Home Online Manual
Top
Back: modulo (plural)
Forward: nc_algebra
FastBack: Data types (plural)
FastForward: Mathematical background (plural)
Up: Functions (plural)
Top: Singular Manual
Contents: Table of Contents
Index: Index
About: About this document

7.3.15 mres (plural)

Syntax:
mres ( ideal_expression, int_expression )
mres ( module_expression, int_expression )
Type:
resolution
Purpose:
computes a minimal free resolution of an ideal or module M with the Groebner basis method. More precisely, let A=matrix(M), then mres computes a free resolution of $coker(A)=F_0/M$

\begin{displaymath}...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F... ...er{\longrightarrow} F_0\longrightarrow F_0/M \longrightarrow 0,\end{displaymath}

where the columns of the matrix $A_1$are a (possibly) minimal set of generators of $M$. If the int expression k is not zero, then the computation stops after k steps and returns a resolution consisting of modules $M_i={\tt module} (A_i)$, $i= 1 \ldots k$.
mres(M,0) returns a resolution consisting of at most n+2 modules, where n is the number of variables of the basering. Let list L=mres(M,0); then L[1] consists of a minimal set of generators M, L[2] consists of a minimal set of generators for the first syzygy module of L[1], etc., until L[p+1], such that ${\tt L[i]}\neq 0$ for $i \le p$, but L[p+1] (the first syzygy module of L[p]) is 0 (if the basering is not a qring).
Note:
Accessing single elements of a resolution may require that some partial computations have to be finished and may therefore take some time. Hence, assigning right away to a list is the recommended way to do it.
Example:
 
LIB "ncalg.lib";
def A = makeUsl2();
setring A; // this algebra is U(sl_2)
option(redSB);
option(redTail);
ideal i = e,f,h;
i = std(i);
resolution M=mres(i,0);
M;
==>  1      2      2      1      
==> A <--  A <--  A <--  A
==> 
==> 0      1      2      3      
==> 
list l = M; l;
==> [1]:
==>    _[1]=f
==>    _[2]=e
==> [2]:
==>    _[1]=ef*gen(1)-f2*gen(2)-2h*gen(1)-2*gen(1)
==>    _[2]=e2*gen(1)-ef*gen(2)-h*gen(2)+2*gen(2)
==> [3]:
==>    _[1]=e*gen(1)-f*gen(2)
// see the exactness at this point
size(ideal(transpose(l[2])*transpose(l[1])));
==> 0
print(matrix(M[3]));
==> e,
==> -f
// see the exactness at this point
size(ideal(transpose(l[3])*transpose(l[2])));
==> 0
See ideal (plural); minres (plural); module (plural); nres (plural).