# Questions tagged [kahler-differentials]

Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.

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### Universal derivations in complex analytic geometry

In algebraic geometry, we are used to the following universal property of K"ahler differential forms: Let $f: X \rightarrow S$ be a morphism of (Noetherian) schemes. Then there is a relative ...

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### Is there an analogue of the module of differentials for "higher order derivations" in the Hochschild/cyclic senses?

Note added by YC: the definition below of the cyclic sub-complex is incorrect; and the "higher order derivations" referred to here are traditionally known (since the 1940s) as n-cocycles.
$\...

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### How to get a concrete description of $\pi_*\Omega_{X/S}(Z)|_Z$, when $X \supset Z \to S$ is a finite extension of Dedekind schemes?

Let $X/S$ be a proper, smooth relative curve over a Dedekind scheme $S$, for example, $X = \mathbb{P}^1_S \xrightarrow{\pi} S$.
Suppose that $Z \to X$ is a horizontal effective Cartier divisor such ...

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### Logarithmic differentials on an arithmetic surface, and Poincaré residue

Suppose that $X$ is an arithmetic surface, i.e. $\pi: X \to S$ flat and relative dimension 1 over a Dedekind scheme $S$, and assume $X$ smooth.
Let $Y \subset X$ be a horizontal effective Cartier ...

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### Is the cohomology $H^1(X, \mathcal{E}^\nabla)$ trivial, for the sheaf of constants of an algebraic connection $\nabla$?

Suppose that $\pi:X\to S$ is a flat morphism between Noetherian, integral schemes (of characteristic zero, if need be).
Let $\mathcal{E}$ be a locally free sheaf on $X$, and
$$\nabla:\mathcal{E} \to \...

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### Exact sequence for low-degree terms of relative de Rham cohomology

Let $\varphi:X\to Y$ be a morphism of schemes, smooth of relative dimension 1. The de Rham cohomology $H^\bullet(X/Y)$ is the result of applying the derived functor $R^\bullet\varphi_*$ to the de Rham ...

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### Conditions for long exact sequence for line bundles on curve to degenerate?

Let $\varphi:X\to Y$ be a morphism of schemes of relative dimension 1, and $\mathcal{L}' \xrightarrow{g} \mathcal{L}$ an injection of line bundles on $X$.
The sequence
$$0\to \mathcal{L}' \xrightarrow{...

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1
answer

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### Confusion about the (Grothendieck–Poincaré) double dual of reflexive differentials vs usual differentials on a normal Cohen–Macaulay scheme

$\DeclareMathOperator\Hom{Hom}$Let $\mathcal{A}$ be an abelian category, my question is about the case when $\mathcal{A}$ is the category of quasi-coherent sheaves on a scheme $X$. There is a fully ...

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### Jet Nestruev's proof that the exterior derivative $d$ on a real line is not a Kähler differential $d_{C^\infty(\mathbb{R})/\mathbb{R}}$

The relation between the exterior derivative $d:C^\infty(\mathbb{R})\to\Omega^1\mathbb(\mathbb{R})$ and the Kähler differential $d_{C^\infty(\mathbb{R})/\mathbb{R}}:C^\infty(\mathbb{R})\to\Omega_{C^\...

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### Sufficient conditions for $b\not\in I^2$ given that $b\in I$

Let $I$ be an $R$-ideal in a commutative algebra $B$ over a commutative ring $R.$ Given $b\in I$ I want to prove that $b\not \in I^2$.
Are there any sufficient conditions for showing that $b\not\in I^...

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### $\Omega^1_{B_\bullet/A_\bullet}$ is acyclic if $A_\bullet \to B_\bullet$ is quasi-isomorphism

Let $A_\bullet \to B_\bullet$ be a quasi-isomorphism of simplicial rings in the sense of (P.62, I.3.1.7, Complexe Cotangent et Déformations I, Luc Illusie).
Then, we define the simplicial $B_\bullet$-...

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### Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism

If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\...

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1
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### The left exactness of conormal sequence when $X$ is singular

When $X$ is a nonsingular variety over a field $k$ and Z is a closed nonsingular subvariety, it is known that the conormal sequence
$$
0\to\mathscr{I}/\mathscr{I}^2\to \mathscr{O}_Z \otimes_{\mathscr{...

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### Kähler differentials on an Artinian local ring

Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the ...

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### Existence theorem for symmetric nondegenerate forms over a ring

There exists a rich theory for inner product spaces (i.e. vector spaces with a symmetric nondegenerate bilinear form) over fields, and it can be discussed in the context of local rings and free ...

