The flatness tag has no usage guidance.

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### Do finite flat sheaves define families of $0$-cycles?

Let $X$ be a smooth projective $\mathbb C$-variety and let $X^{(n)}$ denote the symmetric product $X^n/S_n$, parametrizing effective $0$-cycles of degree $n$ on $X$.
Question. Let $S$ be a ...

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187 views

### Homotopical interpretation of flatness?

I have read a discussion (in a less common language) which discussed a homotopical interpretation of flatness, which went something like:
A map of commutative algebras is flat if pushing it out ...

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204 views

### Is the zero locus of a global section flat?

Let $f:X \to Y$ be a surjective, smooth projective morphism of noetherian schemes. Let $\mathcal{L}$ be an inverible sheaf on $X$ satisfying $f_*\mathcal{L}$ is locally free and $s \in H^0(\mathcal{L})...

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180 views

### A Hartogs-type criterion for flatness

Let $U$ be a smooth affine connected variety over $\mathbb C$ and let $V\subset U$ be an open whose complement is of codimension at least two.
Now, let $Y$ be a smooth quasi-affine connected variety ...

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**1**answer

88 views

### Torsion free sheaves in flat families

Let $R$ be a dvr, $X$ a flat, projective, integral, normal $R$-scheme such every closed fiber is again integral, normal. Let $F$ be a torsion-free coherent sheaf on $X$, flat over $R$. Is it true that ...

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107 views

### Reflexive sheaf and flatness

Let $f:X \to Y$ be a proper, flat morphism of noetherian schemes. Let $\mathcal{F}$ be a coherent sheaf on $X$ non-zero on every fibers of $f$. Is it true that $\mathcal{F}^{\vee \vee}$ is going to be ...

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87 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 ...

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160 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 ...

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**1**answer

204 views

### Euler characteristic on flat families of quasi-projective schemes

Let $A$ be a noetherian integral domain (may be regular). Let $\pi:X \to \mathrm{Spec}(A)$ be a flat morphism. Suppose that each fiber of $\pi$ are quasi-projective. Let $\mathcal{F}$ be a coherent ...

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80 views

### Are “vector spaces” over a smooth scheme with constant fiber dimension locally free?

I've got the follow question which drives me almost crazy as the answer seems to be simple. Given a morphism $p:V\to S$ of schemes of finite type over some base field. Assume that $p$ has all the ...

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305 views

### Failure of universal flatification

Raynaud and Gruson proved a beautiful "flatification" theorem (5.2.2): If $S$ is a quasicompact, quasiseparated scheme, and $X$ is a finitely presented $S$-scheme, $M$ is an $\mathcal O_X$-module of ...

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125 views

### Extend a vector bundle on a flat family

Let $f: X\to T$ be a flat family, and $\mathcal{F}_t$ is a vector bundle on $X_t$ for some $t\in T$. Can this $\mathcal{F}_t$ be extended to a vector bundle $\mathcal{F}$ on $f^{-1}(U)$ for some open ...

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105 views

### A naive question on rational equivalence of varieties

Let $X$ be a projective scheme and $\pi:\mathcal{Z} \to \mathbb{P}^1$ a surjective morphism of finite type such that for any pair $t_0, t_1 \in \mathbb{P}^1$, the fibers $\mathcal{Z}_{t_0}$ and $\...

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**1**answer

152 views

### Control of a meromorphic function according to distance between its zeros

My question is rather philosophical : can a meromorphic function of normalized norm with simple zeros on the flat torus stay close to zero on a large set when its zeros are far from each other ?
The ...

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**1**answer

196 views

### When is the flatness locus non-empty

Let $k$ be an algebraically closed field, $f:X \to Y$ be a surjective proper $k$-morphism locally of finite presentation between irreducible noetherian schemes. Assume that $Y$ is reduced. Under what ...

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**1**answer

73 views

### A confusion about covering flatness

I'm reading this entry on nLab. But I'm confusing with the notion of covering-flatness. More precisely, I meet some trouble when I try to show that the $Sets$-valued flatness is a special case of ...

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802 views

### Is there a notion of “flat vector bundle over a topological space”?

I am reading this paper and at the top of page 5 the author makes reference to categories consisting of flat complex vector bundles over $X$ where $X$ is an arbitrary topological space. However, the ...

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542 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, ...

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**1**answer

103 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) \...

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**1**answer

183 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 ...

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302 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 ...

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### Ultraproducts and subobjects of projectives

Usually the question whether the class of projective algebras in a given variety is closed under taking subalgebras seems to be quite hard. In varieties with well understood dual geometry (e. g. ...

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687 views

### Does projective imply flat?

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...

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175 views

### Sections of proper, flat morphism

Let $f:X \to Y$ be a proper, flat morphism of projective scheme and $Y$ is an irreducible, non-singular surface. Assume further that there exists a Zariski open subset $U$ of $Y$ whose complement is ...

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93 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 ...

