Lech-Hironaka inequalities for flat couples of local rings (Q921063)

Let (A,m) be a local ring with Hilbert series \(H^ 0_ A=\sum^{\infty}_{0}\dim_{A/m}(m^ d/m^{d+1})T^ d \), and series of i-th sums \(H^ i_ A=(1-T)^{-i}H^ 0_ A\), and let f: (A,m)\(\to (B,n)\) be a flat local homomorphism of local rings. The general problem is to discover when \(H^ i_ A\leq H^ i_ B\) in the sense that the inequality holds between coefficients. This is a stronger inequality than the 1959 suggestion of \textit{C. Lech} [see Ark. Mat. 4, 63-86 (1960; Zbl 0192.139)] that the multiplicities satisfy e(A)\(\leq e(B)\). Lech sketches a proof of the stronger inequality for \(i=1\) when the fibre B/mB is a zero-dimensional complete intersection. Later \textit{H. Hironaka} [J. Math. Kyoto Univ. 10, 151-187 (1970; Zbl 0214.200)] proved the same inequality for \(B=A[x]/(F).\) In this note the author reconstructs a complete version of Lech's proof and gives a different proof, using Cohen's structure theorems and an inequality of Bennett, to establish \(H_ A^{d+1}\leq H^ 1_ B\) when d is the dimension of the complete intersection B/mB. Similar arguments are used to establish the same inequality in a more general setting.