Questions tagged [kahler-differentials]
7 questions
3
votes
1 answer
Maps between Differential/Kähler Forms in Algebraic Geometry
Let $f: X \to Y$ be a morphism of connected schemes over base field $k$ of characteristic zero.
By universal property of Kähler differentials & functoriality there are maps $f^*_j:\Omega_Y^j \to f_*\Omega_X^j$ for each $j \in \Bbb N$ inducing maps…
user267839
- 9,217
2
votes
1 answer
Kähler potential for $SU(n)$ coadjoint orbit
The coadjoint orbit of the Lie group $SU(n)$ is given by $\mathcal{O}_{SU(n)} = SU(n)/\mathbb{T}_{n-1} \cong \mathbb{C}\mathbb{P}^{n-1} \ltimes \mathcal{O}_{SU(n-1)}$.
It is well known that this orbit $\mathcal{O}$ is a homogeneous Kähler manifold,…
FieldTheorist
- 253
1
vote
1 answer
If $B'=B\otimes_AA'$, then the Kähler differentials satisfy $\Omega_{B/A}\otimes_A A'\cong \Omega_{B'/A'}$
Let $B$ and $A'$ be commutative $A$ algebras, and set $B'=B\otimes_AB'$. I am trying to show that as $B$ modules, the modules of Kähler differentials satisfy:
$$\Omega_{B/A}\otimes_AA'\cong \Omega_{B'/A'}$$
Now, via the base change property of…
Chris
- 5,420
1
vote
1 answer
Proving universal property in Kähler differentials
I have recently been introduced to Kähler differentials. Our lecturer gave a sketch of the proof of how to construct such object. He said that if $A$ is a $k$-algebra, then we can construct the following free module
$$\Omega_A^1:=\bigoplus_{a\in…
kubo
- 2,198
1
vote
1 answer
What means “top exterior power of the sheaf of differentials $\Omega^1_{X/Y}$” when the rank of $\Omega^1_{X/Y}$ is globally unbounded?
In Conrad's Grothendieck Duality and Base Change, p. 6, it is said:
When $f:X\to Y$ is a smooth map of schemes, then $\omega_{X/Y}$ denotes the top exterior power of the locally free finite rank sheaf $\Omega_{X/Y}^1$ on $X$.
My question is: for…
0
votes
0 answers
Sections of the Cotangent Sheaf
I'm struggling to prove that the sections for an affine open set $U$ of the cotangent sheaf $\Omega_X^1$ are isomorphic to the Kähler-differential $\Omega_{\mathcal{O}(X)\vert\mathbb{C}}$.
First I want to describe my case and my assumptions. I…
nico_r
- 1
0
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0 answers
Book on Commutative algebra.
I just learned a little bit about commutative algebra from A term of Commutative Algebra by Allen B. ALTMAN and Steven L. KLEIMAN
it's great and I like the writing style. However the book doesn't have a section on kahler differential. I also tried…
