Recent content by MathLearner123

I want to prove that ##\mathcal{O}_{\mathbb{A}^n}(\mathbb{A}^n \setminus \{0\}) = k[X_1, \ldots, X_n]## for ##n \ge 2## and some user on another forum gave this proof: "We know that ##\mathcal{O}_{\mathbb{A}^n}(\mathbb{A}^n \setminus \{0\}) = \bigcap_{i = 1}^n \mathcal{O}(D(X_i))##, where...

I think that we rotate the coordinate system.

I'm reading "From Holomorphic Functions to Complex Manifolds" - Fritzsche & Grauert and I have something that I don't understand very well: If ##\nu \in \mathbb{N}_0^n, t \in \mathbb{R}^n_+## and ##z \in \mathbb{C}^n##, write ##\nu = (\nu_1, \nu'), t = (t_1, t')## and ##z = (z_1, z')##. \ An...

Thanks for your help and sorry if I bothered you with my questions!

Suppose that I have a representation of a group ##G##, and ##g \in G## with ##f(g) = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}##. Let ##V = \mathcal{M}_{1,2}(\mathbb{R})## (line matrices) and take the bases ##B_1 = \{(1,0), (0,1)\}## and ##B_2 = \{(3,4), (2,5)\}##. Also, take ##v = (7,11) \in...

Sorry if I answer too late after you replied. Ok. What I've referred to by saying "different structure" was the fact that if I choose fixed vector ##v \in V## and a basis ##\{b_1, \ldots, b_n\}## then I can get ##v \cdot g = v_1 \in V##. If I choose another basis, let it be ##\{B_1, \ldots...

Thanks for all the answers. I understand that I need to put ##GL(V)##. But I think that it's sufficient to add to the statement of the proposition that ##\{v_1, \ldots, v_n\}## is a fixed basis for ##V## and then I can take ##v \cdot g## to be the unique vector from ##v## which have the...

Definition. Let ##F## be a field and ##G## be a group. An ##FG##-module is a finite-dimensional vector space ##V## on which ##G## acts (from the right: ##V \times G \to V, (v,g)\mapsto v\cdot g##) such that the next conditions hold: 1) ##(v \cdot g)\cdot h = v \cdot (g \cdot h)## 2) ##v \cdot e...

I want to prove following (Big Picard Theorem forms):\ Theorem. The followings are equivalent:\ a) If ##f \in H(\mathbb{D}\setminus\{0\})## and ##f(\mathbb{D}') \subset \mathbb{C} \setminus \{0, 1\}##, then ##f## has a pole of an removable singularity at ##0##.\ b) Let ##\Omega \subset...

@FactChecker ##\mathbb{R}^* = \mathbb{R} \setminus \{0\}## and ##\mathbb{H}## is the upper half-plane

Let ##\alpha \in \mathbb{R}^*,\, p:\mathbb{H} \to \mathbb{D} \setminus \{0\}, \, p(z) = e^\frac{2 \pi i z}{|a|}##. I want to show that ##p## is a covering map but I dont't know how to make this. I think I need to start with an ##y \in \mathbb{D} \setminus \{0\}## and take an open disk ##D...

I saw that thesis. Has MANY TYPOS, wrong transcriptions. It is copy-paste from Complex Made Simple by David Ullrich (the same book from where the theorems from my post are) but the guy who wrote it didn't understood anything.

I need help please! So I'm reading 'Complex made simple' by David C. Ullrich. I made all the requirements for this proof but the author don't give the proof of this final theorem, instead it gives a similar proof for another set of theorems. Let ##\mathbb{D}' = \mathbb{D} \setminus \{0\}##...

Thanks for all help! I think I got it!

Theorem 1 Suppose that ##p : X \to Y## is a covering map. Suppose ##\gamma_0, \gamma_1 : [0, 1] \to Y## are continuous, ##x_0 \in X## and ##p(x_0) = \gamma_0(0) = \gamma_1(0)##. Fie ##\tilde{\gamma}_0## and ##\tilde{\gamma}_1## be the continuous functions mapping [0,1] to X such that...