Linked Questions
14 questions linked to/from Kahler differentials and Ordinary Differentials
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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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37
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3
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Do these properties characterize differentiation?
Let $L: C^\infty(\mathbb{R}) \to C^\infty(\mathbb{R})$ be a linear operator which satisfies:
$L(1) = 0$
$L(x) = 1$
$L(f \cdot g) = f \cdot L(g) + g \cdot L(f)$
Is $L$ necessarily the derivative? ...
- 12.1k
17
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4
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2k
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Can you tell the volume of a symplectic manifold from the Poisson brackets?
Suppose $(X^{2n},\omega)$ is a compact symplectic manifold. Knowing the algebra $C^\infty(X)$ is equivalent to knowing the manifold $X$, and knowing the Poisson bracket $\{\cdot,\cdot\}:C^\infty(X)\...
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24
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1
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Real manifolds and affine schemes
I noticed the following strange (to me) fact. If $M$ is a real manifold (smooth or not) and $R = C(X, \mathbb{R})$ is the ring of real functions (smooth functions in the smooth case) then the affine ...
- 3,625
18
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1
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682
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De Rham via topoi
Étale cohomology of schemes $X$ is constructed as follows: one associates to $X$ the so-called étale topos of $X$, and then one just takes the sheaf cohomology of that topos.
Is it possible to ...
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12
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1
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417
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Are algebras of smooth functions formally smooth?
Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$?
If it helps, feel free to assume that $M$ is compact.
(This is not a joke ...
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5
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1
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344
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Sufficient conditions for $\mathrm{Der}_k(A)$ to be f.g. projective
Let $k$ be a field and $A$ a commutative $k$-algebra. What are sufficient conditions for the module of derivations $\mathrm{Der}_k(A)$ to be finitely generated projective?
I'm looking for conditions ...
- 6,406
5
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1
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580
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The complex of Kahler differentials and de Rham complex
Let $A$ be a unital commutative algebra (say over complex numbers). Consider the multiplication map $m:A \otimes A \to A$ and put $\Omega^1_u(A)=\ker m$ to be the space of universal differential forms....
- 9,330
5
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1
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578
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Lie algebra of a compact Lie group and derivations of the Hopf algebra of representative functions
Let $\mathcal{G}$ be a compact (real) Lie group. We know that the Lie algebra $\mathfrak{g}$ of $\mathcal{G}$ is, by definition, the space of all left-invariant (smooth) vector fields over $\mathcal{G}...
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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 ...
user13113
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Sheaf of Kähler differentials for complex manifold
Let $X \subset \mathbb{C}^n$ be an analytic set which means that it is the zero locus of holomorphic functions $f_1,f_2,\dotsc,f_n$ on $\mathbb{C}^n$ and suppose that there is a singular locus ...
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2
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436
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Why is $\Omega_k(C^\infty(M))\to\Omega^1(M)$ surjective?
Let $M$ be a smooth manifold and let $A=C^\infty(M).$
We consider module of Kahler differentials $\Omega_k(A)$ and module of 1-forms $\Omega^1(M).$ Denote Kahler differential by $d_k$ and classical ...
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4
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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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4
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Are the canonical maps from $\Omega^1_k(C^\infty(M))$ into $\Omega^1(M)$ and into $\Omega^1_k(C^\infty(M))^{**}$ compatible?
Let $M$ be a smooth manifold and let $A=C^\infty(M)$. In this question, it is observed that the map $\Omega^1_k(A)\to \Omega^1(M)$ from the Kähler differentials of $A$ to the 1-forms of $M$ is not an ...
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