Commutative algebra

This course is about commutative algebra, loosely following the textbook of Atiyah-Macdonald. Homework will be due weekly on Thursdays, starting Thursday January 20. Office hours will take place Tuesdays after class, in my office, Boyd 321C.

The course meets from 1245pm-2pm in Poultry Sciences 0238.

HW 1 (due Jan 20): Read AM Chapter 1, do exercises 1, 8, 10, 13, 15, 16, and 19.

HW 2 (due Jan 27): Read AM Chapter 2, do exercises 1, 2, 9, 11, 14, 16, and 20

HW 3 (due Feb 3): Read Chapter 3, exercises 1, 4, 12. Show that $\mathbb{Q}$ is a flat $\mathbb{Z}$-module which is not free. Prove that if $B, C$ are $A$-algebras, the tensor product algebra $B\otimes_A C$ has the following universal property: an algebra homomorphism $B\otimes_A C\to S$ is the same as a pair of algebra homomorphisms $B\to S, C\to S$.

HW 4 (due Feb 10):

Let $f: M\to N$ be a map of modules over a local ring $A$, with $N$ finitely generated. Show that if the induced map $M/\mathfrak{m}M\to N/\mathfrak{m}N$ is surjective, the same is true for $f$. Does a similar statement hold for injectivity?
Let $M$ be a finitely-generated module over a ring $A$, and let $\mathfrak{p}$ be a prime ideal of $A$. Suppose $M_\mathfrak{p}=0$. Show that there exists a finite set $x_1, \cdots, x_n$ of elements of $A\setminus \mathfrak{p}$ such that the localization of $M$ at the multiplicative set generated by the $x_i$ is zero.
Let $M$ be a finitely-generated module over a Noetherian ring $A$ (that means any submodule of a finitely-generated module is finitely generated), and let $\mathfrak{p}$ be a prime ideal of $A$. Suppose $M_\mathfrak{p}$ is free. Show that there exists a finite set $x_1, \cdots, x_n$ of elements of $A\setminus \mathfrak{p}$ such that the localization of $M$ at the multiplicative set generated by the $x_i$ is free.
Give an example of a module $M$ over a ring $A$ such that $M_\mathfrak{p}$ is free for each prime ideal $\mathfrak{p}$ of $A$, but $M$ itself is not free. Such modules are called locally free.
Give an example of a flat module which is not projective.
Write a careful proof of the Cayley-Hamilton theorem over an arbitrary field.

HW 5 (due Feb 17): AM Chapter 2, exercises 24, 25, 26

Let $M$ be an $A$-module and let $F_\bullet: \cdots\to F_2\to F_1\to F_0$ be a flat resolution of $M$, i.e. a chain complex with each $F_i$ flat, and such that $H_i(F_\bullet)=0$ for $i>0$ and $H_0(F_\bullet)=M$. Show that for any $A$-module $N$, $H_i(F_\bullet\otimes_A N)=\operatorname{Tor}_i^A(M, N)$.
Let $M$ be a finitely-generated flat module over a Noetherian local ring $A$. Show that $M$ is free.
Carefully check that Ext is well-defined, i.e. independent of the choice of injective resolution in the definition.
Compute $\operatorname{Ext}^i_\mathbb{Z}(\mathbb{Z}/p\mathbb{Z}, \mathbb{Z})$ for each prime $p$.

HW 6 (due Feb 24): AM Chapter 5, exercises 1, 2, 10, 12, 14, and Chapter 6, exercises 2 and 5

HW 7 (due Thursday, March 17, after Spring Break):

Let $M$ be an $A$-module, and $f\in A$. Construct an isomorphism between $M_f$ and $\varinjlim M\overset{\cdot f}{\to} M\overset{\cdot f}{\to} M \overset{\cdot f}{\to} \cdots$.
Construct a module $M$ over a ring $A$ such that for each prime ideal $\mathfrak{p}$ of $A$, $M_{\mathfrak{p}}$ is finitely generated, but $M$ is not finitely-generated.
Let $M$ be a finitely-generated module over a Noetherian ring. Show that $M=0$ if and only if the support of $M$ is empty.
Suppose $\text{Spec}(A)=V_1\sqcup V_2 $, where $V_1, V_2 $ are clopen disjoint subsets. Show that there exists a direct sum decomposition $A=A_1\oplus A_2$ such that the natural quotient maps $A\to A_i$ induce isomorphisms $\text{Spec}(A_i)\to V_i$ for $i=1,2$.
Show that exactness of a long exact sequence is a local property.
Let $A$ be a Noetherian local domain with residue field $k$ and fraction field $K$, and $M$ a finitely-generated $A$-module. Show that the following are equivalent:
1. $M$ is free
2. $\text{dim}_k M\otimes_A k=\text{dim}_K M\otimes_A K$.
Let $A$ be a Noetherian ring and $M$ finitely-generated. Show that the following are equivalent:
1. $M$ is locally free
2. $M$ is projective
3. $M$ is flat.
Show that any Artinian ring is Noetherian
Show that if $A$ is a Noetherian ring such that $\text{Spec}(A)$ is Hausdorff, then $A$ is Artinian.

HW 8 (due March 31)

Give an example (with proof) of a rank one locally free module over a Dedekind domain that is not free.
Give an example (with proof) of a Noetherian domain of Krull dimension one which is not a Dedekind domain.
Let $M$ be a finitely generated module over a Dedekind domain $A$. Show that $M$ has a projective resolution of length two. Conclude that $\text{Tor}_i^A(M, -)$ and $\text{Ext}^i_A(M, -)$ equal zero for $i>1$.
Let $M,N$ be a finitely generated modules over a Dedekind domain $A$. Show that $\text{Tor}_1^A(M,N)$ is a torsion $A$-module. Can you identify its support?
Give an example of a Dedekind domain with uncountable Picard group.
Let $k$ be a field. Show that the Picard group of $k[t]$ is trivial, i.e. any rank one locally free module over $k[t]$ is free.
Give an example of a domain with a maximal non-zero ideal $\mathfrak{m}$ such that $\mathfrak{m}^2=\mathfrak{m}$.

HW 9 (due by the end of the semester)

Give an example (with proof) of a finitely generated $k$-algebra $A$ which is an integral domain, such that $\Omega_{A/k}$ is not locally free.
Let $A\to B\to C$ be maps of commutative $k$-algebras. Show that we have an exact sequence $$C\otimes \Omega_{B/A}\to \Omega_{C/A}\to \Omega_{C/B}\to 0.$$
Compute the Hilbert dimension of $k[x,y]/(y^2-x^3)$ using the definition.
Suppose you have three inverse systems $A_n, B_n, C_n$, and short exact sequences of inverse systems $$0\to A_n\to B_n\to C_n\to 0.$$ Show that there is a natural exact sequence $$0\to \varprojlim_n A_n\to \varprojlim_n B_n\to \varprojlim_n C_n.$$ Give an example where this sequence is not right exact.
Check carefully that the Kahler differentials of a polynomial ring in $n$ variables are free of rank $n$.
Give an example of a surjection of $k$-algebra $A\to B$ with kernel $I$, where the natural map $I/I^2\to \Omega_{A/k}\otimes B$ discussed in class is not injective.
Let $L/k$ be a field extension of finite degree. Show that $L/k$ is separable if and only if $\Omega_{L/k}$ is zero.