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109.58 Commutative Algebra, Final Exam, Fall 2016

These were the questions in the final exam of a course on Commutative Algebra, in the Fall of 2016 at Columbia University.

Exercise 109.58.1 (Definitions). Let $R$ be a ring. Provide definitions of the italicized concepts.

  1. the local ring of $R$ at a prime $\mathfrak p$,

  2. a finite $R$-module,

  3. a finitely presented $R$-module,

  4. $R$ is regular,

  5. $R$ is catenary,

  6. $R$ is Cohen-Macaulay.

Exercise 109.58.2 (Theorems). Precisely state a nontrivial fact discussed in the lectures related to each item.

  1. regular rings,

  2. associated primes of Cohen-Macaulay modules,

  3. dimension of a finite type domain over a field, and

  4. Chevalley's theorem.

Exercise 109.58.3. Let $A \to B$ be a ring map such that

  1. $A$ is local with maximal ideal $\mathfrak m$,

  2. $A \to B$ is a finite1 ring map,

  3. $A \to B$ is injective (we think of $A$ as a subring of $B$).

Show that there is a prime ideal $\mathfrak q \subset B$ with $\mathfrak m = A \cap \mathfrak q$.

Exercise 109.58.4. Let $k$ be a field. Let $R = k[x, y, z, w]$. Consider the ideal $I = (xy, xz, xw)$. What are the irreducible components of $V(I) \subset \mathop{\mathrm{Spec}}(R)$ and what are their dimensions?

Exercise 109.58.5. Let $k$ be a field. Let $A = k[x, x^{-1}]$ and $B = k[y]$. Show that any $k$-algebra map $\varphi : A \to B$ maps $x$ to a constant.

Exercise 109.58.6. Consider the ring $R = \mathbf{Z}[x, y]/(xy - 7)$. Prove that $R$ is regular.

Given a Noetherian local ring $(R, \mathfrak m, \kappa )$ for $n \geq 0$ we let $\varphi _ R(n) = \dim _\kappa (\mathfrak m^ n/\mathfrak m^{n + 1})$.

Exercise 109.58.7. Does there exist a Noetherian local ring $R$ with $\varphi _ R(n) = n + 1$ for all $n \geq 0$?

Exercise 109.58.8. Let $R$ be a Noetherian local ring. Suppose that $\varphi _ R(0) = 1$, $\varphi _ R(1) = 3$, $\varphi _ R(2) = 5$. Show that $\varphi _ R(3) \leq 7$.

[1] Recall that this means $B$ is finite as an $A$-module.

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