Questions tagged [flatness]
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197 questions
3 votes
1 answer
183 views
Rings for which every module of projective dimension at most 1 is projective
Let $R$ be an associative ring with identity. Recall that a pair of classes of modules $(\mathcal{A}, \mathcal{B})$ in $R\text{-Mod}$ is called a cotorsion pair if $$ \mathcal{A} = {}^{\perp}\mathcal{...
- 247
1 vote
1 answer
141 views
Do quasi-coherent crystals of $\mathcal{O}_{S/\Sigma}$-modules yield fpqc sheaves?
I am looking at the first chapter of Berthelot-Breen-Messing's Théorie de Dieudonné Cristalline II. Towards the end of the first section, in paragraph 1.1.18, they make a deduction that I can't follow ...
- 513
4 votes
0 answers
119 views
When is every $s$-pure exact sequence pure, and every $s$-flat module flat?
Let $R$ be a ring. Consider a short exact sequence of left $R$-modules: $$ \eta: \quad 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 . $$ We say that $\eta$ is $s$-pure if it remains exact ...
- 247
5 votes
1 answer
370 views
Do forms of $\mathbb{G}_a$ split over perfect rings?
I'll try to pursue my understanding of forms of $\mathbb{G}_a^d$ over general bases. Let $R$ be a ring of positive characteristic $p$ and let $G$ be a smooth affine group scheme over $R$ that is a ...
- 1,573
3 votes
2 answers
395 views
Endomorphisms of groups schemes involving $\mathbb{G}_a$ and flat base change
Let $R$ be a ring of positive characteristic. Let $G$ be a commutative affine and smooth group scheme over $R$. I consider two abelian groups: $$M(G):=\mathrm{Hom}(G,\mathbb{G}_{a,R})$$ and $$N(G):=\...
- 1,573
0 votes
1 answer
115 views
Induced map on Tor via a canonical projection between cyclic groups
In a homological algebra course, I encountered the following claim: Let $p$ be a prime number. Consider the cyclic groups: $C_{p^2}=\mathbb{Z} / p^2 \mathbb{Z}$, $C_p=\mathbb{Z} / p \mathbb{Z}$, and ...
- 247
0 votes
0 answers
120 views
Example of a morphism of $\mathbb{Z}$-modules that kills maps from simples but not from some finitely presented module
I am trying to construct an example of a morphism of $\mathbb{Z}$-modules $$ f: M \longrightarrow N $$ satisfying the following properties: $f$ is nonzero, $M$ and $N$ are not projective (i.e., not ...
- 247
3 votes
1 answer
185 views
How to characterize morphisms $f: M \rightarrow N$ such that $\operatorname{Tor}_1(S, f)=0$ for all simple right $R$-modules $S$?
Let $R$ be a ring (not necessarily commutative), and let $f: M \rightarrow N$ be a morphism of left $R$-modules. We say that $f$ is an S -morphism if for every simple right $R$-module $S$, the induced ...
- 247
2 votes
0 answers
196 views
Zariski's main theorem + Raynaud-Gruson’s platification
If $X\to Y$ is a quasi-finite map of finite presentation between qcqs schemes and $U\subseteq X$ is open such that $U\to Y$ is flat, then we have the following two results: Raynaud-Gruson’s ...
- 407
3 votes
1 answer
281 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 ...
- 2,202
1 vote
1 answer
279 views
Flatness of "derived local system sheaves"
Let $f: Y\longrightarrow X$ be a smooth proper map of smooth proper schemes over $\mathbb{Q}$, and let $\mathcal{F} = R^1_\text{ét}\overline{f}_*\mathbb{Q}_p$ denote the derived pushforward of $\...
- 3,346
2 votes
1 answer
308 views
Direct product of direct sum of a flat module
In the book "Rings and Categories of Modules" by Anderson & Fuller, this problem is given: If $V^A$, i.e. the direct product of the module $V$ by the index set $A$, is flat for all sets $...
- 355
4 votes
0 answers
235 views
Vanishing of all higher direct images for a non-flat morphism
Let $f: X \to Y$ be a proper morphism of algebraic varieties which is not flat. Then for a line bundle $L$ on $X$, is the vanishing of all higher direct images ($R^i f_* L = 0$ for all $i \geq 0$) ...
- 3,454
6 votes
2 answers
580 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
2 votes
0 answers
266 views
Proposition 4.3.8 Qing Liu about flat morphisms of schemes
I have a problem with a detail of Qing Liu's proof of Proposition 4.3.8 (pag. 137 of "Algebraic Geometry and Arithmetic Curves"). The statement is: Let $Y$ be a scheme having only a finite ...
- 149
8 votes
2 answers
710 views
Faithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following: Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...
- 564
1 vote
0 answers
188 views
When flat base change is reduced
I am sorry, if this a very standard fact. Let $f\colon X\to S$ be a morphism between varieties over field of characteristic zero. Let $\psi \colon S'\to S$ be a flat morphism. Is it true that $X\...
- 229
3 votes
1 answer
263 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 ...
- 4,612
2 votes
0 answers
318 views
Can we check smoothness of a morphism after base change to the algebraic closure?
