Status: this sheet is the script of a brief interactive demo during the :ref:`crm.2017.equivariant-combinatorics`.
Let us define an ideal:
sage: P = QQ['a,b,c,d,e'] sage: P.inject_variables() Defining a, b, c, d, e sage: p1 = 3*c^2 - 4*b*d + a*e sage: p2 = -2*b*c*d + 3*a*d^2 + 3*b^2*e - 4*a*c*e sage: p3 = 8*b^2*d^2 - 9*a*c*d^2 - 9*b^2*c*e + 9*a*c^2*e + 2*a*b*d*e - a^2*e^2 sage: I = Ideal([p1, p2, p3]) sage: a in I False sage: (p1*a - b * p2) in I True sage: I.dimension() 3
The calculations are actually carried out by Singular. Many more advanced features are not directly exposed in Sage, in which case one need to call singular directly. Here we follow the instructions from Singular's manual to compute a free resolution of this ideal:
sage: res = I._singular_().mres(0); res [1]: _[1]=3*c^2-4*b*d+a*e _[2]=2*b*c*d-3*a*d^2-3*b^2*e+4*a*c*e [2]: _[1]=2*b*c*d*gen(1)-3*a*d^2*gen(1)-3*b^2*e*gen(1)+4*a*c*e*gen(1)-3*c^2*gen(2)+4*b*d*gen(2)-a*e*gen(2) [3]: _[1]=0 [4]: _[1]=gen(1) [5]: _[1]=0
And its Betti numbers:
sage: res.betti() 1 0 0 0 1 0 0 1 0 0 0 1
TODO: explore how to get the nice pretty printing provided by Singular:
sage: res.betti().print("betti") # todo: not implemented
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