# how does Macaulay2 computes analytic spread for non-local rings?

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.

The analytic spread is equal for both rings you mentioned since it is a finitely generated $k$-algebra, where $k = R/$maximal ideal.
In fact, since your ideal is principal, the analytic spread is either $0$ or $1$. If it were $0$, then $yz$ is nilpotent, but it is not (cf. $yz \notin (x))$. Therefore, analytic spread of $I$ is $1$. In other words, it is its own minimal reduction.