1
vote
0answers
194 views

R[[X]] flat as a R[X]-module?

I assume $R[X]\rightarrow R[[X]]$ is not flat in general, but I was wondering if any conditions on a commutative ring $R$ are known such that $R[[X]]$ is flat as a $R[X]$-module. Would $R$ noetherian ...
5
votes
1answer
171 views

Injective flat module

Let $R$ be a (right noetherian) ring. Is there always a right $R$-module which is both flat and injective? If $R$ is an integral domain, then the answer is indeed yes, as the quotient field is such. ...
4
votes
2answers
444 views

Faithful-flatness for maps of formal power series rings

Let $R$ be a ring (commutative with unit).Let $f_1,...,f_n\in R$ elements that generate the unit ideal. The map $R\to R_{f_1}\times ...\times R_{f_n}$ is faithfully flat, since this is just a Zariski ...
2
votes
1answer
86 views

on smoothness of morphisms on an artinian base

Let $A$, $B$ two smooth $R$-algebras of finite type for a artinian local ring $R$. Let $I$ an ideal such that $I^{2}=0$ and $\bar{R}=R/I$. We assume that the map $Spec B/IB\rightarrow Spec A/IA$ is ...
7
votes
1answer
310 views

flatness of power series rings

It is known that $A[[X]]$ is flat if $A$ is noetherian (see for example Bourbaki, Algèbre commutative, Ch. III, §3, Cor. 3 p. 146). What happens if A is not noetherian? Is there an easy ...
3
votes
1answer
326 views

Formal criterion of flatness

Let $k$ be a field, $S$ and $R$ be local $k$-algebras with residue field $k$ and $\phi:S\to R$ be a local homomorphism. Then $\phi$ induces (obviously) a natural transformation of "functors of points" ...
2
votes
1answer
290 views

Are irreducible components of a flat family flat?

Let $f:X\rightarrow Y$ be a flat morphism of schemes of finite type over a field $k$, and assume $Y$ is irreducible. Let $X_1, \dots, X_n$ be the scheme-theoretic irreducible components of $X$ (i.e., ...
9
votes
1answer
369 views

Can flatness be specified by a natural coherent sheaf?

More precisely: Given a finite-type morphism $f \colon X \to Y$ of nice schemes (say, both of finite type over a field), is there a "natural" coherent sheaf $\mathcal F_f$ on $X$ such that the ...
11
votes
4answers
830 views

Does smoothness descend along flat morphisms?

Suppose $f:X\to Y$ is a flat morphism of schemes. If $X$ is smooth at $x$, must $Y$ be smooth at $f(x)$? If $f$ is locally finitely presented, then it is open (using EGA IV 1.10.4), so after ...
6
votes
2answers
507 views

Is the support of a flat sheaf flat?

Note: in the following, all scheme/algebra morphisms should be assumed essentially of finite type. Geometric version: Let $X$ be a scheme flat over $S$ (both noetherian), and let $\mathscr{F}$ be ...
2
votes
4answers
1k views

Subtle examples of morphisms that are finite but not flat

Let $R$ be a ring (commutative noetherian with unit), and let $K(R)$ be its total ring of fractions (obtained by inverting all nonzerodivisors). Thus, $R \hookrightarrow K(R)$. Let $a \in K(R)$ be ...
3
votes
1answer
444 views

Alternative module-theoretic characterization of flatness

Let $A \to B$ be a homomorphism of commutative rings. I would like to find a criterion for the flatness of $A \to B$ which does not involve the notion of kernels; it should rather involve cokernels. ...
4
votes
3answers
888 views

Quotient of flat module is flat - a property in Mumford's Red book

Hi, In Mumford's "Red Book of Varieties and Schemes", Chapter III, Paragraph 10 (entitled "Flat and smooth morphisms"), the following property is stated: Let $M$ be a $B$-module, and $B$ an algebra ...
6
votes
3answers
570 views

Torsion-free tensor powers

Let $R$ be an integral domain. If $M$ is an $R$-module such that every tensor power of $M$ over $R$ is $R$-torsion-free, then is $M$ necessarily flat as an $R$-module? If not, then does this ...
2
votes
1answer
335 views

Commutator tensors and submodules

Let $k$ be a commutative ring with $1$, and let $B$ be a submodule of a $k$-module $A$. For every $n\in\mathbb N$ and every $k$-module $V$, let $K^n\left(V\right)$ denote the kernel of the canonical ...
3
votes
0answers
211 views

Flatness of a homogeneous quasi-coherent sheaf on the formal plane from flatness on lines through the origin?

(This is a follow-up to a previous question; as often happens, I didn't include enough hypotheses, so I'm asking a new, more-likely-to-be-true question). Consider the formal plane ...
1
vote
1answer
233 views

Flatness on the formal plane from flatness on lines through the origin?

Consider the formal plane $\operatorname{Spec}\mathbb C[[t,h]]$, and let $\mathcal F$ be a quasi-coherent sheaf. Now assume that $\mathcal F$ is flat over infinitely many lines through the origin of ...
4
votes
1answer
532 views

The inverse limit of locally free module

A is an I-adic complete Noetherian ring. M is a finitely generated A module. For any n>0, $M/I^nM$ is a finitely generated locally free A/I^n-module. Is M necessarily a locally free A-module?
28
votes
4answers
2k views

Are submersions of differentiable manifolds flat morphisms?

Let $\pi \colon M\to N$ be a smooth map between real smooth manifolds. Then $C^\infty(M)$ forms a module over $C^\infty(N)$ (via pullback). Is this module flat when $\pi$ is a submersion? Recall that ...
14
votes
5answers
3k views

Can a quotient ring R/J ever be flat over R?

If R is a ring and J⊂R is an ideal, can R/J ever be a flat R-module? For algebraic geometers, the question is "can a closed immersion ever be flat?" The answer is yes: take J=0. For a less ...