# CoherentSheaf ^** ZZ -- tensor power

## Synopsis

• Operator: ^**
• Usage:
M^**i
• Inputs:
• Outputs:
• , the i-th tensor power of M

## Description

The second symmetric power of the canonical sheaf of the rational quartic:
 i1 : R = QQ[a..d]; i2 : I = monomialCurveIdeal(R,{1,3,4}) 3 2 2 2 3 2 o2 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c) o2 : Ideal of R i3 : X = variety I o3 = X o3 : ProjectiveVariety i4 : KX = sheaf(Ext^1(I,R^{-4}) ** ring X) o4 = cokernel {1} | c 0 -d 0 -b | {1} | b c 0 a 0 | {1} | 0 d c b a | 3 o4 : coherent sheaf on X, quotient of OO (-1) X i5 : K2 = KX^**2 o5 = cokernel {2} | c 0 -d 0 -b 0 0 0 0 0 0 0 0 0 0 c 0 -d 0 -b 0 0 0 0 0 0 0 0 0 0 | {2} | b c 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 -d 0 -b 0 0 0 0 0 | {2} | 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 -d 0 -b | {2} | 0 0 0 0 0 c 0 -d 0 -b 0 0 0 0 0 b c 0 a 0 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 b c 0 a 0 0 0 0 0 0 0 0 0 0 0 b c 0 a 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b c 0 a 0 | {2} | 0 0 0 0 0 0 0 0 0 0 c 0 -d 0 -b 0 d c b a 0 0 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 b c 0 a 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 d c b a | 9 o5 : coherent sheaf on X, quotient of OO (-2) X i6 : prune K2 o6 = cokernel {1} | c2 bd ac b2 | {2} | -d -c -b -a | 1 1 o6 : coherent sheaf on X, quotient of OO (-1) ++ OO (-2) X X
Notice that the resulting sheaf is not always presented in the most economical manner. Use prune to improve the presentation.