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3
votes
1answer
207 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 ...
2
votes
0answers
96 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, ...
1
vote
0answers
194 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 ...
5
votes
1answer
171 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. ...
4
votes
1answer
150 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 ...
2
votes
0answers
54 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 $\pi(x,t)=t$, ...
1
vote
1answer
153 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 ...
0
votes
0answers
128 views

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: ...
4
votes
2answers
444 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 ...
2
votes
1answer
86 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 ...
4
votes
0answers
138 views

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 $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ ...
2
votes
1answer
190 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?
2
votes
1answer
380 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 ...
0
votes
0answers
161 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, ...
9
votes
1answer
369 views

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 ...
0
votes
1answer
186 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 ...
7
votes
1answer
310 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 ...
3
votes
1answer
326 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" ...
2
votes
2answers
210 views

flatness criterion on normal bases

Let $f:X\rightarrow Y$ a $k$-morphism of finite type from a Cohen-Macaulay scheme to a normal scheme such that all the fibers have the same dimension, do we have that $f$ is flat? We know, that it's ...
3
votes
1answer
270 views

On the m-th power of the Hodge bundle and Arakelov's theorem

Let $S$ be a smooth projective curve over $\mathbf C$ and let $f:X\to S$ be a projective flat morphism with "semi-stable" fibres (i.e., the fibres are reduced and strict normal crossings divisors) and ...
2
votes
1answer
290 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., ...
2
votes
2answers
257 views

Are quotients of stacks flat?

Let $\cal X$ be a DM stack of finite type over a field (if necessary, I will assume that $k=\mathbb{C}$ and $\cal X$ is a scheme, or even a variety) and $G$ be a finite group. Then we have a quotient ...
9
votes
1answer
369 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 ...
1
vote
1answer
290 views

1st-flat cohomology group for elliptic curves

Let $E$ be an elliptic curve over an algebraically closed field $k$ of characteristic $p$. Is there any nice computation for the group $H^1(E,\alpha_p)$ and $H^1(E,\mathbb{G}_a)$? The cohomology is ...
1
vote
0answers
153 views

Existence of the universal family for the Hilbert scheme of plane curves

Given a finitely generated $k$-algebra $A$ over alg. closed $k$, a family of curves of degree $d$ is defined to be a subscheme $X\subset \mathbb P^2_A$ flat over $A$ whose fibers over closed points of ...
3
votes
2answers
172 views

is intersection of a curve and a family of curves generically constant as a scheme?

(everything below is defined over an algebraically closed field) Let $D$ be a (smooth) surface, and let $X \subset T \times D$ be a flat family of curves on $D$, where $T$ is irreducible. Let $E$ be ...
3
votes
3answers
745 views

Flatness for family of hypersurfaces

Let $X \to Y$ be a family of hypersurfaces in a constant $\mathbb{P}^n$, i.e. $X \subset Y \times \mathbb{P}^n$ is locally on $Y$ given by one equation of degree $d$ in $\mathbb{P}^n$. Is $X \to Y$ ...
1
vote
2answers
797 views

properties of the fibers of a flat morphism

Let $X$ be an arbitrary scheme of finite type over $\mathbb{C}$ and $f:X \rightarrow \mathbb{A}_{\mathbb{C}}^1$ be a flat morphism. I suppose that the special fiber have some nice property, I would ...
8
votes
3answers
780 views

Relationship between monodromy representations and isomorphism of flat vector bundles

This question is somehow related to this one. Let $M$ be a smooth (compact, if you wish) connected manifold. Then, it is well known that there is an equivalence between the isomorphism classes of ...
3
votes
1answer
318 views

Spreading out flat morphisms of schemes

In EGA IV, Chapter 8, projective systems of schemes (and morphisms between them) are considered. Let $(S_{\lambda})_{\lambda \in L}$ be a projective system of schemes and let $S$ be the projective ...
7
votes
1answer
220 views

Triviality of direct multiples of flat complex vector bundles

Atiyah Patodi and Singer [Spectral asymmetry and Riemannian geometry III] write that if $E$ is a complex flat bundle (non holomorphic, just smooth and complex) on a compact manifold $X$ (more ...
11
votes
4answers
830 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 ...
6
votes
2answers
507 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 ...
1
vote
1answer
224 views

Relative generic flatness.

