# Questions tagged [flatness]

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb E_1$-...
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### 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 $J$ ...
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### 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 ...
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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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### 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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### 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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### 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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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 $K[X_1,... • 5,786 1 vote 1 answer 226 views ### Non-unique completion of a flat family of smooth projective varieties Let$\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)$. What is ... 10 votes 0 answers 487 views ### How general are Gröbner degenerations? While working with flat degenerations of flag varieties and Schubert varieties I've noticed that among the numerous known constructions there doesn't seem to be a single one that doesn't turn out to ... • 3,495 8 votes 3 answers 515 views ### Lifting a complete intersection in$\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$that is a complete intersection. In other words, it can be defined ... • 121 3 votes 0 answers 57 views ### When flatness of$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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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{\...