All Questions
Tagged with flatness ac.commutative-algebra
58 questions
3
votes
1
answer
166
views
Finite flat maps
Let $f : A \to B$ be a finite, finitely presented, flat map of (commutative) rings. It is a known consequence of Chevalley's theorem (on constructible sets) that the induced map $Spec B \to Spec A$ is ...
- 19.6k
13
votes
4
answers
2k
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 ...
- 761
28
votes
5
answers
9k
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Can a quotient ring R/J ever be flat over R?
If $R$ is a ring and $J\subset R$ 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=...
- 6,367
3
votes
1
answer
238
views
Flatness of certain subrings
The following question appears, more or less, here:
Let $k$ be an algebraically closed field of characteristic zero and let $S$ be a commutative $k$-algebra
(I do not mind to further assume that $S$ ...
- 28.7k
3
votes
1
answer
194
views
Flatness over regular local rings of dimension 3
Let $R$ be a regular local ring of dimension $n$. Let $i:\text{Flat}\to\text{Fin}$ be the fully faithful inclusion of the category of flat finitely generated type $R$-modules into all finitely ...
- 19.6k
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 $J$ ...
- 6,262
3
votes
1
answer
563
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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. ...
- 2,821
4
votes
4
answers
3k
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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 ...
- 12.9k
3
votes
1
answer
591
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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 \...
- 1,144
1
vote
1
answer
233
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Flatness criterion for $I$-adic ring: $I$-torsion free
Let $R$ be an $I$-adically separated and complete valuation ring, with $I$ finitely generated.
It is used a few times in Bosch, Lectures on Formal and Rigid Geometry e.g. first lines of pg. 164, Cor. ...
- 1,313
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
1
answer
264
views
Flat and algebraic (non-integral) local rings extension $R \subseteq S$ with $m_RS=m_S$
Let $R \subseteq S$ be two Noetherian local rings, not necessarily regular, which are integral domains,
with $m_RS=m_S$, namely, the ideal in $S$ generated by $m_R$ (= the maximal ideal of $R$) is $...
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0
votes
0
answers
191
views
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
3
votes
0
answers
152
views
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
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
0
votes
1
answer
429
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Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras
For commutative rings $R \subseteq S$,
recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.
...
- 28.7k
3
votes
1
answer
366
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flatness and reduction
Let $\mathcal J$ be an ideal sheaf on a (Noetherian) $Y$-scheme $X$, and let $\mathcal I$ be the unique primary ideal in a primary decomposition $\mathcal J$ corresponding to a minimal associated ...
- 30.5k
0
votes
1
answer
208
views
Separable non-flat simple ring extension
Let $R \subseteq S$ be two commutative $\mathbb{C}$-algebras such that:
(1) $R$ and $S$ are integral domains.
(2) $Q(R)=Q(S)$, namely, their fields of fractions are equal.
(3) $S=R[w]$, for some $w \...
- 661
3
votes
1
answer
208
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Faithful flatness of left adjoint to almostification of algebras
I have been reading Bhatt's notes on perfectoid spaces and I have stumbled upon a fact whose proof I am unable to understand. Specifically, in Remark 4.2.8 Bhatt describes the functor $A\mapsto A_{!!}$...
- 28.2k
2
votes
1
answer
294
views
The local flatness criterion
I am self studying the book "Commutative Ring Theory" by H. Matsumura. The main theorem of section 22 is the theorem 22.3, which characterizes flatness of a module $M$ over any ring $A$. The (part of ...
- 1,356
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
1
answer
830
views
Is the support of a flat module generically flat?
Let $X$ be an affine, complex variety, $A$ be a $\mathbb{C}$-algebra (not necessarily noetherian) and $F_A$ is a coherent sheaf over $X \times \mbox{Spec}(A)$, flat over $\mbox{Spec}(A)$. Denote by $Y ...
