Use R ::= QQ[x,y,z];
Depth(R/Ideal(0)); -- the (x,y,z)-depth of the entire ring is 3
3
-------------------------------
I := Ideal(x^5,y^3,z^2);
-- one can check that it is zerodimensional and CM this way
Dim(R/I);
0
-------------------------------
Depth(R/I);
0
-------------------------------
N := Module([x^2,y], [x+z,0]);
Depth(I, R^2/N); --- a max reg sequence would be (z^2,y^3)
2
-------------------------------
Use R ::= QQ[x,y,z,t,u,v];
-- Cauchy-Riemann system in three complex vars!
N := Module([x,y],[-y,x],[z,t],[-t,z],[u,v],[-v,u]);
--- is it CM?
Depth(R^2/N);
3
-------------------------------
Dim(R^2/N);
3
-------------------------------
--- yes!
M := Module([x,y,z],[t,v,u]);
Res(R^3/M);
0 --> R^2(-1) --> R^3
-------------------------------
Depth(R^3/M); -- using Auslander Buchsbaum 6-1=5
5
-------------------------------
Dim(R^3/M); -- not CM
6
-------------------------------
Depth(Ideal(x,y,z,t), R^2/N);
2
-------------------------------
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