The analytic spread of a module is the dimension of its special fiber ring. When I is an ideal (and more generally, with the right definitions) 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 the book Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series, 336. Cambridge University Press, Cambridge, 2006, by Craig Huneke and Irena Swanson.
i1 : R=QQ[a,b,c,d,e,f]
o1 = R
o1 : PolynomialRing
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i2 : M=matrix{{a,c,e},{b,d,f}}
o2 = | a c e |
| b d f |
2 3
o2 : Matrix R <--- R
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i3 : analyticSpread image M
o3 = 3
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i4 : specialFiberIdeal image M
o4 = ideal (f, e, d, c, b, a)
o4 : Ideal of R[w , w , w ]
0 1 2
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