Geometric Auslander criterion for flatness

2012

Abstract. We prove that, if F is a coherent sheaf of OX-modules over a morphism ϕ: X → Y of complex-analytic spaces, where Y is smooth, then the stalk Fξ at a point ξ ∈ X is flat over R: = O Y,ϕ(ξ) if and only if the n-fold analytic tensor power of Fξ over R (where n = dimR) has no vertical elements. The result implies that if F is a finite module over a morphism ϕ: X → Y of complex algebraic varieties, where Y is smooth and dimY = n, then Fξ is R-flat if and only if its n-fold tensor power is a torsionfree R-module. The latter generalizes a classical freeness criterion of Auslander to modules that are not necessarily finitely generated over the base ring. Contents

2016

We present a constructive criterion for flatness of a morphism of analytic spaces ϕ : X → Y (over K = R or C) or, more generally, for flatness over O Y of a coherent sheaf of O X-modules F. The criterion is a combination of a simple linear-algebra condition "in codimension zero" and a condition "in codimension one" which can be used together with the Weierstrass preparation theorem to inductively reduce the fibre-dimension of the morphism ϕ.

Proceedings of the American Mathematical Society, 2012

We present a constructive criterion for flatness of a morphism of analytic spaces ϕ : X → Y (over K = R or C) or, more generally, for flatness over O Y of a coherent sheaf of O X-modules F. The criterion is a combination of a simple linear-algebra condition "in codimension zero" and a condition "in codimension one" which can be used together with the Weierstrass preparation theorem to inductively reduce the fibre-dimension of the morphism ϕ.

Cornell University - arXiv, 2016

Let R be a commutative ring. Roughly speaking, we prove that an R-module M is flat iff it is a direct limit of R-module affine algebraic varieties, and M is a flat Mittag-Leffler module iff it is the union of its R-submodule affine algebraic varieties.

Communications in Algebra, 2006

We prove various extensions of the Local Flatness Criterion over a Noetherian local ring R with residue field k. For instance, if Ω is a complete R-module of finite projective dimension, then Ω is flat if and only if Tor R n (Ω, k) = 0 for all n = 1,. .. , depth(R). In low dimensions, we have the following criteria. If R is onedimensional and reduced, then Ω is flat if and only if Tor R 1 (Ω, k) = 0. If R is twodimensional, then in order for Ω to be flat, it suffices that it is separated, that its projective dimension is finite and that Tor R 1 (Ω, k) = 0. Many of these criteria have global counterparts and in particular, it is shown that the aadic completion of a flat module of finite projective dimension over an arbitrary Noetherian ring is again flat.

Bulletin of the London Mathematical Society, 2013

We give a topological analogue for openness of a criterion for flatness that originates with Auslander. Over a normal base of dimension n, failure of openness is detected by a vertical component in the n'th fibred power of the morphism.

Journal of Pure and Applied Algebra, 1997

Let I) : A H B be a homomorphism of finitely generated algebras over a field k or over Z. This note is concerned with methods and tools to ascertain whether +!I makes B a flat module over A. Morphisms with such properties are very common and desirable in the study of mappings between algebraic varieties. We show how three pillars of the study, a computable generic flatness, the local criterion and tricks in reducing the dimension of the ring, combine to allow for several tests. @ 1997 Elsevier Science B.V.

Kodai Mathematical Journal, 1998

Journal of Pure and Applied Algebra, 1983

Flat morphisms from YI to B (commutative and unitary rings) such that the multiplication B@,., B-B is flat, have I Iany of the properties of ind-etale morphisms. They don't raise weak dimension. As a consequence they preserve integral closure. In the local case they are the extensions of A that have the same strict henselian extensions as A.