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### non constant regular function derivative is zero

Let $R$ be a Noetherian regular $k$-algebra (where $k$ any field of char = 0) of dimension greater than 0. Is it true that $H^{0}_{dR}(R \lvert k) = k$?.
More generally we could ask, Is it true that $...

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### Commutative square of module of differential is cartesian?

Is it true that the following square is Cartesian? $\require{AMScd}$
\begin{CD}
R @>{d}>> \Omega^{1}_{R} \\
@VVV @VVV\\
\widehat{R} @>{ \widehat{d}}>> \Omega^{1}_{\widehat{R}}
\end{...

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### Categorical Kähler differentials and the Leibniz rule

From nlab, the module of Kähler differentials over some category $\mathcal{C}$ is the free functor:
$$\Omega: \mathcal{C} \to \mathsf{Mod_{\mathcal{C}}}$$
left-adjoint to the (forgetful) embedding:
$$...

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1
answer

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### When is the module of Kahler differentials free?

As the title says, when is the module of Kahler differentials a free module? In particular, are there known conditions or criterions that could be met that ensures that it will be free?
For example, ...

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### Is the module of Kähler differentials a coend?

Let $\phi\colon R\to S$ be a ring map. The module of Kähler differentials $\Omega_{S/R}$ of $\phi$ can be constructed as the following coequaliser:
$$\left(\bigoplus_{(a, b)\in S^2} S[(a, b)]\right) \...

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### Kähler differential of completion of algebra

Let $(R, \mathfrak{m}) $ be a local $k$- algebra and $\Omega^{1}_{R}$ denote the module of Kahler differential. Does the canonical map $ \Omega^{1}_{R} \otimes_{R} \hat{R} \rightarrow \Omega^{1}_{...

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1
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### Residue of the canonical sheaf along subvariety

Let $S$ be a smooth projective surface over an
algebraically closed field $k$ and $C \subset S$ a singular curve. Let us denote by $K_S$ the class of canonical divisor of $S$ and $\mathcal{O}(K_S)$ ...

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### Module of Kahler differentials for manifolds [duplicate]

Let $A$ be a $k$-algebra and let $\mathcal{M}_A$ be the set of all $A$-modules. In $\mathcal{M}_A$, there exists a universal object $\Omega_{A/k}$, called the module of Kahler differentials, and a $k$...

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### Kaehler differentials of tensor products over another ring

Is this result well-known (or even true)?
Let
\begin{align*}
f:A & \to B\\
g:A & \to C
\end{align*}
be homomorphisms of finitely generate $k$-algebras and let
$R=B\otimes_{k}C$ and
$S=B\...

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answers

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### Are Kähler differentials the same as one-forms for compact manifolds?

Let $M$ be a manifold and let $A = \mathcal{C}^\infty(M)$ be the ring of smooth real-valued functions.
An old posting asks about the relationship of Kähler differentials and ordinary differential ...

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### Quotient of a smooth curve by a finite group and differentials

Let $X$ be a proper smooth connected curve over an algebraically closed field $k$ of characteristic $0$, and suppose that $X$ is equipped with a $k$-linear action of a finite group $G$. It makes sense ...

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### Why should the Kaehler form be closed? [closed]

As the question says, why should the Kaehler form be closed? Like people start from a fundamental 2-form (say, a 2-from $\mathcal{K}$) and then set they set the condition that in order for the ...

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votes

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### Differentials for algebraic stacks

Let $S$ be a base scheme. For which algebraic stacks $X$ over $S$ can we define a sheaf of differentials $\Omega^1_{X/S}$ (classifying derivations)? Probably it works when $X$ is Deligne Mumford ...

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### A question on Kähler differentials and cotangent spaces on schemes

I have the following question (should be easy for those who know something about the field):
On page 92 (97 of the old edition) of Mumford's book "Abelian varieties", the author talks about an ...

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0
answers

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### Surjectivity of global sections of sheaf of Kaehler differentials

By a curve I mean a scheme of pure dimension $1$.
Let $C_1, C_2$ be a local complete intersection curve in $\mathbb{P}^3$ such that $C_1 \cap C_2$ are finitely many points. Assume further that $C:=...

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1
answer

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### How can one interpret homology and Stokes' Theorem via derived categories?

I am very far removed from being an expert on derived categories. Every few months, however, I read a different introductory text with the hope that eventually I will have some basic grasp on this ...

