UserFunction FreeResolution

FreeResolution}(u:DPOLID(C,v,O)|MODULE(DPOLY(C,vs,O),ORD)

The single argument $u$ has the type $DPOLID$ which abbreviates
{\em Distributive Polynomial Ideal} or $MODULE$ which abbreviates {\em VectorModule}. The meaning of its parameters is as follows.

$C$: Coefficient type.

$v=\{v_1,v_2,\ldots \}$; Distributive variables in decreasing order.

$O:LEX|GRLEX|GREVLEX$: Term ordering.

$ORD:TOP|POT$: Module term order (see Adams and Loustaunau, page 142).

Specification. Returns the free resolution of $u$.

Examples. The input is given in Reduce algebraic mode syntax. The output for each
minisession is returned in a separate window.

gs:={x**2-x,x*y,y**2-y};

FreeResolution(gs|DPOLID(Q,{y,x},LEX)|);

Groebner Basis for given Ideal in $A\equiv{\bf Q}[x,y].$
Term Order $LEX,x>y.$ Module Order $TOP$.
$I\equiv<x^{2}-x,xy,y^{2}-y>$
Free Resolution: $0\longrightarrow A^2\stackrel{\phi_2}\longrightarrow A^3\stackrel{\phi_1 \longrightarrow I\longrightarrow 0$ $0\longrightarrow A^2\stackrel{\phi_2}$

 
\begin{array}{cc}
-y &
0\\
x-1 &
-y+1\\
0 &
x\\
\end{array}
\right]$
$\phi_1\hat{=}\left[
\begin{array}{ccc}
x^{2}-x &
xy &
y^{2}-y\\
\end{array}
\right]$
$    $

Related Functions. JanetResolution.

References.

W.W. Adams, P. Loustaunau, An Introduction to Groebner Bases, American Mathematical Society, 1994.

D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms Using Algebraic Geometry,
Undergraduate Texts in Mathematics, Springer, 1992 and 1998.

G.M. Greuel, G. Pfister, A Singular Introduction to Commutative Algebra, Springer, 2002.

M. Kreuzer, L. Robbiano, Using Computational Commutative Algebra I and II, Springer, 2000.