The main point of this lecture will be to understand projective space (and projective schemes) from the point of view of homogeneous ideals. Review the theory of graded rings and homogeneous ideals from Algebra 2, if it is not fresh in your mind. In particular, review the definition of a Z-graded ring, homogeneous ideal.

If S_* is a graded ring, then we will construct a scheme Proj(S_*) — first as a set, then as a topological space, and finally as a scheme. Read 4.5.7 very carefully. Pages 149 and 150 are also very important.

- Check that the Proj construction applied to the graded ring k[x0,…,xn] and with coordinate charts indexed by f = xi, gives the construction of P^n discussed earlier.
- Let g be a homogeneous polynomial in k[x0,…,xn] (k algebraically closed). Apply the Proj construction to k[x0,…,xn]/(g), with coordinate charts indexed by f = xi. Show that the set of closed points of the resulting scheme can be identified with the set of nonzero solutions [a0:…:an] of g(a0,…,an) = 0.
- Exercise 4.5.E.
- How are the definitions of V(T) and D(f) modified in the projective setting?
- Check that the sets V(I) satisfy the axioms of a closed set of a topological space.
- 4.5.G. Show that the sets D(f) form a base of the Zariski topology.
- Mimic proofs in the affine case, to do Exercises 4.5.H and 4.5.I.
- We will continue to do exercises on page 150 until time runs out.