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@@ -50,11 +50,13 @@ |
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(endowed with the Zariski topology) and, for all $U \subseteq V$ open, |
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(endowed with the Zariski topology) and, for all $U \subseteq V$ open, |
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\[ |
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\[ |
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\mathcal{O}_V(U) \coloneqq |
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\mathcal{O}_V(U) \coloneqq |
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\{ f \colon U \to k \mid \forall x \in U \exists U_x \subseteq |
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\text{ open neighbourhood of $x$ and polynomials} |
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P, Q \text{ sucht that } \forall z \in U \cap U_x, |
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\left\{ f \colon U \to k\ \middle \vert |
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\begin{array}{l} |
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\forall x \in U \exists x \in U_x \text{ open}, |
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P, Q \in k[x_1, \ldots, x_n] \text{ such that }\\ \text{for } z \in U \cap U_x, |
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Q(z) \neq 0 \text{ and } f(z) = \frac{P(z)}{Q(z)} |
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Q(z) \neq 0 \text{ and } f(z) = \frac{P(z)}{Q(z)} |
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\} |
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\end{array} |
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\right\} |
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.\] |
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.\] |
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\item $(M, \mathcal{C}^{\infty}_M)$ where |
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\item $(M, \mathcal{C}^{\infty}_M)$ where |
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$M = \varphi^{-1}(0)$ is a non-singular level set of a $\mathcal{C}^{\infty}$ |
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$M = \varphi^{-1}(0)$ is a non-singular level set of a $\mathcal{C}^{\infty}$ |
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