# Singular          ##### 7.5.5.0. deRhamCohomIdeal
Procedure from library `dmodapp.lib` (see dmodapp_lib).

Usage:
deRhamCohomIdeal (I[,w,eng,k,G]);
I ideal, w optional intvec, eng and k optional ints, G optional ideal

Return:
ideal

Assume:
The basering is the n-th Weyl algebra D over a field of characteristic
zero and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1
holds, i.e. the sequence of variables is given by
x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator
belonging to x(i).
Further, assume that I is of special kind, namely let f in K[x] and
consider the module K[x,1/f]f^m, where m is smaller than or equal to
the minimal integer root of the Bernstein-Sato polynomial of f.
Since this module is known to be a holonomic D-module, it has a cyclic
presentation D/I.

Purpose:
computes a basis of the n-th de Rham cohomology group of the complement
of the hypersurface defined by f

Note:
The elements of the basis are of the form f^m*p, where p runs over the
entries of the returned ideal.
If I does not satisfy the assumptions described above, the result might
have no meaning. Note that I can be computed with `annfs`.
If w is an intvec with exactly n strictly positive entries, w is used
in the computation. Otherwise, and by default, w is set to (1,...,1).
If eng<>0, `std` is used for Groebner basis computations,
otherwise, and by default, `slimgb` is used.
Let F(I) denote the Fourier transform of I wrt w.
An integer smaller than or equal to the minimal integer root of the
b-function of F(I) wrt the weight (-w,w) can be specified via the
optional argument k.
The optional argument G is used for specifying a Groebner Basis of F(I)
wrt the weight (-w,w), that is, the initial form of G generates the
initial ideal of F(I) wrt the weight (-w,w).
Further note, that the assumptions on I, k and G (if given) are not
checked.

Theory:
(SST) pp. 232-235

Display:
If printlevel=1, progress debug messages will be printed,
if printlevel>=2, all the debug messages will be printed.

Example:
 ```LIB "dmodapp.lib"; ring r = 0,(x,y,z),dp; poly F = x^3+y^3+z^3; bfctAnn(F); // Bernstein-Sato poly of F has minimal integer root -2 ==> : ==> _=-1 ==> _=-4/3 ==> _=-5/3 ==> _=-2 ==> : ==> 2,1,1,1 def W = annRat(1,F^2); // so we compute the annihilator of 1/F^2 setring W; W; // Weyl algebra, contains LD = Ann(1/F^2) ==> // coefficients: QQ ==> // number of vars : 6 ==> // block 1 : ordering dp ==> // : names x y z Dx Dy Dz ==> // block 2 : ordering C ==> // noncommutative relations: ==> // Dxx=x*Dx+1 ==> // Dyy=y*Dy+1 ==> // Dzz=z*Dz+1 LD; // K[x,y,z,1/F]F^(-2) is isomorphic to W/LD as W-module ==> LD=x*Dx+y*Dy+z*Dz+6 ==> LD=z^2*Dy-y^2*Dz ==> LD=z^2*Dx-x^2*Dz ==> LD=y^2*Dx-x^2*Dy ==> LD=x^3*Dz+y^3*Dz+z^3*Dz+6*z^2 ==> LD=x^3*Dy+y^3*Dy+y^2*z*Dz+6*y^2 deRhamCohomIdeal(LD); // we see that the K-dim is 2 ==> _=-x^3*Dx*Dy*Dz ==> _=-x*y*z*Dx*Dy*Dz ``` 