All Questions
Tagged with ac.commutative-algebra flatness
16 questions with no upvoted or accepted answers
10
votes
0
answers
575
views
How general are Gröbner degenerations?
While working with flat degenerations of flag varieties and Schubert varieties I've noticed that among the numerous known constructions there doesn't seem to be a single one that doesn't turn out to ...
- 3,513
7
votes
0
answers
181
views
Self-flat modules
(This is inspired by this question and asked out of pure curiosity.)
Let $R$ be a commutative ring. Let $M$ and $N$ be $R$-modules. Then, $N$ is called $M$-flat if whenever $M'\rightarrow M$ is a ...
- 6,700
7
votes
0
answers
275
views
Lifting flat modules over ring quotients
Let $R$ be a commutative ring, $I$ its ideal, and $\bar{R}=R/I$. For which flat $\bar{R}$-modules $\bar{F}$ is there a flat $R$-module $F$ such that $F \otimes_R R/I \simeq \bar{F}$?
By Lazard's ...
- 409
3
votes
0
answers
152
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Flatness of certain $R \subseteq \mathbb{C}[x,y]$
The two-dimensional Jacobian Conjecture over $\mathbb{C}$ says the following:
Let $p,q \in \mathbb{C}[x,y]$ satisfy $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
Then $\mathbb{C}[p,q]=...
- 2,837
3
votes
0
answers
59
views
When flatness of $S$ over $R_i$ implies flatness of $S$ over the ring generated by $R_1,R_2$
The following question I have asked in MSE, but have not received an answer, so I ask it here; I really apologize if it is not suitable for MO.
Let $k$ be a field of characteristic zero and let $R_1,...
- 2,837
3
votes
0
answers
130
views
Flatness of $Hom_R(M,M)$
The question is simple and clear: Let $M$ be an $R$-module, where $R$ is a commutative ring with an identity. When the $R$-module $Hom_R(M,M)$ is flat?
3
votes
0
answers
313
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Exterior power of a torsion-free sheaf on a DVR
Let $R$ be a discrete valuation ring and $X$ be a regular, integral. projective $R$-scheme, flat over $R$. Let $F$ be a torsion-free coherent sheaf on $X$ of rank $n$, flat over $\mathrm{Spec}(R)$. Is ...
- 2,126
3
votes
0
answers
458
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 ...
- 31
3
votes
0
answers
240
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 $\operatorname{Spec}...
- 44.7k
2
votes
0
answers
352
views
Regular rings and finite flat modules
Let $A$ be a Noetherian domain. Assume $f:A\rightarrow B$ is an injective homomorphism making $B$ into a finite flat module over $A$. If $B$ is regular is $A$ regular as well? I played with some ...
user145520
2
votes
0
answers
423
views
Flatness of modules over dual numbers
Let $X$ be a smooth, affine complex surface, and $M$ be a coherent $\mathcal{O}_X$-module. Denote by $D:=\mbox{Spec}(\mathbb{C}[t]/(t^2))$ and $X_D:=X \times_{\mathbb{C}} D$, the trivial deformation ...
- 1,593
2
votes
0
answers
106
views
Descent of flatness from algebras to monoids II
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
- 6,700
1
vote
0
answers
365
views
Flatness over a local noetherian ring
Let $(R,\mathfrak m)$ be a local noetherian ring, and $M$ an arbitrary $R$-module. Suppose that $\mathrm{Tor}_1(M,R/\mathfrak m)=0$. Does it follow that $M$ is flat?
The answer is positive when $M$ ...
- 1,313
1
vote
0
answers
65
views
Non-minimal Krull associated primes of a PF-ring
A commutative ring $R$ is said to be a PF-ring if every principal ideal of $R$ is a flat $R$-module. Also, a prime ideal $P$ of $R$ to be a Krull associated prime of $R$ if
for every element $x\in P$ ,...
- 51
0
votes
0
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191
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When $K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ is a flat morphism
Let $K$ be a field and $\varphi: K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ a polynomial $K$-algebra morphism. Assume $n, m \ge 2$. By definition $\varphi$ endows $K[Y_1,Y_2,...,Y_m]$ with a $K[X_1,...
- 6,028
0
votes
0
answers
176
views
Flatness of a simple ring extension
Assume $A \subseteq B=A[b]$ are integral domains, $b \in B$ is algebraic over $A$ (but not necessarily integral over $A$), and $A$ and $B$ have the same field of fractions.
(Notice that $b=u/v$ for ...
- 2,837