The tag has no usage guidance.

learn more… | top users | synonyms

2
votes
1answer
184 views

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 ...
6
votes
0answers
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 ...
2
votes
2answers
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})...
7
votes
0answers
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 ...
2
votes
1answer
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 ...
0
votes
0answers
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 ...
2
votes
0answers
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 ...
2
votes
1answer
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 ...
0
votes
1answer
205 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 ...
2
votes
0answers
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 ...
6
votes
1answer
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 ...
4
votes
0answers
126 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 ...
1
vote
0answers
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 $\...
2
votes
1answer
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 ...
4
votes
1answer
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 ...
0
votes
1answer
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 ...
10
votes
3answers
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 ...
13
votes
2answers
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, ...
2
votes
1answer
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) \...
2
votes
1answer
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 ...
0
votes
2answers
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 ...
2
votes
0answers
82 views

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. ...
14
votes
2answers
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 ...
0
votes
1answer
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 ...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
2
votes
1answer
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 ...
3
votes
1answer
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 $[...
0
votes
0answers
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 $X_2$ ...
1
vote
0answers
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 ...
4
votes
1answer
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 ...
6
votes
2answers
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 ...
5
votes
1answer
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 ...
3
votes
1answer
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 ...
2
votes
0answers
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, ...
1
vote
0answers
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 ...
5
votes
1answer
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. ...
4
votes
1answer
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 ...
1
vote
1answer
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 $S$?
2
votes
0answers
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 $\pi(x,t)=t$, ...
0
votes
1answer
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 ...
0
votes
0answers
174 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
547 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
99 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
152 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
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?
2
votes
1answer
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 ...
0
votes
0answers
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, ...
9
votes
1answer
457 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
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 ...