Commutative Algebra

Excercises 6

1. Suppose that k is a field having a primitive nth root of unity ζ. This means that ζ^{n} = 1, but ζ^{m} ≠ 1 for 0 < m < n.

(a) Show that the characteristic of k is prime to n.

(b) Suppose that a ∈ k is an element of k which is not an dth power in k for any divisor d of n, in ≥ d > 1. Show that k[x]/(x^{n} − a) is a field. (Hint: Consider a splitting field for x^{n} − a and use Galois theory.)

2. Let ν : k[x] \ {0} → ℤ be a map with the following properties: ν(fg) = ν(f) + ν(g) whenever f, g not zero, and ν(f + g) ≥ min(ν(f), ν(g)) whenever f, g, f + g are not zero, and ν(c) = 0 for all c ∈ k^{∗}.

(a) Show that if f, g, and f + g are nonzero and ν(f) ≠ ν(g) then we have equality ν(f + g) = min(ν(f), ν(g)).

(b) Show that if f = ∑ a_{i}x^{i}, f ≠ 0, then ν(f) ≥ min({iν(x)}_{a_{i}≠0}). When does equality hold?

(c) Show that if ν attains a negative value then ν(f) = −n deg(f) for some n ∈ ℕ.

(d) Suppose ν(x) ≥ 0. Show that {f | f = 0, or ν(f) > 0} is a prime ideal of k[x].

(e) Describe all possible ν.

3. Let (A, 𝔪, k) be a local ring and let k ⊂ k′ be a finite field extension. Show there exists a flat, local map of local rings A → B such that 𝔪_{B} = 𝔪B and B/𝔪B is isomorphic to k′ as k-algebra. (Hint: first do the case where k ⊂ k′ is generated by a single element.)

Remark. The same result holds for arbitrary field extensions k ⊂ K.

4. Let R be a ring and let M be a finitely presented R module. Recall this means that there is an exact sequence

R^{⊕r} ⟶ R^{⊕n} ⟶ M ⟶ 0.

This is called a presentation of M. Note that the map R^{⊕n} → M is given by a sequence of elements x_{1}, …, x_{n} of M. The elements x_{i} are generators of M. The map R^{⊕r} → R^{⊕n} is given by a n × r matrix A with coefficients in R. The columns of A are said to be the relations. Any vector (r_{i}) ∈ R^{⊕r} such that ∑ r_{i}x_{i} = 0 is a linear combination of the columns of A. Of course any module has a lot of different presentations. We define P res(M) to be the collection of matrices you can obtain in this way (meaning all matrices A of any size n × r that occur in some presentation of M).

(a) Show that if A ∈ P res(M) has size n×r then the matrix Ã of size n×(r +1) obtained from A by adding a column of zeros occurs in P res(M). (Hint: this corresponds to adding a trivial relation.)

(b) Show that if A ∈ P res(M), then any Ã ∈ P res(M), where Ã is obtained from A by replacing ith column vector A_{i} by A_{i} + ∑_{j≠i} r_{j}A_{j} for any r_{j} ∈ R. (Hint: This corresponds to replacing a relation by itself plus a linear combination of other relations.)

(c) Show that if A ∈ P res(M) has size n × r then the matrix Ã of size (n + 1) × (r + 1) obtained from A by setting

\begin{aligned}ã_{ij} & = a_{ij}, i < n + 1, j < r + 1, \\ ã_{n+1j} & = 0, j < r + 1, \\ ã_{ir+1} & = 0, i < n + 1, \\ ã_{n+1r+1} & = 1 \end{aligned}

occurs in P res(M). (Hint: This corresponds to adding x_{n+1} = 0 and the trivial relation x_{n+1} = 0.)

(d) Show that if A ∈ P res(M), then Ã ∈ P res(M) where Ã is obtained from A by replacing the jth row by a sum consisting of itself and a linear combination of other rows (with coefficients in R). (Hint: This corresponds to replacing x_{j} by x_{j}+∑_{i≠j} r_{i}x_{i} and adjusting the relations accordingly.)

We say that matrices A and A′ with coefficients in R are obtained from each other by a sequence of elementary moves if there is a sequence of matrices A = A_{0}, A_{1}, A_{2}, …, A_{n} = A′ such that for each 0 ≤ ℓ < n the pair (A_{ℓ}, A_{ℓ+1}) is the pair (A, Ã) or (Ã, A) for one of the operations on matrices described in (a)-(d) above.

(e) Show that any two matrices in P res(M) are obtained from each other by a sequence of elementary moves. (Hint: First show this holds if A, A′ in P res(M) are matrices of relations among the same set of generators.)

(f) Let k be an integer. Suppose that A, A′ are obtained from each other by a sequence of elementary moves. Say A has size n × r and A′ has size n′ × r′. Show that the ideal generated by the (n − k) × (n − k) minors of A agrees with the ideal generated by the (n′ − k) × (n′ − k) minors of A′. [[Convention: If k ≥ n then we say the ideal generated by the (n − k) × (n − k)-minors is R. In other words, the determinant of a matrix of size 0 × 0, −1 × −1, etc is defined to be 1.]]

This defines the kth fitting ideal of M. Notation F it_{k}(M).

(g) Show that F it_{0}(M) ⊂ F it_{1}(M) ⊂ F it_{2}(M) ⊂ . . .. (Hint: Use that a determinant can be computed by expanding along a column.)

(h) Show that M = (0) if F it_{0}(M) = R.

(i) Show that M if F it_{0}(M) = (0) and F it_{1}(M) = R, then M is locally free of rank 1. (This is slightly tricky.)