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votes

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### How to prove this algebra is flat?

Hi,
Let $S = R[T_1,\dots,T_n]/(f_1,\dots,f_r)$ where $\det(\partial f_i/\partial T_j)_{i,j=1,\dots,r}\in S^\times$. Then $S$ is flat over $R$. How to prove it?
I am not looking for an answer like: "$...

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votes

3
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### algebraic de Rham cohomology of singular varieties

Hi,
Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where
the $H^*_{dR}(X/\mathbb C) = H^*(\Omega^\bullet_{A/\mathbb C})$ differs from the singular cohomology $H^*_{...

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0
answers

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### de Rham complex of closed immersion between smooth schemes

Hi,
Let $R$ be a $\mathbb Q$-algebra and let $P$ and $Q$ be (EDIT: smooth) $R$-algebras such that there is a surjective
map of $R$-algebras $Q\to P$. The following proof cannot possibly be correct, ...

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0
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### smooth algebras and triviality of de Rham complex

Hi,
Let $R$ be a $\mathbb Q$-algebra and let $A$ be a smooth $R$-algebra. If $A$ is a polynomial algebra
$A = R[T_1,\dots,T_n]$, then it is easy to see that the natural map
$R \to \Omega^\bullet_{A/R}...

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answers

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### Generalized Euler sequence on a projective scheme

Let $\mathcal{E}$ be a quasi-coherent sheaf on a scheme $S$. Consider the projective scheme $p : \mathbb{P}(\mathcal{E}) \to S$ and the canonical epimorphism $p^*(\mathcal{E}) \to \mathcal{O}_{\mathbb{...

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1
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### recurrence formula for *i*-th Chern class of $CP^n$

one can show that the relation between first Chern class and second Chern class of $CP^n$ is
$\frac{2(n+1)}{n} c_2 (M)=c_1 (M)^2$
here $c_1 (M)^2=c_1 (M)∧c_1 (M)$.
So is there any recurrence ...

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### Do smooth ind schemes have Dualizing sheafs?

Say I have an ind scheme $X = \cup_i X_i$ over a field $k$. I have its tangent bundle $\hom_k(k[\epsilon], X)$ which I can think of as ind scheme via $\cup_i \hom_k(k[\epsilon],X_i)$. The problem is ...

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### If $p=0$ and $df=0$, is $f$ a $p$th power?

This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...

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### Maximal Ideals and Kahler Differentials

For an algebraic variety $V$, denote its ring of regular functions by ${\cal O}(V)$. The Kahler differentials of $V$ are the quotient of the kernel $M$ of the multiplication map
$$
m: {\cal O}(V) \...

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votes

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answers

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### Flatness of sheaf of relative Kahler differentials

Suppose we have a projective flat non-smooth morphism of Noetherian schemes $g: X \rightarrow S$. My question regards when the sheaf of relative Kahler differentials $\Omega_{X/S}$ is flat over $S$. ...

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### wedge product of second chern class and kahler form on Calabi-Yau 3-folds.

Let $X$ be a smooth Calabi-Yau 3-fold with Kahler form $w$,
It is true that $\int c_2(TX) \wedge w \geq 0$ (for any Kahler form $w$ on $X$).
Proof via algebraic geometry is rather difficult. Some ...

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votes

1
answer

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### complex gradient of a function

Let $M$ be a complex $n-$dim manifold and $u : M \rightarrow \mathbb{R}$ be some smooth function. On $M$ assume that we have a Kaehler metric $h$. How is the complex gradient vectorfield defined with ...

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### Are Kahler differentials the same on the affine closure on a quasi-affine scheme?

Let $X$ be a quasi-affine scheme; that is, the natural map
$$X\rightarrow \overline{X}:=Spec(\Gamma(X,\mathcal{O}_X))$$
is an inclusion. Each scheme has a quasi-coherent sheaf of Kahler ...

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votes

2
answers

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### Diagonal map and "infinitesimal points"

Let $f:X\to Y$ be a morphism between schemes. To construct the relative sheaf of differentials on $X$ (relative to $Y$), we first consider the diagonal map $\Delta: X \to X\times_Y X$ and then define $...

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### Kahler differentials and the m-adic filtration

Let $A$ be a comm. local $k$-algebra with max. ideal $m$. Let $gr(-)$ be the associated graded algebra or module for the $m$-adic filtration. The canonical derivation $d:A \to \Omega^1_A$ satisfies $...