In the polynomial ring k[x, y, z], which of the following is a regular sequence?
A(x, y - x, z - x)
B(xy, z)
C(0, x, y)
D(x + y, x + y, z)
The sequence (x, y - x, z - x) is regular: x is not a zero divisor on k[x,y,z]; the image of y - x is not a zero divisor on k[x,y,z]/(x) ≅ k[y,z]; and the image of z - x is not a zero divisor on k[y,z]/(y) ≅ k[z]. Option B fails because xy is a zero divisor on k[x,y,z]/(xy) in the next step. Option C fails at the start since 0 is a zero divisor. Option D fails because the second x + y is zero in the quotient.
Question 2 True / False
In a Noetherian local ring (R, m), the depth of R is always at most the Krull dimension of R.
TTrue
FFalse
Answer: True
The depth of R (length of a maximal regular sequence in m) satisfies depth(R) ≤ dim(R). This is because each element of a regular sequence reduces dimension by exactly 1 (by the Hauptidealsatz), so you cannot have more regular elements than the dimension. When equality holds, R is Cohen-Macaulay.
Question 3 Short Answer
What is the depth of the local ring k[x, y]_(x,y) / (x^2, xy)?
Think about your answer, then reveal below.
Model answer: 0, because every element of the maximal ideal is a zero divisor on this ring.
In R = k[x,y]/(x^2, xy), the element x is nonzero but x times any element of (x, y) is zero: x*x = x^2 = 0 and x*y = xy = 0. So x is annihilated by the entire maximal ideal, making (x, y) ⊆ Ass(R). This means every element of the maximal ideal is a zero divisor, so no regular sequence of length ≥ 1 exists. The depth is 0.
Question 4 True / False
A regular local ring is Cohen-Macaulay.
TTrue
FFalse
Answer: True
A regular local ring of dimension d has maximal ideal generated by exactly d elements, and these generators form a regular sequence (a 'regular system of parameters'). So depth = d = dim, which is the Cohen-Macaulay condition. Regular local rings are the 'smoothest' local rings; Cohen-Macaulay is a weaker condition that still ensures good behavior.
Question 5 Short Answer
Explain why Cohen-Macaulay rings are important for intersection theory.
Think about your answer, then reveal below.
Model answer: In a Cohen-Macaulay ring, the height of an ideal generated by r elements equals r whenever the ideal has height r (no embedded components at the 'wrong' codimension). This means intersections are well-behaved: codimensions add, and Bezout-type formulas hold without correction terms.
The technical statement is that Cohen-Macaulay rings are 'unmixed': every associated prime of an ideal generated by a regular sequence is minimal, with no embedded primes. This makes intersection multiplicities well-defined and ensures that Serre's intersection formula gives the expected answer. Non-Cohen-Macaulay rings can have embedded components that distort intersection counts.