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The A_f condition and relative conormal spaces for functions with non-vanishing derivative
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Abstract
We introduce a join construction as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom's A_f condition.
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Let be an affine Noetherian scheme, and be a pair of finitely generated R-modules. Suppose either that and are locally free at the generic point of each irreducible component of X or that the modules are contained in a free R-module. Let and be the nth homogeneous components of the Rees algebras of and , and let be the integral closure of in . We prove that and are asymptotically stable, generalizing known results for the case where is an ideal or where is a free module. When X is universally catenary, we prove a generalization of a classical result due to McAdam and obtain a geometric classification of the points appearing in . Notably, we show that if for some n, then x is the generic point of a codimension-one component of the closed sets in X defined by certain Fitting ideals of or x is a generic point of an irreducible set in X where the fiber dimension jumps. Our approach is novel and geometric. Also, we recover, strengthen, and prove a sort of converse of an important result of Kleiman and Thorup about integral dependence of modules.
We give a complete description of the relationship between the vanishing cycles of a complex of sheaves along a function f and Thom’s af condition. 1
We establish the principle of specialization of integral dependence for submodules of finite colength of free modules, as
part of the general algebraic-geometric theory of the Buchsbaum–Rim multiplicity. Then we apply the principle to the study
of equisingularity of ICIS germs, obtaining results for Whitney’s Condition A and Thom’s Condition Af. Notably, we describe these equisingularity conditions for analytic families in terms of various numerical invariants, which,
for the most part, depend only on the members of a family, not on its total space.
Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem with the multiplicity of an ideal or module is that it is only defined for modules and ideals of finite colength. In this paper we use pairs of modules and their multiplicities as a way around this difficulty. A key tool is the multiplicity-polar theorem which applies to families of pairs of modules, linking the multiplicity at a general pair of the family to the multiplicity of the pair at the special fiber. The specific problem we work on is: Suppose X,0 is an equidimensional complex analytic set in C^n, and T^*_X(C^n) is the closure of the conormal vectors to the smooth part of X, s, a section of T^*(C^n), the cotangent bundle of C^n restricted to X. Suppose the intersection of the image of s, denoted im(s), with T^*_X(C^n) is isolated. Then we calculate the intersection multiplicity im(s)\cdot T^*_X(C^n) using the multiplicity of pairs. It is easy to see that if X=C^n and s(x)=df(x), where f has an isolated singularity at the origin, then this number is the Milnor number of f. Variations of this number appear in several different situations, such as, the theory of differential forms on singular spaces, the theory of D-modules, the Milnor fiber of a function with an isolated singularity, the A_f stratification condition of Thom, and the theory of the Euler invariant. We discuss the application of our result to all of these situations.
Let A be a Noetherian local domain, N be a finitely generated torsion-free module, and M a proper submodule that is generically equal to N. Let A[N] be an arbitrary graded overdomain of A generated as an A-algebra by N placed in degree 1. Let A[M] be the subalgebra generated by M. Set C ≔ Proj(A[M]) and r ≔ dim C. Form the (closed) subset W of Spec(A) of primes p where A[N]p is not a finitely generated module over A[M]p, and denote the preimage of W in C by E. We prove this: (1) dim E = r − 1 if either (a) Nis free and A[N] is the symmetric algebra, or (b) W is nonempty and A is universally catenary, and (2) E is equidimensional if (a) holds and A is universally catenary. Our proof was inspired by some recent work of Gaffney and Massey, which we sketch; they proved (2) when A is the ring of germs of a complex-analytic variety, and applied it to improve a characterization of Thom's Af-condition in equisingularity theory. From (1), we recover, with new proofs, the usual height inequality for maximal minors and an extension of it obtained by the authors in 1992. From the latter, we recover the authors' generalization to modules of Böger's criterion for integral dependence of ideals. Finally, we introduce an application of (1), being made by Thorup, to the geometry of the dual variety of a projective variety, and use it to obtain an interesting example where the conclusion of (1) fails and A[N] is a finitely generated module over A[M].
We compare and contrast various notions of the "critical locus" of a complex analytic function on a singular space. After choosing a topological variant as our primary notion of the critical locus, we justify our choice by generalizing L\^e and Saito's result that constant Milnor number implies that Thom's a_f condition is satisfied.
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