listDetComplex -- This function calculates the list with the determinants of some minors of the maps of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree.

Synopsis

• Usage:

  listOfTheDetOfTheComplex = listDetComplex(aChainComplex, varList, complexDegree)

• Inputs:

  • aChainComplex, a chain complex, a chain complex of free modules over a polynomial ring
  • varList, a list, list of variables of the polynomial ring R to take into account for elimination
  • complexDegree, an integer, integer corresponding to the degree of the strand of the chain complex.
• Outputs:

  • aPolynomialList, a list, a list with the determinant polynomials of the maps computed by 'minorsComplex'

Description

i1 : R=QQ[a,b,c,x,y] 

o1 = R

o1 : PolynomialRing

i2 : f1 = a*x^2+b*x*y+c*y^2 

        2              2
o2 = a*x  + b*x*y + c*y

o2 : R

i3 : f2 = 2*a*x+b*y 

o3 = 2a*x + b*y

o3 : R

i4 : M = matrix{{f1,f2}} 

o4 = | ax2+bxy+cy2 2ax+by |

             1       2
o4 : Matrix R  <--- R

i5 : l = {x,y} 

o5 = {x, y}

o5 : List

i6 : dHPM = listDetComplex (koszul M,l,2)

           2     2
o6 = {- a*b  + 4a c, 1}

o6 : List

i7 : dHPM = listDetComplex (koszul M,l,3)

           2      2 2
o7 = {- a*b c + 4a c , c}

o7 : List

i8 : R=QQ[a,b,c,d,e,f,g,h,i,x,y,z] 

o8 = R

o8 : PolynomialRing

i9 : f1 = a*x+b*y+c*z 

o9 = a*x + b*y + c*z

o9 : R

i10 : f2 = d*x+e*y+f*z 

o10 = d*x + e*y + f*z

o10 : R

i11 : f3 = g*x+h*y+i*z 

o11 = g*x + h*y + i*z

o11 : R

i12 : M = matrix{{f1,f2,f3}} 

o12 = | ax+by+cz dx+ey+fz gx+hy+iz |

              1       3
o12 : Matrix R  <--- R

i13 : l = {x,y,z} 

o13 = {x, y, z}

o13 : List

i14 : dHPM = detComplex (koszul M,l,1)

o14 = - c*e*g + b*f*g + c*d*h - a*f*h - b*d*i + a*e*i

o14 : frac(R)

i15 : dHPM = detComplex (koszul M,l,2)

o15 = - c*e*g + b*f*g + c*d*h - a*f*h - b*d*i + a*e*i

o15 : frac(R)

See also

• degHomPolMap -- given a subset of variables 'var' of the polynomial ring R, it returns the base of monomials on these variables, and the matrix of coefficients of a morphism of free modules f:R(d1)+...+R(dn)->R_d with respect to these variables
• detComplex -- This function calculates the determinant of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree.
• minorsComplex -- This function calculates some minors of the maps of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree. The choice of the minors is according to the construction of the determinant of a complex
• mapsComplex -- This function calculates the maps of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree.