Computation in Local Rings
Many questions in algebraic geometry involve a study of local properties of varieties, that is, properties of a single point, or of a suitably small neighborhood of a point. For example, in analyzing V(I) for a zero-dimensional ideal I ⊂ k[x 1,..., x n ], even when k is algebraically closed, it sometimes happens that V(I) contains fewer distinct points than the dimension d = dim k[x 1,..., x n ]/ I. In this situation, thinking back to the consequences of unique factorization for polynomials in one variable, it is natural to ask whether there is an algebraic multiplicity that can be computed locally at each point in V(I), with the property that the sum of the multiplicities is equal to d. Similarly in the study of singularities of varieties, one major object of study are local invariants of singular points. These are used to distinguish different types of singularities and study their local structure. In §1 of this chapter, we will introduce the algebra of local rings which is useful for these both types of questions. Multiplicities and some first invariants of singularities will be introduced in §2. In §3 and §4, we will develop algorithmic techniques for computation in local rings parallel to the theory of Gröbner bases in polynomial rings.
KeywordsLocal Ring Maximal Ideal Standard Basis Tangent Cone Local Order
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