# truncate(List,Module) -- truncation of the graded ring, ideal or module at a specified degree or set of degrees

## Synopsis

• Function: truncate
• Usage:
truncate(degs, M)
• Inputs:
• degs, a list, an integer or a list, a single degree, a multidegree, or a list of degrees or multidegrees
• M, , , an ideal, or a ring,
• Optional inputs:
• MinimalGenerators => , default value true, indicates whether the result should be trimmed
• Outputs:
• , or an ideal, the truncation submodule $M_{\ge degs}$

## Description

The truncation to degree $d$ in the singly graded case of a module (or ring or ideal) is generated by all homogeneous elements of degree at least $d$ in $M$. The resulting truncation is minimally generated (assuming that $M$ is graded).

 i1 : R = ZZ/101[a..c]; i2 : truncate(2, R) 2 2 2 o2 = ideal (c , b*c, a*c, b , a*b, a ) o2 : Ideal of R i3 : truncate(2, R^1) o3 = image | c2 bc ac b2 ab a2 | 1 o3 : R-module, submodule of R i4 : truncate(2, R^{0,-3}) o4 = image {0} | c2 bc ac b2 ab a2 0 | {3} | 0 0 0 0 0 0 1 | 2 o4 : R-module, submodule of R

The coefficient ring of $R$ may be $\ZZ$ or another polynomial ring. Over $\ZZ$, the generators may not be minimal, but they do generate.

 i5 : A = ZZ[x,y,z]; i6 : truncate(2, ideal(3*x, 5*y, 15)) 2 2 2 o6 = ideal (15z , 5y*z, 3x*z, 5y , x*y, 3x ) o6 : Ideal of A

If a multi-degree $d$ is given, then the result is the submodule generated by elements of degree $d+\NN\mathcal C$ where $\mathcal C$ is either a generating set for the degree semigroup of $R$ or the Nef cone of the toric variety.

The following example finds the 11 generators needed to obtain all graded elements whose degrees are at least $\{7,24\}$.

 i7 : S = ZZ/101[x,y,z, Degrees => {{1,3},{1,4},{1,0}}]; i8 : trunc = truncate({7,24}, S^1 ++ S^{{-8,-20}}) o8 = image {0, 0} | y6z y7 xy6 x2y5 x3y4 x4y3 x6y2 x7y x8 0 0 | {8, 20} | 0 0 0 0 0 0 0 0 0 y x2 | 2 o8 : S-module, submodule of S i9 : degrees trunc o9 = {{7, 24}, {7, 28}, {7, 27}, {7, 26}, {7, 25}, {7, 24}, {8, 26}, {8, 25}, ------------------------------------------------------------------------ {8, 24}, {9, 24}, {10, 26}} o9 : List

Given a list of multi-degrees $D$, then the result is the submodule generated by elements of degree $d+\NN\mathcal C$ for any $d\in D$.

The following example finds the generators needed to obtain all graded elements whose degrees at least $\{3,0\}$ or at least $\{0,1\}$. The resulting module is also minimally generated.

 i10 : S = ZZ/101[x,y,z, Degrees => {{1,3},{1,4},{1,0}}]; i11 : trunc = truncate({{3,0}, {0,1}}, S^1 ++ S^{{-8,-20}}) o11 = image {0, 0} | y x z3 yz2 xz2 y2z xyz x2z y3 xy2 x2y x3 0 | {8, 20} | 0 0 0 0 0 0 0 0 0 0 0 0 1 | 2 o11 : S-module, submodule of S i12 : degrees trunc o12 = {{1, 4}, {1, 3}, {3, 0}, {3, 4}, {3, 3}, {3, 8}, {3, 7}, {3, 6}, {3, ----------------------------------------------------------------------- 12}, {3, 11}, {3, 10}, {3, 9}, {8, 20}} o12 : List

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.

 i13 : B = R[x,y,z, Join => false]; i14 : degrees B o14 = {{1}, {1}, {1}} o14 : List i15 : truncate(2, B^1) o15 = image | c2 bc ac b2 ab a2 cz bz az cy by ay cx bx ax z2 yz xz y2 xy x2 | 1 o15 : B-module, submodule of B i16 : truncate(4, ideal(b^2*y,x^3)) 2 3 2 2 2 2 2 3 3 3 3 o16 = ideal (b c*y, b y, a*b y, b y*z, b y , b x*y, c*x , b*x , a*x , x z, ----------------------------------------------------------------------- 3 4 x y, x ) o16 : Ideal of B

If the coefficient variables have degree 0:

 i17 : A1 = ZZ/101[a,b,c, Degrees => {3:{}}]; i18 : degrees A1 o18 = {{}, {}, {}} o18 : List i19 : B1 = A1[x,y]; i20 : degrees B1 o20 = {{1}, {1}} o20 : List i21 : truncate(2, B1^1) o21 = image | y2 xy x2 | 1 o21 : B1-module, submodule of B1 i22 : truncate(2, ideal(a^3*x, b*y^2)) 2 3 3 2 o22 = ideal (b*y , a x*y, a x ) o22 : Ideal of B1

## Caveat

The behavior of this function has changed as of Macaulay2 version 1.13. This is a (potentially) breaking change. Before, it used a less useful notion of truncation, involving the heft vector, and was often not what one wanted in the multi-graded case. Additionally, in the tower ring case, when the coefficient ring had variables of nonzero degree, sometimes incorrect answers resulted.

Also, the function expects a graded module, ring, or ideal, but this is not checked, and some answer is returned.