
B.7 System Variables
B.8 Libraries
B.9 SINGULAR and Maple
B.10 SINGULAR and Mathematica

References (ps-file)
Index (ps-file)
Algorithms

### SINGULAR Examples

 algebraic dependence, 87 annihilator, 186 Betti numbers, 135 - graded, 137 classification of singularities, 493 computation - in fields, 5 - in polynomial rings, 7 - in quotient rings, 25 - of d(I,K[x]), 222 - of Hom, 106 - of the dimension, 211 - of Tor, 340 computing with radicals, 27 counting nodes, 489 creating ring maps, 8 cyclic decomposition, 159 deformation of singularities, 495 degree, 289 - of projection, 469 - of projective variety, 478 diagonal form, 154 dimension, 289 - embedding, 304 - of a module, 302 elimination - and resultant, 431 - of module components, 180 - of variables, 71 - projective, 466 equidimensional - decomposition, 263 - part, 261 estimating the determinacy, 491 finite maps, 196 finiteness test, 324 Fitting ideal, 186, 345 flat locus, 356 flatness test, 369 flattening stratification, 352 global versus local rings, 35 graded - Betti numbers, 137 - rings and modules, 116 highest corner, 60 Hilbert - function, 289 - polynomial, 299 Hilbert-Poincare series, 282 homogeneous resolution, 137 ideal membership, 68 image of module homomorphism, 99 independent set, 220 initial ideal, 299 injective, 420 integral - closure of an ideal, 202 - elements, 195 intersection - of ideals, 79 - of submodules, 102, 181 inverse of a power series, 316 Jacobian criterion, 304 Jordan normal form, 163 kernel - of a ring map, 85 - of module homomorphism, 99, 187 Koszul complex, 378 leading data, 11 linear combination of ideal members, 68 local and global dimension, 472 lying over theorem, 225 maps induced by Hom, 96 matrix operations, 94 Milnor and Tjurina number, 488 minimal - associated primes, 209 - presentations, 109 module - annihilator, 186 - membership, 178 - presentation of, 104 - quotient, 102 - radical and zerodivisors, 184 monomial orderings, 16 morphisms of projective varieties, 455 multiplicity, 480 Noether normalization, 216 non-normal locus, 232 normal form, 51, 123 normalization, 230 Poincare series, 299 presentation of a module, 104 primary - decomposition, 258 - test, 252 projective - closure, 443 - elimination, 466 - Nullstellensatz, 437 - subschemes, 446 properties of ring maps, 20 quotient - of ideals, 81 - of submodules, 102, 183 radical, 184, 265 - membership, 78 realization of rings, 42 reduction to zero-dimensional case, 257 regular - sequences, 372 - system of parameters, 304 regularity test, 400 resolution, 135 - homogeneous, 137 saturation, 83, 446 Schreyer resolution, 150 singular locus, 309 solving equations, 76 - linear, 189 standard bases, 59, 124 subalgebra membership, 87 submodules, 104 - intersection of, 102, 181 - of An, 98 sum of submodules, 102 surface plot, 407, 413 surjective, 420 syzygies, 141 tangent cone, 480 tensor product - of maps, 170 - of modules, 172 - of rings, 175 test - for Cohen-Macaulayness, 386, 392, 394 - for flatness, 354 - for local freeness, 347 Weierstrass polynomial, 321 Zariski closure of the image, 74 zero-dimensional primary decomposition, 253 zerodivisors, 184 z-general power series, 320