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Let $I$ be a homogeneous ideal in a graded local commutative ring $R$, $S$ be its minimal homogeneous system of generators. So, we know that the cardinality of $S$ is unique as the dimension of the vector space $I/\mathfrak{m}I$, where $\mathfrak{m}$ is the maximal graded ideal of $R$.

My question is the following: Is the degree of each element of $S$ uniquely determined by $I$?

Thank for reading my question.

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    There is in general no "**the** minimal homog. system of generators": there are usually many minimal such things.2012-10-08

2 Answers 2