- 63.9k
-1
votes
1
answer
209
views
Flatness of certain quotient rings
Let $p,q \in \mathbb{C}[x,y]$ be two polynomials such that $p_xp_yq_xq_y \neq 0$
(namely, each partial derivative is non-zero).
Assume that the following four conditions are satisfied:
(1) $\frac{\...
- 28.7k
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 ...
CommunityBot
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41
votes
5
answers
3k
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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 ...
CommunityBot
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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 $x\in P$ ,...
- 4,985
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
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
10
votes
1
answer
256
views
Iteration of a morphism and flatness
Let $A$ be a Noetherian local ring, $f:A \rightarrow A$ be a local ring morphism. Assume some power of $f$ is a flat morphism, must $f$ be flat as well?
Motivation: Kunz's theorem shows the result is ...
- 19.6k
7
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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
11
votes
2
answers
2k
views
Idea behind Grothendieck's proof that formally smooth implies flat?
From this answer I learned that Grothendieck proved the following result.
Theorem. Every formally smooth morphism between locally noetherian schemes is flat.
The book Smoothness, Regularity, and ...
- 23.3k
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?
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 ...
CommunityBot
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5
votes
2
answers
236
views
Descent of flatness from algebras to monoids
Consider a morphism of commutative monoids $u\colon M\rightarrow N$. We say that $u$ is flat, if the tensor product functor $\bullet\otimes_MN$ from the category of $M$-modules to the category of $N$-...
- 38.6k
4
votes
1
answer
512
views
Being Cohen-Macaulay open in Hilbert scheme?
Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) ...
- 4,111
2
votes
1
answer
203
views
Is flatness preserved under exterior power
Let $\phi:A \to B$ be a flat ring homomorphism, $M$ be a $B$-module which is flat when considered as an $A$-module. Is the tensor product $M \otimes_B M \otimes_B ... \otimes_B M$ flat over $A$? If ...
- 35.2k
3
votes
0
answers
313
views
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
16
votes
2
answers
2k
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Every finitely generated flat module over a ring with finitely many minimal primes is projective
Over a commutative ring $R$, a finite type locally free (weak sense) module for which the rank function is locally constant is projective.
If we notice that for each minimal prime $p$ of the ring, ...
CommunityBot
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2
votes
1
answer
354
views
Projective dimension of a quotient ring
Assume $A$ and $B$ are commutative algebras with $1$, $B = A[z] = A[Z]/(h(Z))$,
$Z$ an indeterminate.
The first comment in this question says that, if $A$ is noetherian, then
$pd_{B\otimes_A B}(B) \...
CommunityBot
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2
votes
1
answer
284
views
Deciding whether a non-f.g. non-divisible flat module is projective or not
Assume $S= R[T]/(f)= R[w]$ is a flat non-divisible $R$-module, where $R$ is a noetherian UFD, $T$ is an indeterminate over $R$, and $f\in R[T]$ is a non-monic polynomial of positive degree.
Can we ...
CommunityBot
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2
votes
2
answers
789
views
When $mB \neq B$? $m$ is a maximal ideal of $A$, $A \subseteq B$
The following is a question I have asked here without receiving any comments, therefore I post it here:
Let $A \subseteq B$ be commutative rings, $m$ a maximal ideal of $A$.
When $mB \neq B$?
This ...
CommunityBot
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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 ...
- 1,355
2
votes
1
answer
246
views
Does the going-up theorem hold between flat algebras?
Let $R$ be a commutative Noetherian ring with unit and $S$ a flat $R$-algebra. Does the going-up theorem hold between $R$ and $S$?
- 1,355
3
votes
0
answers
458
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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
7
votes
1
answer
751
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.
...
- 121
5
votes
2
answers
801
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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 ...
- 281
6
votes
3
answers
948
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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 ...
CommunityBot
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4
votes
1
answer
1k
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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" ...
- 1,254
2
votes
1
answer
110
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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 ...
- 19.6k
11
votes
1
answer
1k
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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 counter-...
CommunityBot
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