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111.64 Commutative Algebra, Final Exam, Fall 2021

These were the questions in the final exam of a course on commutative algebra, in the Fall of 2021 at Columbia University.

Exercise 111.64.1 (Definitions). Provide brief definitions of the italicized concepts.

  1. a multiplicative subset of a ring $A$,

  2. an Artinian ring $A$,

  3. the spectrum of a ring $A$ as a topological space,

  4. a flat ring map $A \to B$,

  5. the height of a prime ideal $\mathfrak p$ in $A$, and

  6. the functors $\text{Tor}^ A_ i(-, -)$ over a ring $A$.

Exercise 111.64.2 (Theorems). Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

  1. Artinian rings,

  2. flatness and prime ideals,

  3. lengths of $A/\mathfrak m^ n$ for $(A, \mathfrak m)$ Noetherian local,

  4. the dimension formula for universally catenary Noetherian rings,

  5. completion of a Noetherian local ring, and

  6. Matlis duality for Artinian local rings.

Exercise 111.64.3 (Units). What is the structure of the group of units of $\mathbf{Z}[x, 1/x]$ as an abelian group? No explanation necessary.

Exercise 111.64.4 (Ideals). Let $A = \mathbf{F}_2[x, y]/(x^2, xy, y^2)$ and denote $\overline{x}$ and $\overline{y}$ the images of $x$ and $y$ in $A$. List the ideals of $A$. No explanation necessary.

Exercise 111.64.5 (Tor and Ext). Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Set $\varphi (n) = \dim _\kappa \mathfrak m^ n/\mathfrak m^{n + 1}$.

  1. Show that $\text{Tor}_1^ A(A/\mathfrak m^ n, \kappa )$ has dimension $\varphi (n)$ as a $\kappa $-vector space.

  2. Show that $\text{Ext}^1_ A(A/\mathfrak m^ n, \kappa )$ has dimension $\varphi (n)$ as a $\kappa $-vector space.

Exercise 111.64.6 (Two vectors). Let $A = \mathbf{Z}[a_1, a_2, a_3, b_1, b_2, b3]$. Set $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ in $A^{\oplus 3}$. Consider the set

\[ Z = \{ \mathfrak p \in \mathop{\mathrm{Spec}}(A) \mid a, b \text{ map to linearly dependent vectors of } \kappa (\mathfrak p)^{\oplus 3}\} \]

  1. Prove the $Z$ is a closed subset of $\mathop{\mathrm{Spec}}(A)$.

  2. What is the dimension $\dim (Z)$ of $Z$?

  3. What would happen to $\dim (Z)$ if we replaced $\mathbf{Z}$ by a field?

Exercise 111.64.7 (Injectives). Let $(A, \mathfrak m, \kappa )$ be an Artinian local ring. Assume $A$ is injective as an $A$-module. Show that $\mathop{\mathrm{Hom}}\nolimits _ A(\kappa , A)$ has dimension $1$ has a $\kappa $-vector space.

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