i1 : R = ZZ/101[a..c]; |
i2 : truncate(2,R^1) o2 = image | a2 ab ac b2 bc c2 | 1 o2 : R-module, submodule of R |
i3 : truncate(2,R^1 ++ R^{-3}) o3 = image {0} | a2 ab ac b2 bc c2 0 | {3} | 0 0 0 0 0 0 1 | 2 o3 : R-module, submodule of R |
i4 : truncate(2, ideal(a,b,c^3)/ideal(a^2,b^2,c^4)) o4 = subquotient (| ab ac bc c3 |, | a2 b2 c4 |) 1 o4 : R-module, subquotient of R |
i5 : truncate(2,ideal(a,b*c,c^7)) 2 7 o5 = ideal (a , a*b, a*c, b*c, c ) o5 : Ideal of R |
The base may be ZZ, or another polynomial ring. In this case, the generators may not be minimal, but they do generate.
i6 : A = ZZ[x,y,z]; |
i7 : truncate(2,ideal(3*x,5*y,15)) 2 2 2 o7 = ideal (3x , 3x*y, 3x*z, 5x*y, 5y , 5y*z, 15z ) o7 : Ideal of A |
i8 : trim oo 2 2 2 o8 = ideal (15z , 5y*z, 3x*z, 5y , x*y, 3x ) o8 : Ideal of A |
i9 : truncate(2,comodule ideal(3*x,5*y,15)) o9 = subquotient (| x2 xz y2 yz z2 |, | 3x 5y 15 |) 1 o9 : A-module, subquotient of A |
If i is a multi-degree, then the result is the submodule generated by all elements of degree exactly i, together with all generators of M whose first degree is higher than the first degree of i. The following includes the generator of degree 8,20.
i10 : S = ZZ/101[x,y,z,Degrees=>{{1,3},{1,4},{1,-1}}]; |
i11 : truncate({7,24}, S^1 ++ S^{{-8,-20}}) o11 = image {0, 0} | x4y3 0 | {8, 20} | 0 1 | 2 o11 : S-module, submodule of S |
The coefficient ring may also be a polynomial ring. In this example, the coefficient variables also have degree one. The given generators will generate the truncation over the coefficient ring.
i12 : B = R[x,y,z, Join=>false] o12 = B o12 : PolynomialRing |
i13 : degree x o13 = {1} o13 : List |
i14 : degree B_3 o14 = {1} o14 : List |
i15 : truncate(2, B^1) o15 = image | x2 xy xz y2 yz z2 | 1 o15 : B-module, submodule of B |
i16 : truncate(4, ideal(b^2*y,x^3)) 2 2 2 2 4 3 3 o16 = ideal (b x*y, b y , b y*z, x , x y, x z) o16 : Ideal of B |
If the coefficient variables have degree 0:
i17 : A1 = ZZ/101[a,b,c,Degrees=>{3:{}}] o17 = A1 o17 : PolynomialRing |
i18 : degree a o18 = {} o18 : List |
i19 : B1 = A1[x,y] o19 = B1 o19 : PolynomialRing |
i20 : truncate(2,B1^1) o20 = image | x2 xy y2 | 1 o20 : B1-module, submodule of B1 |
i21 : truncate(2, ideal(a^3*x, b*y^2)) 3 2 3 2 o21 = ideal (a x , a x*y, b*y ) o21 : Ideal of B1 |