Published online by Cambridge University Press: 06 July 2010
These lectures provide a glimpse of the applications of toric geometry to singularity theory. They illustrate some ideas and results of commutative algebra by showing the form which they take for very simple ideals of polynomial rings: monomial or binomial ideals, which can be understood combinatorially. Some combinatorial facts are the expression for monomial or binomial ideals of general results of commutative algebra or algebraic geometry such as resolution of singularities or the Brianç– Skoda theorem. In the opposite direction, there are methods that allow one to prove results about fairly general ideals by continuously specializing them to monomial or binomial ideals.
We shall also meet a property of finitely generated ideals that is stronger than principality, namely that given any pair of generators, one divides the other. This implies principality (exercise), but is stronger in general: take an ideal in a principal ideal domain such as Z, or a nonmonomial ideal in k[u]. I shall call this property strong principality. Integral domains in which every finitely generated ideal is strongly principal are known as valuation rings. Most are not noetherian
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