Flatness testing and torsionfree morphisms

Mathematische Zeitschrift, 2012

Let R be a commutative Noetherian ring. We give criteria for flatness of R-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if R has characteristic p, or more generally if it has a locally contracting endomorphism. Dualizing, we give criteria for injectivity of R-modules in terms of coassociated primes and (h-)divisibility of certain Hom-modules. Along the way, we develop tools to achieve such a dual result. These include a careful analysis of the notions of divisibility and h-divisibility (including a localization result), a theorem on coassociated primes across a Hom-module base change, and a local criterion for injectivity.

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.

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.

St. Petersburg Mathematical Journal, 2014

The generalization of the well-known criterion for flatness of a projective morphism of Noetherian schemes involving Hilbert polynomial, is given for the case of nonreduced base of the morphism.

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.

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.

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.

Journal of Algebra, 1970

Our aim here is to give a structure theorem for flat extensions of a commutative noetherian ring R-that is, those R-algebras which are flat when viewed as R-modules-which arc obtained, essentially, by adjoining a single element to R. Such an extension S is best described by an exact sequence of R-homomorphisms (*I O-zI-tR[Xl+S+O;

1970

Transactions of the American Mathematical Society, 1970

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules-unmixed-is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.