# Questions tagged [kahler-differentials]

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

30
questions

**4**

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**1**answer

108 views

### 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:
$$...

**2**

votes

**1**answer

147 views

### 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, ...

**5**

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**0**answers

105 views

### 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) \...

**1**

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93 views

### 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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59 views

### Does Kahler differential commute with completion

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}_{\hat{R}} \rightarrow \varprojlim \Omega^{1}_{...

**3**

votes

**1**answer

197 views

### 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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78 views

### 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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87 views

### 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\...

**9**

votes

**0**answers

373 views

### 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 ...

**10**

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**3**answers

1k views

### 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 ...

**4**

votes

**1**answer

238 views

### 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 ...

**6**

votes

**2**answers

724 views

### 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 ...

**3**

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170 views

### 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

139 views

### 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:=...

**15**

votes

**1**answer

1k views

### 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 ...

**3**

votes

**1**answer

244 views

### 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: "$...

**14**

votes

**3**answers

2k views

### 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^*_{...

**1**

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**0**answers

121 views

### 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, ...

**1**

vote

**0**answers

74 views

### 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}...

**3**

votes

**0**answers

804 views

### 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{...

**0**

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**1**answer

367 views

### 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 ...

**11**

votes

**2**answers

843 views

### 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 ...

**5**

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419 views

### 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 ...

**2**

votes

**1**answer

349 views

### 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) \...

**3**

votes

**2**answers

820 views

### 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 $S$. ...

**4**

votes

**3**answers

742 views

### 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 ...

**3**

votes

**1**answer

2k views

### 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 ...

**6**

votes

**1**answer

437 views

### 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 ...

**7**

votes

**2**answers

1k views

### 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 $...

**4**

votes

**0**answers

180 views

### 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 $...