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The proof given in the text I'm reading is (here $S=K[x_1, \dots , x_n]$):

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The bit I'm having problems with understanding is how does $v=u_i w$ imply that supp$(f) \subset N$? It would make sense to me if the implication were that supp$(f)\subset I$ because $u_i$ are in $I$ and $w \in$ supp$(f_i) \subset$ Mon$(S) \subset I$. I'm new and still learning and sorry if this is a trivial question.

Mark
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1 Answers1

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I suppose that $\text{supp}(f)$ is the set of monomials which effectively appear in $f$. When write $f=\sum_{i=1}^mf_iu_i$ the monomials of $f$ are of the form $wu_i$ with $w$ a monomial from $f_i$. Since $u_i$ is a monomial from $I$ then the same holds for $wu_i$.

user26857
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