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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 ...
Jakob's user avatar
  • 2,040
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 ...
Matthieu Romagny's user avatar
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 $J$ ...
feder's user avatar
  • 73
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 \...
user337580's user avatar
1 vote
1 answer
234 views

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. ...
Bryan Shih's user avatar
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 $M$ ...
user26857's user avatar
  • 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 $...
user237522's user avatar
  • 2,837
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]=...
user237522's user avatar
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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 $S$ ...
user237522's user avatar
  • 2,837
3 votes
1 answer
366 views

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 ...
Dima Sustretov's user avatar
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 \...
user237522's user avatar
  • 2,837
0 votes
1 answer
429 views

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: $f(s_1 \otimes_R s_2)=s_1s_2$. ...
user237522's user avatar
  • 2,837
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 ...
KkD's user avatar
  • 23
3 votes
1 answer
208 views

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 $A\mapsto A_{!!}$...
Wojowu's user avatar
  • 28.2k
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,...
user267839's user avatar
  • 6,038
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 ...
Igor Makhlin's user avatar
  • 3,513
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,...
user237522's user avatar
  • 2,837
-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{\...
user237522's user avatar
  • 2,837
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 ...
user avatar
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 $x\in P$ ,...
user140640's user avatar
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 ...
Chen's user avatar
  • 1,593
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 ...
Chen's user avatar
  • 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 ...
Fred Rohrer's user avatar
  • 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 ...
Zhiyu's user avatar
  • 6,622
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 ...
Fred.Fred's user avatar
  • 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 ...
Arrow's user avatar
  • 10.5k
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?
Peyman Nasehpour's user avatar
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 ...
Fred Rohrer's user avatar
  • 6,700
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 $N$-...
Fred Rohrer's user avatar
  • 6,700
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) ...
Hans's user avatar
  • 3,031
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 ...
Ron's user avatar
  • 2,126
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 ...
Ron's user avatar
  • 2,126
16 votes
2 answers
2k views

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, ...
brunoh's user avatar
  • 1,128
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) \...
user237522's user avatar
  • 2,837
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 ...
user237522's user avatar
  • 2,837
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 ...
user237522's user avatar
  • 2,837
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 ...
user237522's user avatar
  • 2,837
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 ...
Garland's user avatar
  • 31
7 votes
1 answer
752 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. ...
Fred.Fred's user avatar
  • 409
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 $S$?
user 1's user avatar
  • 1,355
5 votes
2 answers
801 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 ...
user41340's user avatar
2 votes
1 answer
110 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 ...
prochet's user avatar
  • 3,472
11 votes
1 answer
1k 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 counter-...
Baptiste Calmès's user avatar
4 votes
1 answer
1k 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" ...
Piotr Achinger's user avatar
7 votes
1 answer
949 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., ...
quim's user avatar
  • 1,811
11 votes
1 answer
631 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 ...
Charles Staats's user avatar
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 ...
Anton Geraschenko's user avatar
16 votes
2 answers
1k 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 a ...
Charles Staats's user avatar
4 votes
4 answers
3k 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 ...
Charles Staats's user avatar
3 votes
1 answer
563 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. ...
Martin Brandenburg's user avatar