Math and stuff

Proposition All finite subsets of $k^n$ are affine varieties. Prove that a single point $(a_1, \cdots, a_n) \in k^n$ is an affine variety. Prove that every finite subset of $k^n$ is an affine variety. Solution 1 Let $(a_1, \cdots, a_n) \in k^n$ be given. Let $S = V(x_1 - a_1,...

Proposition A variety $X$ is irreducible if $I(X)$ is prime. Solution Let a variety $X$ be given. Suppose that $I(X)$ is a prime ideal. Let $X_1, X_2$ be two varieties such that $X = X_1 \cup X_2$. If $X_1 = X$ or $X_2 = X$, we are done. Suppose otherwise....

When I first learned $\mathbb{R}\mathbb{P}^2$, I could understand the definition, but I could not at all understand why that would ever be useful. Here is a list of videos that made me appreciate $\mathbb{R}\mathbb{P}^2$. Projective geometry and homogeneous coordinates This video explains how the real projective plane includes a plane...

Proposition Let $A$ be an integral domain with field of fractions $K$, and suppose that $f \in A$ is nonzero and not a unit. Prove that $A[1/f]$ is not a finite $A$-module. Solution Suppose $A[1/f]$ is a finite $A$-module. Let ${ \frac{a_1}{f^{b_1}}, \cdots, \frac{a_n}{f^{b_n}} }$ be a generating set where...

Proposition Give an example of a ring $A$ and ideals $I, J$ such that $I \cup J$ is not an ideal; in your example, what is the smallest ideal containing $I$ and $J$? Solution Let $A = \mathbb{C}[x], I = (x + 1), J = (x)$. $I \cup J$ contains...