2025/11/18 by Puthenpurakal, Tony J.
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Advanced Combinatorial Mathematics #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.2511.14383
Let (A,\mathfrakm) be an excellent local complete intersection ring and let I = (a1, …, ar) be an ideal of positive height. Let R(I) = A[It] be the Rees algebra of I. Consider the map ψ\colon S = A[X1, …, Xr] → R(I) which maps Xi ↦ ait for all i. Let J = ker ψ and let H_*(J) be the Koszul homology of J. We prove that the following assertions are equivalent: (i) Proj R(I) is a complete intersection. (ii) (a) D3(R(I)|A, R(I))n = 0 for n ≫ 0 and, (ii) (b) For P ∈ Proj R(I) we have H1(J)P is a free R(I)P-module. Here D3(R(I)|A, R(I)) is the third André-Quillen homology of R(I) with respect to A → R(I). We prove an analogous result for the extended Rees algebra \widehatR = A[It, t-1]. When A is a Cohen-Macaulay domain (not necessarily a complete intersection) we compute that rank of H1(J) and hence compute its free locus.