extIsOnePolynomial -- check whether the Hilbert function of Ext(M,k) is one polynomial

Synopsis

Usage:

(p,t) = extIsOnePolynomial M
Inputs:
- M, a module, module over a complete intersection
Outputs:
- p, a ring element, p(z)=pe(z/2), where pe is the Hilbert poly of Ext^{even}(M,k)
- t, a Boolean value, true if the even and odd polynomials match to form one polynomial

Description

Computes the Hilbert polynomials pe(z), po(z) of evenExtModule and oddExtModule. It returns pe(z/2), and compares to see whether this is equal to po(z/2-1/2). Avramov, Seceleanu and Zheng have proven that if the ideal of quadratic leading forms of a complete intersection of codimension c generate an ideal of codimension at least c-1, then the betti numbers of any module grow, eventually, as a single polynomial (instead of requiring separate polynomials for even and odd terms.) This script checks the result in the homogeneous case (in which case the condition is necessary and sufficient.)

i1 : R1=ZZ/101[a,b,c]/ideal(a^2,b^2,c^5)

o1 = R1

o1 : QuotientRing

i2 : R2=ZZ/101[a,b,c]/ideal(a^3,b^3)

o2 = R2

o2 : QuotientRing

i3 : extIsOnePolynomial coker random(R1^{0,1},R1^{3:-1})

      1 2   1
o3 = (-z  - -z + 3, true)
      2     2

o3 : Sequence

i4 : extIsOnePolynomial coker random(R2^{0,1},R2^{3:-1})

o4 = (3z - 2, false)

o4 : Sequence

Ways to use extIsOnePolynomial :

extIsOnePolynomial(Module)

For the programmer

The object extIsOnePolynomial is a method function.

extIsOnePolynomial -- check whether the Hilbert function of Ext(M,k) is one polynomial

Synopsis

Description

See also

Ways to use extIsOnePolynomial :

For the programmer