Geometric characterization of flat modules

1970

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.

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.

2017

We consider $R$-modules as module functors in the following way: if $M$ is a (left) $R$-module, let $\mathcal M$ be the functor of $\mathcal R$-modules defined by $\mathcal M(S) := S\otimes_R M$ for every $R$-algebra $S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\mathcal M\to \mathcal M^{**}$ is an isomorphism. We give functorial characterizations of finitely generated projective modules, flat modules and flat Mittag-Leffler modules.

Journal of Mathematical Sciences, 2005

American Journal of Mathematics, 2013

We prove that, if F is a coherent sheaf of O X-modules over a morphism ϕ : X → Y of complex-analytic spaces, where Y is smooth, then the stalk F ξ at a point ξ ∈ X is flat over R := O Y,ϕ(ξ) if and only if the n-fold analytic tensor power of F ξ over R (where n = dim R) 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 dim Y = n, then F ξ 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 1. Introduction 1 2. Analytic tensor product and fibred product 7 3. Homological properties of almost finitely generated modules 9 4. Vertical components and variation of fibre dimension 12 5. Proof of the main theorem 13 References 20 Key words and phrases. flat, torsion free, fibred power, vertical component, analytic tensor product, complex analytic geometry.

Journal of Mathematical Sciences, 1999

Communications in Algebra, 2003

For a ring S, let K 0 (FGFl(S)) and K 0 (FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat S-modules and the category of all finitely generated projective S-modules respectively. We prove that a semilocal ring R is semiperfect if and only if the group homomorphism K 0 (FGFl(R)) → K 0 (FGFl(R/J(R))) is an epimorphism and K 0 (FGFl(R)) = K 0 (FGPr(R)).

arXiv (Cornell University), 2018

Let R be an associative ring with unit. Given an R-module M , we can associate the following covariant functor from the category of R-algebras to the category of abelian groups: S → M ⊗ R S. With the corresponding notion of dual functor, we prove that the natural morphism of functors M → M ∨∨ is an isomorphism. We prove several characterizations of the functors associated with flat modules, flat Mittag-Leffler modules and projective modules.