Kodaira-Spencer maps in local algebra. (English) Zbl 0809.13011

Herzog, Bernd

Lecture Notes in Mathematics. 1597. Berlin: Springer-Verlag. xvii, 176 p. (1994).

M. Nagata proved in 1955 that in a local ring \(A\) for any prime \(P\), such that \(A/P\) is analytically unramified, holds \(e_ 0 (A_ P) \leq e_ 0 (A)\). In 1959 C. Lech asked if for any flat local homomorphism \(f:(A,{\mathfrak m}) \to (B, {\mathfrak n})\) the inequality \(e_ 0 (A) \leq e_ 0 (B)\) holds, and proved it for some special cases, and in 1964 he asked if there exists an \(i\) such that for the sum transform of the Hilbert functions it is true that \(H^ i_ A (n) \leq H^ i_ B (n)\) for all \(n\), and proved this for \(n=1\), or if the fibre \(B/{\mathfrak m} B\) is a complete intersection. In a geometric setting, let \(f:(X,x) \to (Y,y)\) be a flat morphism of Noetherian schemes, is \(e_ 0 (Y,y) \leq e_ 0(X,x)\) and \(H^ i_{Y,y} \leq H^ i_{X,x}\) (coefficientwise)? From this point of view it is natural to look at flat morphisms \(f:(X,x) \to (Y,y)\) with fixed special fibre \((X_ y,x) = f^{-1} (y)\) equal to \((X_ 0, x_ 0)\). The author has shown (without flatness assumption) that \(H^ 1_{Y,y} H^ 0_{X_ y,x} \geq H^ 1_{X,x}\). If \(f\) is tangentially flat, i.e. the induced morphism \(df:C(X,x) \to C(Y,y)\) of tangent spaces is flat, there is an equality \(H^ 1_{Y,y} H^ 0_{X_ y,x} = H^ 1_{X,x}\), which gives \(H^ 1_{Y,y} \leq H^ 1_{X,x}\). If Schlessinger’s \(T^ 1\) of the tangent cone has no elements of degree less than \(-1\), all deformations are tangentially flat, which enables the author to find many classes of singularities satisfying the inequality \(H^ 0_{Y,y} \leq H^ 0_{X,x}\).
One main aim with this monograph is to prove a somewhat weaker version of the inequality, namely \[ H^ 1_{Y,y} H^ 0_{X_ y,x} = H^ 1_{X,x} \prod^ \infty_{d = 2} \bigl( (1 - T^ d)/(1 - T) \bigr)^{n( d)} \] for residually separable flat morphisms. Here \(n(d) = \dim T^ 1_{C (X_ y,x)} (-d)\). In order to do so, the author has to carefully study more general filtrations than the usual ones, so-called Artin-Rees filtrations.

Reviewer: R.Fröberg (Stockholm)

MSC:

13D40	Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13D10	Deformations and infinitesimal methods in commutative ring theory
13-02	Research exposition (monographs, survey articles) pertaining to commutative algebra
14B12	Local deformation theory, Artin approximation, etc.
14A05	Relevant commutative algebra
16S80	Deformations of associative rings
13H15	Multiplicity theory and related topics

Keywords:

Kodaira-Spencer maps; transform of Hilbert functions; tangentially flat morphisms; deformations; Artin-Rees filtrations