Macaulay2 computes analytic spread for R=QQ[a,b,c,d,e,f] which is not a local ring.
In the books like
"*Cohen-Macaulay rings* (*Bruns-Herzog*)", "*Integral Closure of Ideals, Rings, and Modules* (*Irena Swanson and Craig Huneke*)" and "*Equimultiplicity and Blowing up* (*M. Herrmann S. Ikeda U. Orbanz*)", analytic spread is defined for local rings. then they reach to an equivalent condition:
"the analytic spread of I is the smallest number of generators of an ideal J such that I is integral over J." See for example see Huneke-Swanson, Proposition 8.3.7.
I have two Questions:
Question1: how does Macaulay2 computes analytic spread for non-local rings?
Question2: if Macaulay2 computes analytic spread for ideal $I=yzR$ in $R=k[x,y,z]/(x)\cap (y,z)$ will this be equal to the analytic spread of ideal $I=yzR$ in $R=k[[x,y,z]]/(x)\cap (y,z)$? (this is an example I don't want analytic spread of this $I$)
thank you for reading this long Question.