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Let $R$ be an associative ring with identity. Recall that a pair of classes of modules $(\mathcal{A}, \mathcal{B})$ in $R\text{-Mod}$ is called a cotorsion pair if $$ \mathcal{A} = {}^{\perp}\mathcal{...
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I am looking at the first chapter of Berthelot-Breen-Messing's Théorie de Dieudonné Cristalline II. Towards the end of the first section, in paragraph 1.1.18, they make a deduction that I can't follow ...
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Let $R$ be a ring. Consider a short exact sequence of left $R$-modules: $$ \eta: \quad 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 . $$ We say that $\eta$ is $s$-pure if it remains exact ...
Mourad Khattari's user avatar
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I'll try to pursue my understanding of forms of $\mathbb{G}_a^d$ over general bases. Let $R$ be a ring of positive characteristic $p$ and let $G$ be a smooth affine group scheme over $R$ that is a ...
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Let $R$ be a ring of positive characteristic. Let $G$ be a commutative affine and smooth group scheme over $R$. I consider two abelian groups: $$M(G):=\mathrm{Hom}(G,\mathbb{G}_{a,R})$$ and $$N(G):=\...
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In a homological algebra course, I encountered the following claim: Let $p$ be a prime number. Consider the cyclic groups: $C_{p^2}=\mathbb{Z} / p^2 \mathbb{Z}$, $C_p=\mathbb{Z} / p \mathbb{Z}$, and ...
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I am trying to construct an example of a morphism of $\mathbb{Z}$-modules $$ f: M \longrightarrow N $$ satisfying the following properties: $f$ is nonzero, $M$ and $N$ are not projective (i.e., not ...
Mourad Khattari's user avatar
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Let $R$ be a ring (not necessarily commutative), and let $f: M \rightarrow N$ be a morphism of left $R$-modules. We say that $f$ is an S -morphism if for every simple right $R$-module $S$, the induced ...
Mourad Khattari's user avatar
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If $X\to Y$ is a quasi-finite map of finite presentation between qcqs schemes and $U\subseteq X$ is open such that $U\to Y$ is flat, then we have the following two results: Raynaud-Gruson’s ...
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Let $f : A \to B$ be a finite, finitely presented, flat map of (commutative) rings. It is a known consequence of Chevalley's theorem (on constructible sets) that the induced map $Spec B \to Spec A$ is ...
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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 $\...
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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 $...
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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 $i \geq 0$) ...
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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 ...
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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 ...
Ricky Bern's user avatar
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$\DeclareMathOperator\Spec{Spec}$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 $...
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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\...
Galois group's user avatar
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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 ...
Matthieu Romagny's user avatar
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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 ...
TCiur's user avatar
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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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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 $\mathbb E_1$-...
Z. M's user avatar
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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 $J$ ...
feder's user avatar
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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 F}...
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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 ...
Jake Wetlock's user avatar
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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 ...
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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 $A$ ...
nolatos's user avatar
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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 ...
did's user avatar
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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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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 $S$ (...
nolatos's user avatar
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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 $(\...
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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 ...
Tim Montegue's user avatar
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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, ...
Tim Montegue's user avatar
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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 ...
Tim Montegue's user avatar
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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? ...
Jakob Werner's user avatar
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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 \...
user337580's user avatar
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1 answer
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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. ...
Bryan Shih's user avatar
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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 ...
Z. M's user avatar
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431 views

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 $M$ ...
user26857's user avatar
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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 ...
Boris Henriques's user avatar
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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 ...
user237522's user avatar
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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. ...
user237522's user avatar
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365 views

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 $...
user237522's user avatar
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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]=...
user237522's user avatar
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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 ...
John117's user avatar
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1 answer
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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 $S$ ...
user237522's user avatar
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3 votes
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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 ...
Dima Sustretov's user avatar
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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 \...
user237522's user avatar
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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: $f(s_1 \otimes_R s_2)=s_1s_2$. ...
user237522's user avatar
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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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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 ...
user267839's user avatar
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