It is known that any morphism is flat at an open set of points. I'd like to know if there is a relative version of this fact. Let $f: X \rightarrow Y$ and $g:Y \rightarrow S$ be morphisms of ...
7
votes
2answers
376 views

Resolution of singularities for flat families.

Is there a resolution of singularities for flat families? More precisely, if $X \rightarrow \mathbb{A} ^n$ is a flat map, does there exist a map $Y \rightarrow X$ such that, for every $p \in ...
2
votes
4answers
1k 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 ...
7
votes
1answer
376 views

Extending faithfully flat covers of closed subschemes to open neighborhoods

I am curious about the analogue of this question, as stated in the title. Namely, If $Z \subset X$ is a closed subscheme and $Y \to Z$ is faithfully flat (let's also say of finite presentation), ...
1
vote
0answers
103 views

Cuspidal stable curves

I have seen this many times but never know a rigorous proof. In a flat family of stable curves, if an elliptic tail is contracted then we get a cuspidal curve, if an elliptic bridge is contracted, we ...
5
votes
1answer
370 views

sections of morphisms of complex spaces

A smooth morphism of schemes $f: X \to Y$ admits an étale-local section through any point $x \in X$. One might wonder if this fact is true in the more general context of complex spaces (i.e. things ...
3
votes
1answer
444 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. ...
2
votes
1answer
387 views

Flatness over non-reduced schemes : no geometric characterisation

I know someone already asked about flatness over non-reduced schemes, but I think my question is different. I'm reading Bosch, Lütkebohmert and Raynaud's "Néron models", and in the second chapter, ...
4
votes
3answers
888 views

Quotient of flat module is flat - a property in Mumford's Red book

Hi, In Mumford's "Red Book of Varieties and Schemes", Chapter III, Paragraph 10 (entitled "Flat and smooth morphisms"), the following property is stated: Let $M$ be a $B$-module, and $B$ an algebra ...
2
votes
2answers
472 views

Flatness of sheaf of relative Kahler differentials

Suppose we have a projective flat non-smooth morphism of Noetherian schemes $g: X \rightarrow S$. My question regards when the sheaf of relative Kahler differentials $\Omega_{X/S}$ is flat over $S$. ...
2
votes
1answer
269 views

can one define the pullback between stacks of coherent sheaves for non-flat morphisms?

Consider a morphism $f: Y \to X$ between two varieties and consider the stacks parametrizing coherent sheaves on them $\mathcal{M}_X, \mathcal{M}_Y$. Does one have for free an induced pullback ...
4
votes
2answers
2k views

Rank 2 flat bundles on an elliptic curve, via extensions

I have some hopefully elementary questions about rank 2 flat bundles on an elliptic curve $E$. Take $p\in E$, and consider the exact sequence $$0\to \mathcal{O}(-p) \to V \to \mathcal{O}(p)\to 0$$ ...
6
votes
3answers
570 views

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 ...
0
votes
1answer
509 views

Projectivity and faithfully flatness (module theory) [closed]

Is it true that every projective module is faithfully flat, if not what is a counter example. Thanks!
5
votes
2answers
369 views

When can a finite map be blown up to a flat one?

Let $f:X\to Y$ be a generically finite proper morphism of varieties. There is some locus in $Y$ over which the fiber of $f$ is positive dimensional, so we blow it up, along with the preimage of it in ...
2
votes
1answer
335 views

Commutator tensors and submodules

Let $k$ be a commutative ring with $1$, and let $B$ be a submodule of a $k$-module $A$. For every $n\in\mathbb N$ and every $k$-module $V$, let $K^n\left(V\right)$ denote the kernel of the canonical ...
11
votes
2answers
574 views

Given a family of curves, when does there exist a fibered surface over Spec Z parametrizing them?

Let $X_p$ be a projective curve over the finite field $\mathbf{F}_p$ (i.e. a projective $\mathbf{F}_p$-scheme pure of dimension 1) for every prime number $p$. Let $X_\mathbf{Q}$ be a projective curve ...