Math 480, Spring 2014, Assignment 9

Carefully define the following terms, then give one example and one non-example of each:

1. Projective variety (defined by a set of homogeneous polynomials).
2. Dehomogenization (of an element of $\mathsf{k}[x_0,\dots,x_n]$ at $x_i$).
3. Homogenization (of an element of $\mathsf{k}[x_0,\dots,\widehat{x_i},\dots,x_n]$ at $x_i$).
4. Homogeneous ideal.

Carefully state the following theorems:

1. Theorem relating a polynomial to the dehomogenization of its homogenization.
2. Theorem relating a polynomial to the homogenization of its dehomogenization.
3. Theorem characterizing homogeneous ideals (i.e. giving three other conditions equivalent to homogeneity).

Solve the following problems:

1. Determine rigorously whether the ideal $\left\langle x-y^2, x+y^2\right\rangle$ is homogeneous.
2. Prove that the sum of two homogeneous ideals is homogeneous.
3. Prove that the intersection of two homogeneous ideals is homogeneous.
4. Prove that the product of two homogeneous ideals is homogeneous.

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