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Extra info for Commutative Algebra and Its Connections to Geometry: Pan-american Advanced Studies Institute August 3-14, 2009, Universidade Federal De Pernambuco, Olinda, Brazil

Sample text

Un ] with n generators, choose α α = max{αij , βij }, then we have uα i uj = 0 for all i = j. Consider the k-algebras i,j k[u1 + · · · + un−1 + un ] ⊆ k[u1 + · · · + un−1 , un ] ⊆ k[u1 , . . , un−1 , un ]. We claim that (1) k[u1 , . . , un−1 , un ] is integral over k[u1 + · · · + un−1 , un ], and (2) k[u1 + · · · + un−1 , un ] is integral over k[u1 + · · · + un−1 + un ]. α For (1), since uα i uj = 0 for all i = j, by the induction hypothesis, we have that k[u1 , . . , un−1 ] is integral over k[u1 + u2 + · · · + un−1 ] and so k[u1 , .

It can be checked directly t hat II ∗ (I ∗ )2 . Thus, I is not a reduction of I ∗ . On the other hand, we know that K1 = (x4 + y 4 , x2 y + xy 2 ) is a reduction of I, also through a direct computation: K1 I 2 = I 3 . We observe that x4 , y 4 are in Γ(x4 + y 4 ) and the term x2 y +xy 2 is a combination of two monomials corresponding to two points on the left hand side of the line through (4, 0) and (0, 4). ” However, the authors are not able to verify this statement. 7, K2 = (x4 +xy 2 , x2 y+y 4 ) is a minimal reduction of I ∗ while K1 is not a reduction of I ∗ .

I be the ideal generated by fi = All the monomials occurring in the fi for all i together generate a monomial ideal, denoted I ∗ . A simple exercise shows that such monomial ideal is independent from the choices of the generating set {fi }. 1). We focus on finding a sufficient condition under which I becomes a reduction of I ∗ . In such case, the integral closure and, if I is m-primary, the Hilbert-Samuel multiplicity of I can be obtained straightforwardly from the graph of I ∗ as stated in the previous paragraph.

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