Resultant Ideals

We place ourselves in the situation discussed in the last section, now following [43] closely. We consider a regular sequence F ₀,…, F_n of homogeneous polynomials in A[T] = A[T ₀,…,T_n ] of positive degrees δ ₀,…,δ_n over a noetherian ring A and the A-algebra C = A[t]/(F ₀,…,F_n ). Using the homogeneous ideal t(C) of inertia forms, we define the graded residue class algebra C ¯ ≔ C / T ( C ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064265/27e88de8-b4c2-47de-a1e8-3cf9bc1ec436/content/unequ13_105_1.tif"/> which has the same projective spectrum as C. The weights of the indeterminates T ₀,…,T_n are γ₀,…,γ _n . Then the socle determinant Δ = Δ F T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064265/27e88de8-b4c2-47de-a1e8-3cf9bc1ec436/content/inequ13_105_1.tif"/> belongs to C_σ , σ = ∑ j = 0 n δ j − ∑ i = 0 n γ i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064265/27e88de8-b4c2-47de-a1e8-3cf9bc1ec436/content/inequ13_105_2.tif"/> , cf. Section 12.