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. 5...
View ArticleFinitely generated modules over Noetherian local ring that become isomorphic...
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...
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...
View ArticleFaithful 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...
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