Articles of flatness

Flatness and intersection of ideals

This is Exercise 1.2.6(a) in Liu, Algebraic Geometry and Arithmetic Curves Let $B$ be a flat $A$-algebra. Show that for any finite family $\{I_\lambda\}_{\lambda\in \Lambda}$ of ideals of $A$, we have $\cap_{\lambda\in\Lambda}(I_\lambda B)=(\cap_{\lambda\in \Lambda}I_\lambda)B$. How can I prove that? Also does the following holds: Let $B$ be a flat $A$-algebra. Does there exist an infinite […]

When does intersection commute with tensor product

Given two submodules $U,V \subseteq M$ over a (commutative) ring $R$, and a flat $R$-module $A$, I can interpret $U \otimes_R A$ and $V \otimes_R A$ as submodules of $M \otimes_R A$. Is it necessarily true that $$(U \cap V) \otimes_R A \cong (U \otimes_R A) \cap (V \otimes_R A) ?$$ I think it should […]

$M$ is a flat $R$-module if and only if its character module, i.e. $\hom_{\mathbb{Z}}(M,\mathbb{Q/Z})$, is injective.

$M$ is a flat $R$-module if and only if $\hom_{\mathbb{Z}}(M,\mathbb{Q/Z})$ is injective. One direction is easy. Suppose $M$ is flat. We know that $$ \hom_\mathbb{Z}(-\otimes M, \mathbb{Q/Z}) \cong_{\mathbb{Z}} \hom_{\mathbb{Z}}(-,\hom_{\mathbb{Z}}(M,\mathbb{Q/Z}))$$ Since $- \otimes M$ is exact and $\mathbb{Q/Z}$ is injective, the left functor is exact which shows that the right functor is exact, i.e. $\hom_{\mathbb{Z}}(M,\mathbb{Q/Z})$ is […]

Is it true that a flat module is torsion free over an arbitrary ring? Does the reverse implication hold for finitely generated modules?

So when you work over a commutative ring, this result is quite well known. I am wondering if the same holds true for an arbitrary ring; that is, if $R$ is some (possibly noncommutative) ring, does the following implication hold: $$\textit{Flat module} \implies \textit{Torsion free}\ ?$$ In particular, I am considering a ring which has […]

Examples of faithfully flat modules

I’m studying some results about flatness and faithful flatness and I’d like to keep in my mind some examples about faithfully flat modules. In general, free modules are the typical examples. Another (unusual) example of faithfully flat module is the “Zariski Covering”. (Let $R$ be a ring, $(f_1,\dots,f_n)=R$, and let $R_{f_i}$ be a localization $\forall […]

Example of non-flat modules

Let $R = \mathbb{C}[t]$ be a ring of polynomials in variable $t$ with coefficients in the field of complex numbers $\mathbb{C}$ and let $$N = R[x]/(tx-t).$$ I claim that $N$ is not a flat $R$- module. If we consider the exact sequence $$0 \rightarrow(t) \rightarrow R$$ such that the ideal $(t)$ is viewed as an […]

Non-finitely generated, non-projective flat module, over a polynomial ring

Let $R=k[x_1,\ldots,x_n]$. According to the first answer, every finitely generated flat module over an integral domain is necessarily projective. Therefore, the only hope to find a flat non-projective $R$-module $M$ is when $M$ is not finitely generated. Can anyone please suggest such modules? Actually, here there are some examples of flat non-projective modules (but not […]

Is $R/N(R)$ a faithfully flat $R$-module?

I’m studying recently faithfully flat modules and I’d like to know the following: Is $R/N$ faithfully flat as $R$-module, where $R$ is a commutative ring with unit and $N$ is the ideal of nilpotent elements of $R$?

Showing that a certain map is not flat by explicit counterexample

This question already has an answer here: Direct proof of non-flatness 1 answer

A criterion of flat modules

Let $R$ be a commutative ring and $M$ an $R$-module such that for every ideal $I \subset R$ the natural map $I \otimes_R M \rightarrow IM$ is an isomorphism. Why is $M$ flat ? This result is taken from Wikipedia.