Algebraic Geometry

Question 1

What is an algebraic set?

Answer 1

1.1

Question 2

What is the Zariski Topology? What are some of its properties (closed sets)?

Answer 2

1.1

Question 3

Is the Zariski topology Hausdorff? Why? Why not?

Answer 3

1.1

Question 4

Given an algebraic set, what is its vanishing ideal?

Answer 4

1.1

Question 5

State and prove Hilbert’s Nullstellensatz (weak).

Answer 5

1.2

Question 6

State and prove Hilbert’s Basis Theorem.

Answer 6

1.2

Question 7

I(Z(I)) = ? Why?

Answer 7

1.2

Question 8

What is the relationship between ideals of k[x] and algebraic sets?

Answer 8

1.2

Question 9

When is a topological space X reducible?

Answer 9

1.3

Question 10

Prove X is irreducible iff. I(X) is a prime ideal.

Answer 10

1.3

Question 11

Prove X is irreducible iff. I(X) is a prime ideal.

Answer 11

1.3

Question 12

What is a Noetherian topological space? Why are algebraic sets Noetherian?

Answer 12

1.3

Question 13

Define Coordinate Ring A(X).

Answer 13

2.1

Question 14

Define K(X) “field of rational functions on X”.

Answer 14

2.1

Question 15

Define prime ideal and integral domain.

Answer 15

Look up

Question 16

Define local ring.

Answer 16

2.1

Question 17

Define ring of regular functions.

Answer 17

2.1

Question 18

What are distinguished open sets? How are they related to coordinate rings?

Answer 18

2.1

Question 19

Define presheaf.

Answer 19

2.2

Question 20

Define section of presheaf.

Answer 20

2.2

Question 21

Define sheaf.

Answer 21

2.2