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CompleteIntersectionResolutions :: extIsOnePolynomial

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



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

See also

Ways to use extIsOnePolynomial :

For the programmer

The object extIsOnePolynomial is a method function.