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On the equations defining tangent cones

Published online by Cambridge University Press:  24 October 2008

Lorenzo Robbiano  and
Giuseppe Valla
Lorenzo Robbiano
Affiliation:
Università di Genova, Italia
Giuseppe Valla
Affiliation:
Università di Genova, Italia
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Extract

This paper treats the local study of singularities by means of their tangent cones, more specifically the study of graded rings associated to an ideal of a local ring. We recall some basic facts: let (R, ) be a local ring, I, J ideals of R, such that J ⊆ I; then GR/J(I / J), the graded ring associated to I / J, is canonically isomorphic to the quotient of GR(I) modulo a homogeneous ideal, which is called J*, and which is generated by the so-called ‘initial forms’ of the elements of J. Let us consider the following example: Let k be a field, R = k[X, Y, Z](x, y, Z), I = (X, Y, Z)R, J the prime ideal generated by fl, f2 where f1 = Y3 − Z2, f2 = YZ − X4. Then

and it is easily seen that J* properly contains the ideal generated by the initial forms f*1f*2 of f1, f2; namely f*1 = − Z2, f*2 = YZ and (Yf1 + Zf2)* = Y4 ∉ (−Z2, YZ).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 2 , September 1980 , pp. 281 - 297
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

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