-- Examples for Fellowship of the Ring, -- Quadratic Gorensteein algebras and the Koszul property -- Mike Stillman -- 24 April 2020 restart kk = ZZ/32003; -- Quadrics? S = kk[x_1..x_6] F = random(2, S) I = inverseSystem F minimalBetti I -- Cubics? S = kk[x_1..x_6] F = random(3, S) I = inverseSystem F; minimalBetti I -- Quartics? S = kk[x_1..x_6] F = random(4, S); I = inverseSystem F; minimalBetti I -- no quadrics in the ideal -- Matsuda example (r=4, c=7) S = kk[x_0..x_7, x_12, x_23, x_34, x_45, x_56, x_67, x_71] matsuda = ideal(x_1*x_2-x_0*x_12,x_1*x_2-x_0*x_12,x_3*x_2-x_0*x_23,x_3*x_4-x_0*x_34,x_4*x_5-x_0*x_45, x_5*x_6-x_0*x_56,x_6*x_7-x_0*x_67,x_1*x_7-x_0*x_71,x_1*x_23-x_3*x_12,x_2*x_34-x_4*x_23, x_3*x_45-x_5*x_34,x_4*x_56-x_6*x_45,x_5*x_67-x_7*x_56,x_6*x_71-x_1*x_67,x_7*x_12-x_2*x_71) betti res matsuda R = S/matsuda betti res(coker vars R, LengthLimit => 4) -- reg = 4, c >= 7 F = (c) -> ( S = kk[x_1..x_c]; f := sum(1..c-2, i -> x_i * x_(i+1) * x_(i+2)^2); f + x_(c-1) * x_c * x_1^2 + x_c * x_1 * x_2^2 ) f = F 7 I = inverseSystem f (codim I, regularity (S^1/I), degree I) minimalBetti I R = (ring I)/I res(coker vars R, LengthLimit => 4) betti oo -- reg = 4, c = 6 f = F 6 I = inverseSystem f minimalBetti I minimalBetti inverseSystem (f + x_1^2*x_3^2) minimalBetti inverseSystem (f + x_1*x_3^3) g = f + x_1*x_6*x_5^2 + x_1*x_6^3 I = inverseSystem g minimalBetti I R = (ring I)/I betti res(coker vars R, LengthLimit => 4) reduceHilbert hilbertSeries I