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 and...
View ArticleComposition 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, if...
View ArticleWhen 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...
View ArticleA 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...
View ArticleIs $(x^2y,xy^2)$ log smooth?
Consider the map$$f:\mathbb C^2\to\mathbb C^2$$$$(x,y)\mapsto(x^2y,xy^2)$$We can view $f$ as induced by the map of monoids $g:\mathbb Z^2_{\geq 0}\to\mathbb Z^2_{\geq 0}$ given by the matrix...
View ArticleIs local freeness open for curves?
Let $X$ be a complete nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, generated by finitely many global sections and flat over $S$ (via...
View ArticleWhen is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$...
This question is in the interest of answering one part of this question, but I think it is distinct enough to warrant a separate question.Let $X$ be a regular 2-dimensional Noetherian scheme, for...
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Flatness of schemes
I am learning about flatness for the first time and I cannot wrap my head around why the definition with tensor products of a flat module implies geometrically that 1-parameter families of schemes have...
View ArticleEx 1.1c Hartshorne Deformation Theory: Is this family flat?
This comes from my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$...
View ArticleDoes miracle flatness always fail for a non-regular base?
In [1], Huybrechts and Mauri argue that a holomorphic Lagrangian fibration $f: X \to B$ with smooth base $B$ is flat. This is an application of so called miracle flatness [2, Thm 23.1], because...
View ArticleSubrings, submodules, and flatness
Let $R$ be a ring, and $S$ a subring of $R$. Let $M$ be a right $R$-module, and $N$ a right $S$-submodule of $M$. If $N$ is flat (or faithfully flat) as a right $S$-module, does it then follow that the...
View ArticleSubtle 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...
View ArticleAlternative 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....
View ArticleFlatness and locally freeness
Let $X$, $S$ be integral quasi-projective schemes (over $\Bbb C$). Let $\mathcal F$ be a coherent sheaf on $X\times S$, flat on $S$. Suppose that $x\in X$, $s\in S$ are closed points, and ${\mathcal...
View ArticleOn $\text{depth}_S\left(\dfrac {S}{JS+xS}\right)$, when...
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 $J$...
View ArticleFlatness 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...
View ArticleWhy did Ravenel define a ring spectrum to be flat if its smash-square splits...
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...
View ArticleCan we check smoothness of a morphism after base change to the algebraic...
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 argument...
View ArticleFlatness 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...
View ArticleFlatness 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 $S$ is...
View ArticleCan a quotient ring R/J ever be flat over R?
If $R$ is a ring and $J\subset R$ is an ideal, can $R/J$ ever be a flat $R$-module? For algebraic geometers, the question is "can a closed immersion ever be flat?"The answer is yes: take $J=0$. For a...
View ArticleWhen flat base change is reduced
I am sorry, if this a very standard fact.Let $f\colon X\to S$ be a morphism between varieties over field of characteristic zero. Let $\psi \colon S'\to S$ be a flat morphism. Is it true that $X\times_S...
View ArticleFaithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then...
View ArticleProposition 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 number of...
View ArticleDoes 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 replacing...
View ArticleExistence of functorial (K-)flat resolutions?
I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe that...
View ArticleVanishing of all higher direct images for a non-flat morphism
Let $f: X \to Y$ be a proper morphism of algebraic varieties which is not flat. Then for a line bundle $L$ on $X$, is the vanishing of all higher direct images ($R^i f_* L = 0$ for all $i \geq 0$)...
View ArticleDirect product of direct sum of a flat module
In the book "Rings and Categories of Modules" by Anderson & Fuller, this problem is given: If $V^A$, i.e. the direct product of the module $V$ by the index set $A$, is flat for all sets $A$, then...
View ArticleFlatness of "derived local system sheaves"
Let $f: Y\longrightarrow X$ be a smooth proper map of smooth proper schemes over $\mathbb{Q}$, and let $\mathcal{F} = R^1_\text{ét}\overline{f}_*\mathbb{Q}_p$ denote the derived pushforward of...
View ArticleIs 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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