dim(Module) -- compute the Krull dimension

Synopsis

• Function: dim
• Usage:
dim M
• Inputs:
• M,
• Outputs:

Description

Computes the Krull dimension of the module M
 i1 : R = ZZ/31991[a,b,c,d] o1 = R o1 : PolynomialRing i2 : I = monomialCurveIdeal(R,{1,2,3}) 2 2 o2 = ideal (c - b*d, b*c - a*d, b - a*c) o2 : Ideal of R i3 : M = Ext^1(I,R) -- ker (41) called with OptionTable: OptionTable{SubringLimit => infinity} -- ker (41) returned CacheFunction: -*a cache function*- -- ker (41) called with Matrix: 0 -- 2 -- ker (41) returned Module: R assert( ker(map(R^0,R^{{3}, {3}},0)) === (R^{{3}, {3}})) -- ker (42) called with OptionTable: OptionTable{SubringLimit => infinity} -- ker (42) returned CacheFunction: -*a cache function*- -- ker (42) called with Matrix: 0 -- 3 -- ker (42) returned Module: R assert( ker(map(R^0,R^{{2}, {2}, {2}},0)) === (R^{{2}, {2}, {2}})) o3 = cokernel {-3} | c b a | {-3} | d c b | 2 o3 : R-module, quotient of R i4 : dim M o4 = 2 i5 : N = Ext^0(I,R) -- ker (43) called with OptionTable: OptionTable{SubringLimit => infinity} -- ker (43) returned CacheFunction: -*a cache function*- -- ker (43) called with Matrix: {-3} | b -c d | -- {-3} | a -b c | -- ker (43) returned Module: image {-2} | c2-bd | -- {-2} | bc-ad | -- {-2} | b2-ac | assert( ker(map(R^{{3}, {3}},R^{{2}, {2}, {2}},{{b,-c,d}, {a,-b,c}})) === (image(map(R^{{2}, {2}, {2}},R^1,{{c^2-b*d}, {b*c-a*d}, {b^2-a*c}})))) o5 = image {-2} | c2-bd | {-2} | bc-ad | {-2} | b2-ac | 3 o5 : R-module, submodule of R i6 : dim N o6 = 4
Note that the dimension of the zero module is -1.