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63 views

### quantum deformation

The standard quantum groups, say $GL_q(n)$ or $U_q(gl(n))$, depend on the parameter $q$, which in the classical limit tends toward 1. Let $t:=q-1$ be considered as a generic parameter then we can ...

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**1**answer

242 views

### Which base change preserves integrality of schemes

Let $f:X \to Y$ be a flat morphism of projective noetherian integral schemes. Is there any known condition on a morphism $Z \to Y$ under which the resulting fiber product $X \times_Y Z$ is still ...

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192 views

### Linear projection from a point and local complete intersection

Let $p:=[0,0,...,0,1] \in \mathbb{P}^n$ the point whose all the coordinates are zero except for the $n$-th. This defines a linear projection map $\phi:\mathbb{P}^n-p \to \mathbb{P}^{n-1}$, given by $[...

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181 views

### Flatness and intersection of fibers

Let $f:X \to Y$ be a flat, proper, surjective morphism between noetherian schemes. Assume $Y$ is irreducible and smooth over $\mathbb{C}$. Suppose that $X$ is the union of two schemes $X_1$ and $X_2$ ...

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237 views

### Connectedness of fibers for flat, proper morphism

Let $f:X \to Y$ be a flat proper morphism of noetherian schemes of finite type over a field. Assume that $Y$ is an integral scheme and the generic fiber of $f$ is irreducible of dimension $1$. Is it ...

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294 views

### Covering of schemes and flatness

Let $f:X \to Y$ be a finite surjective morphism of quasi-projective schemes over $\mathbb{C}$, $X$ is reduced and $Y$ is integral. Suppose that there exists an integer $n$ such that for every closed ...

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268 views

### Exactness of an additive left Kan extension

Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$
where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...

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105 views

### Tori in Compact Riemannian Symmetric Spaces

The closure of a 1-parameter subgroup in a compact Lie group is a torus. To what extent does this result generalize to compact Riemannian symmetric spaces? In other words, is the closure of a geodesic ...

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287 views

### Affine hulls and base change

Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...

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168 views

### $\delta$-functor and commutativity of pull-back with right derivation

Let $f:X \to Y$ be a faithfully flat projective morphism of noetherial $\mathbb{C}$-schemes. Assume that $Y$ is affine, smooth over $\mathbb{C}$. Let $y \in Y$ be a closed point with residue field, ...

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239 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 ...

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218 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.
...

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252 views

### Simultaneous resolution of singularities in special cases of flat families of projective varieties

Let $\pi:\mathcal{X} \to B$ be a flat family of projective varieties. Assume that $B$ is irreducible. Suppose that $\mathcal{X}$ is smooth except for a closed subscheme, say $Y$ which is isomorphic to ...

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173 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 $S$?

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59 views

### morphism flat and relations

I want to know if the following result is true:
Let $I=<f_1,...,f_r>$ be an ideal of $K[x_1,x_2,...x_n,t_1,..,t_m]$. The morphism $\pi:V(I)\subset{K^{n+m}}\rightarrow{K^m}$, with $\pi(x,t)=t$, ...

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209 views

### Does flatness/smoothness over special fiber imply flatness/smoothness globally?

Let $f:X \to \mbox{Spec } R$ be a projective morphism between irreducible Noetherian schemes. Assume that $R$ is a discrete valuation ring and its residue field is algebraically closed. Suppose now ...

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### Flatness over Hopf subalgebra

Let $A\subset B$ be flat Hopf algebras over a Dedekind ring $R$.
J. Moore has proved in the article
Compléments sur les algèbres de Hopf, John C. Moore, Séminaire Henri Cartan (1959-1960), Volume: ...

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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 ...

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98 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 ...

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### Injectivity criterion for surjective coalgebra maps: does it hold in full generality?

Let $k$ be a commutative ring.
Let $C$ be a filtered $k$-coalgebra. This means a $k$-coalgebra equipped with an increasing $k$-module filtration $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ ...

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221 views

### flat and finite type morphisms

Let $f:X\rightarrow Y$ a faithfully flat morphism between $k$-schemes. We assume that the fibers are locally of finite type, do we have that $f$ is locally of finite type?

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543 views

### Morphism with non-reduced special fibre

Let $X, Y$ be irreducible projective varieties and $Y$ be smooth. Let $f:X \to Y$ be a flat projective morphism. Assume that a special fiber of $f$ is non-reduced i.e., there exists an irreducible ...

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185 views

### on flat morphisms

Let $j:U\rightarrow X$ an open immersion between k-schemes of finite type and $f:X\rightarrow S$ a surjective k-morphism of finite type.
We suppose that $f\circ j:U\rightarrow S$ is faithfully flat, ...

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### Is there a direct proof that affine schemes are fppf quasi-compact?

Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2) each $B_i$ is ...

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253 views

### The locus where a sheaf is supported in a certain dimension

I am trying to understand a particular case of this question.
Let $T$ be an affine scheme of finite type over the ground field C. Let $X \to T$ be a morphism. I'll assume this to be the base change ...