I know that smoothness is fppf local on the base, but this is not enough because taking algebraic closures is not finitely presented. The reason I'm asking this is because I want an easy/quick ...
- 729
8 votes
3 answers
2k views
Why did Ravenel define a ring spectrum to be flat if its smash-square splits into copies of itself?
In appendix A.2 of the orange book, Ravenel defines a ring spectrum $E$ to be flat if $E\wedge E$ is equivalent to a coproduct of suspensions of $E$. (Call this definition (1).) I've seen this ...
- 2,255
6 votes
0 answers
305 views
Flatness of objects in a prestable $\infty$-category
I wonder what is the correct concept of flatness of objects in a prestable $\infty$-category with appropriate conditions? The typical example is the following. Let $R$ be a connective $\mathbb E_1$-...
- 3,358
1 vote
1 answer
134 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
2 votes
0 answers
155 views
Flatness and locally freeness
Let $X$, $S$ be integral quasi-projective schemes (over $\Bbb C$). Let $\mathcal F$ be a coherent sheaf on $X\times S$, flat on $S$. Suppose that $x\in X$, $s\in S$ are closed points, and ${\mathcal F}...
- 779
1 vote
1 answer
161 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
3 votes
1 answer
384 views
Does miracle flatness always fail for a non-regular base?
In [1], Huybrechts and Mauri argue that a holomorphic Lagrangian fibration $f: X \to B$ with smooth base $B$ is flat. This is an application of so called miracle flatness [2, Thm 23.1], because ...
- 1,356
1 vote
0 answers
257 views
Ex 1.1c Hartshorne Deformation Theory: Is this family flat?
This comes from my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ...
- 161
6 votes
1 answer
573 views
Flatness of schemes
I am learning about flatness for the first time and I cannot wrap my head around why the definition with tensor products of a flat module implies geometrically that 1-parameter families of schemes ...
- 667
3 votes
1 answer
343 views
When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?
This question is in the interest of answering one part of this question, but I think it is distinct enough to warrant a separate question. Let $X$ be a regular 2-dimensional Noetherian scheme, for ...
- 1,358
4 votes
1 answer
343 views
Is local freeness open for curves?
Let $X$ be a complete nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, generated by finitely many global sections and flat over $S$ (...
- 161
6 votes
2 answers
710 views
Is $(x^2y,xy^2)$ log smooth?
Consider the map $$f:\mathbb C^2\to\mathbb C^2$$ $$(x,y)\mapsto(x^2y,xy^2)$$ We can view $f$ as induced by the map of monoids $g:\mathbb Z^2_{\geq 0}\to\mathbb Z^2_{\geq 0}$ given by the matrix $(\...
- 19k
4 votes
0 answers
410 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 ...
- 445
1 vote
0 answers
170 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, ...
- 445
3 votes
1 answer
357 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 ...
- 445
4 votes
0 answers
306 views
Is there a notion of „flatness” in point-set topology?
In algebraic geometry, flat morphisms are usually associated with the intuition that they have „continuously varying fibers”. Is there a notion in topology formalizing the same intuition? Consider for ...
- 1,404
3 votes
1 answer
717 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
1 answer
310 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. ...
- 681
4 votes
1 answer
2k views
Flatness of maps of analytic rings
Reference: Lectures on Analytic Geometry Let $f\colon(\mathcal A,\mathcal M)\to(\mathcal B,\mathcal N)$ be a map of analytic ring. There are several possible ways to pose the flatness: Flatness as ...
- 3,358
1 vote
0 answers
427 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 $M$ ...
- 1,323
1 vote
0 answers
108 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
1 vote
0 answers
244 views
Is the following local map unramified?
Let $(R,m)$ and $(S,n)$ be two local rings, $R \subseteq S$, $R$ regular, $S$ a finitely generated and flat $R$-algebra, and $mS=n$. In comments to this question it was claimed that in such situation ...
- 2,875
1 vote
0 answers
146 views
$A \to B$ with $A$ regular imply that $B$ is CM
The answer to this question says the following: "The general statement is if $A \to B$ is finite and injective, and $A$ is noetherian and regular, then $B$ is CM if and only if $A \to B$ is flat. ...
- 2,875
1 vote
1 answer
356 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 $...
- 2,875
3 votes
0 answers
170 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]=...
- 2,875
1 vote
1 answer
571 views
Flat familiy of coherent sheaves over a scheme
I'm studying the moduli problem of locally free sheaves over a connected smooth projective curve on an algebraically closed field, from the Lecture Notes of Victoria Hoskins, and I cannot fully ...
- 435
3 votes
1 answer
299 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 $S$ ...
- 2,875
3 votes
1 answer
479 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 ...
- 4,150
0 votes
1 answer
265 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 \...
- 2,875
0 votes
1 answer
488 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: $f(s_1 \otimes_R s_2)=s_1s_2$. ...
- 2,875
3 votes
1 answer
325 views
Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$
I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3) and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON $\...
- 3,914
0 votes
0 answers
167 views
Projection from closure of locally closed subscheme is Etale
Let $S$ an arbitrary scheme and denote by $\Delta: S \to S \times S$ the diagonal immersion and $p_i: S \times S \to S$ the both projections to first resp second factor. (in following we will wlog ...
- 3,914