All Questions
Tagged with flatness homological-algebra
10 questions
6
votes
2
answers
442
views
Existence of functorial (K-)flat resolutions?
I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe ...
- 91
1
vote
1
answer
106
views
On $\text{depth}_S\left(\dfrac {S}{JS+xS}\right)$, when $\text{depth}_R(R/J)=0$, and $R\to S$ is a certain flat map of local rings
Let $(R, \mathfrak m) \xrightarrow{\phi} (S,\mathfrak n) $ be a flat homomorphism of local rings such that $\mathfrak n=\mathfrak m S +xS$ for some $x\in \mathfrak n \setminus \mathfrak n^2$. Let $J$ ...
- 73
1
vote
1
answer
118
views
Subrings, submodules, and flatness
Let $R$ be a ring, and $S$ a subring of $R$. Let $M$ be a right $R$-module, and $N$ a right $S$-submodule of $M$. If $N$ is flat (or faithfully flat) as a right $S$-module, does it then follow that ...
- 1,144
4
votes
0
answers
363
views
A projective module over a domain that is not faithfully flat?
Let $R$ be a (noncommutative) unital ring which is a domain and let $\mathcal{N}$ be a non-zero projective (right) module. Projectivity of course implies that $\mathcal{N}$ is flat, but does the fact ...
- 435
1
vote
0
answers
134
views
Composition of faithfully flat ring extensions
Let $R$ be a not necessarily commutative, unital, ring, and for simplicity let module always mean right module. We say that a unital ring extension $R \hookrightarrow S$ is flat, or faithfully flat, ...
- 435
3
votes
1
answer
277
views
Faithful flatness for rings
Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes_R -$ from left $R$-modules to abelian groups that preserves ...
- 435
3
votes
1
answer
591
views
Finitely generated modules over Noetherian local ring that become isomorphic after faithfully flat base change
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S$ be a Noetherian ring which is a faithfully flat $R$-algebra. If $M,N$ are finitely generated $R$-modules such that $M\otimes_R S \cong N \...
- 31
1
vote
0
answers
73
views
Faithfull flatness of a module containing the ring as a direct summand
Let $R$ be a not necessarily commutative ring, and let $M$ be a projective left $R$-module.
Question. If $R$ is a direct summand of $M$ as a left $R$-module, then is it
true that $M$ is faithfully ...
- 393
0
votes
1
answer
329
views
How can I show flatness for projective morphisms?
Are there any homological checks I can use to check if a projective morphism is flat? For example, I would expect the following projective morphism to be flat
$$
\textbf{Proj}\left( \frac{\mathbb{C}[s]...
- 1,716
21
votes
0
answers
1k
views
Homotopy flat DG-modules
A right DG-module $F$ over an associative DG-algebra (or DG-category) $A$ is said to be homotopy flat (h-flat for brevity) if for any acyclic left DG-module $M$ over $A$ the complex of abelian groups $...
- 15.6k