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Let $(R,m)$ be a local ring and $I$ an ideal in $R.$ Let $l(I):=\dim \bigoplus_{n\geq 0}(I^n/mI^n)$ and $x\in R\setminus I.$

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what is the relation between $l(I)$ anf $l(I+(x)/(x))?$

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  • $\begingroup$ What kind of relations are you looking for? Of course, there is one which comes from the dimension of the fiber cones. $\endgroup$
    – Youngsu
    Commented Oct 11, 2017 at 20:14

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