Questions tagged [flatness]

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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, ...
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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 ...
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Faithfully flat ring extension and local freeness at a minimal prime

Let $R\subseteq S$ be a faithfully flat extension of Noetherian rings. Let $P$ be a minimal prime ideal of $R$ and $Q$ be a prime ideal of $S$ minimal over $PS$. Let $M$ be a finitely generated $R$-...
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Faithfully flat ring extension of local Cohen--Macaulay rings and extension of prime ideals

Let $(R,\mathfrak m) \subseteq (S,\mathfrak mS)$ be a faithfully flat extension of local Cohen--Macaulay rings. If $P$ is a minimal prime ideal of $R$, then is $PS$ a prime ideal of $S$? If this is ...
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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 ...
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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 \...
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141 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. ...
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187 views

Flatness of affine cone due to semicontinuity theorem

I would like to clarify an important aspect from the discussion in this question. The OP discussed an obstacle to solve part (c) from Exercise 9.5 from Hartshorne's Algebraic Geometry Chap. III page ...
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534 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 ...
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168 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 $M$ ...
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53 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 ...
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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 ...
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$A \to B$ 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. ...
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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 $...
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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]=...
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156 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 ...
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189 views

Proposition on local properties of Quot scheme

I struggle with a couple of problems to understand severtal steps in the proof of Proposition 4.4.4 from the book "Deformations of Algebraic Schemes" by Edoardo Sernesi (pages 223-225). ...
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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 $S$ ...
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204 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 ...
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163 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 \...
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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: $f(s_1 \otimes_R s_2)=s_1s_2$. ...
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Locally freeness of twisted direct images $\pi_* \mathcal{F}(i)$

I have a question about a step in the proof of the Existence of Flattening Stratification I found in Nitsure's paper here: https://arxiv.org/abs/math/0504590 This question is closely related to Local ...
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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 $\...
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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 ...
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Local freeness of $\pi_*F(r)$ from flatness of $F$

In 'Fundamental Algebraic Geometry' by Fantechi there is a lemma in section 5.3.2, page 119: LEMMA 5.5 Let $S$ be a noetherian scheme and let $F$ be a coherent sheaf on $\mathbb{P}^n_S$. Suppose there ...
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Interpolating between curves in different characteristics

Let $p\neq q$ be two primes. For a given integer $g>0$ choose a smooth proper geometrically connected curve of genus $g$ over $\mathbb{F}_p$ and similarly for $q$. Is there a proper flat morphism $...
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Does a flat compactification always exist?

Let $\pi:X\to S$ be a separated flat morphism of finite type of Noetherian schemes. Does $\pi$ necessarily factor as an open immersion followed by a proper flat morphism? The analogue of this question ...
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Proper and flat over $\mathbb{P}^1_{\mathbb{Z}}$ implies locally free

Let $\pi:X\to \mathbb{P}^1_{\mathbb{Z}}$ be a proper flat morphism with $X$ an integral scheme. Is $\pi_*\mathcal{O}_X$ necessarily locally free?
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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 ...
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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 $A\mapsto A_{!!}$...
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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,...
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Non-unique completion of a flat family of smooth projective varieties

Let $\mathbb{k}$ be an algebraically closed field of characteristic 0. Denote $S=\mathrm{Spec}\:\mathbb{k}[t]$, $U=\mathrm{Spec}\:\mathbb{k}[t, t^{-1}]$, $Z=\mathrm{Spec}\:\mathbb{k}[t]/(t)$. What is ...
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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 ...
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Lifting a complete intersection in $\mathbb{P}^n_{\mathbb{F}_p}$ to $\mathbb{Z}_p$

Suppose that you are given a (not necessarily smooth) projective variety $X \subseteq \mathbb{P}^n_{\mathbb{F}_p}$ of codimension $d$ that is a complete intersection. In other words, it can be defined ...
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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,...
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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{\...
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When do generizations ("generalizations") lift uniquely?

If $f : X \to Y$ is proper, then specializations lift along $f$, and uniquely. (This means, if $R$ is a discrete valuation ring with fraction field $K$ and I choose a factorization $\text{Spec}K \to ...
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618 views

Flat connections, curvature and holonomy

Let $A$ be a flat connection on a principal $G$-bundle $G\hookrightarrow P\to M$. Consider an homotopically trivial loop $\gamma \subset M$. For simplicity, suppose $\gamma = \partial D$ is the ...
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Flat connections and global sections of vector bundles

Let $X$ be a (non-singular) complex surface and $(V,\nabla)$ be a vector bundle $V$ equipped with a flat connection $\nabla$ on $X$. Fix a point $x \in X$ and $v_0 \in V_x$ an element in the fiber ...
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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 ...
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Flatness, "Continuously varying fibers", and bordism

It is commonly said that a flat map of schemes $f : X \rightarrow Y$ is like a map with "continuously varying fibers". We see a hint at this in the result that $\text{dim} X_y$ is constant when $f$ is ...
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Push-forward of flat module under a finite, flat morphism

Let $f:X \to Y$ be a finite, faithfully flat morphism of noetherian, affine $\mathbb{C}$-schemes. One can assume $Y$ is non-singular. Let $A$ be a local artinian $\mathbb{C}$-algebra and $f_A:X_A \to ...
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302 views

Isomorphism in fibers and flatness

Let $X$, $Y$ be (reduced) affine varieties and $f:X \to Y$ is a finite morphism which is an isomorphism over an open dense subset (for example a normalization map). Let $A$ be a local noetherian ring ...
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Classification of flat Riemannian three manifold

By a theorem of L. Bieberbach we know that that every closed flat Riemannian manifold is a quotient of a torus via action of a finite group $\Gamma$. In this question we are interested in the ...
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Flat base change in the complex analytic setting

On page 255 of Hartshorne's Algebraic Geometry, it is shown that "cohomology commutes with flat base extension": Proposition III.9.3: Let $f : X \to Y$ be a separated morphism of finite type of ...
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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 $x\in P$ ,...
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Uniqueness of the scheme structure for a given Hilbert polynomial

If we have two lines in $P^3$ which are skewed, then we can take the union of those lines as a subscheme of $P^3$ in order to obtain a subscheme of $P^3$ with a Hilbert Polynomial given by $2m+2$. ...
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153 views

Flat covariant derivative

Is it true that for any flat and torsion-free connection $\nabla : \mathfrak{X} (M) \times \mathfrak{X} (M) \rightarrow \mathfrak{X} (M) $ there exist a local systems of coordinates such that the ...
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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 ...
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290 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 ...