## 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.

a *multiplicative subset* of a ring $A$,

an *Artinian ring* $A$,

the *spectrum of a ring* $A$ as a topological space,

a *flat ring map* $A \to B$,

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

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).

Artinian rings,

flatness and prime ideals,

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

the dimension formula for universally catenary Noetherian rings,

completion of a Noetherian local ring, and

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 $\varphi (n) = \dim _\kappa \mathfrak m^ n/\mathfrak m^{n + 1}$.

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

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

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

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

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

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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