Commutative Algebra

Excercises 7

A numerical polynomial is a polynomial f(x) ∈ ℚ[x] such that f(n) ∈ ℤ for every integer n. A graded module M over a ring A is an A-module M endowed with a direct sum decomposition ⨁_{n∈ℤ} M_{n} into A-submodules. We will say that M is locally finite if all of the M_{n} are finite A-modules. Suppose that A is a Noetherian ring and that φ is a Euler-Poincaré function on finite A-modules. This means that for every finitely generated A-module M we are given an integer φ(M) ∈ ℤ and for every short exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

we have φ(M) = φ(M′) + φ(M′). The Hilbert function of a locally finite graded module M (with respect to φ) is the function χ_{φ}(M, n) = φ(M_{n}). We say that M has a Hilbert polynomial if there is some numerical polynomial P_{φ} such that χ_{φ}(M, n) = P_{φ}(n) for all sufficiently large integers n.

A graded A-algebra is a graded A-module B = ⨁ B_{n} together with an A-bilinear map

B × B ⟶ B, (b, b′) ⟼ bb′

that turns B into an A-algebra so that B_{n} · B_{m} ⊂ B_{n+m}. Finally, a graded module M over a graded A-algebra B is given by a graded A-module M together with a (compatible) B-module structure such that B_{n} · M_{d} ⊂ M_{n+d}. Now you can define homomorphisms of graded modules/rings, graded submodules, graded ideals, exact sequences of graded modules, etc, etc.

1. Let A = k a field. What are all possible Euler-Poincaré functions on finite A-modules in this case?

2. Let A = ℤ. What are all possible Euler-Poincaré functions on finite A-modules in this case?

3. Let A = k[x, y]/(xy) with k algebraically closed. What are all possible Euler-Poincaré functions on finite A-modules in this case?

4. Suppose that A is Noetherian. Show that the kernel of a map of locally finite graded A-modules is locally finite.

5. Let k be a field and let A = k and B = k[x, y] with grading determined by deg(x) = 2 and deg(y) = 3. Let φ(M) = dim_{k}(M). Compute the Hilbert function of B as a graded k-module. Is there a Hilbert polynomial in this case?

6. Let k be a field and let A = k and B = k[x, y]/(x^{2}, xy) with grading determined by deg(x) = 2 and deg(y) = 3. Let φ(M) = dim_{k}(M). Compute the Hilbert function of B as a graded k-module. Is there a Hilbert polynomial in this case?

7. Let k be a field and let A = k. Let φ(M) = dim_{k}(M). Fix d ∈ ℕ. Consider the graded A-algebra B = k[x, y, z]/(x^{d} + y^{d} + z^{d}), where x, y, z each have degree 1. Compute the Hilbert function of B. Is there a Hilbert polynomial in this case?

To be continued.