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{\bf Math 8212\hfill Commutative and Homological Algebra I\hfill  Spring 2022}
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{\bf Homework Assignment 1} Due Saturday 2/12/2022, uploaded to Gradescope.

Each question part is worth 1 point.
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1.  Let $R\subseteq S\subseteq T$ be commutive rings and let $M$ be an $S$-module.

(a) (4.1 of Eisenbud) Show that if $S$ is finite over $R$ and $M$ is finitely generated as an $S$-module, then $M$ is finitely generated as an $R$-module.

(b) Suppose that $S$ is integral over $R$ and $T$ is integral over $S$. Show that $T$ is integral over $R$.
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2. (4.2 of Eisenbud with $R$ and $S$ interchanged.) 
Let $k$ be a field, $R=k[t]$ and suppose $R\subseteq S$ is a containment of rings, where $S$ is supposed to be a domain.

(a) Show that if $S$ is finitely generated as an $R$-module, then $S$ is free as an $R$-module.

(b) Show by giving a basis that if $S=k[x,y]/(x^2-y^3)$ and $t=x^my^n$, then the rank of $S$ as an $R$-module is $3m+2n$.

(c) Assuming again only that the domain $S$ is finitely generated as an $R$-module, let $\bar S$ be the integral closure of $S$ in its field of fractions. Assume Noether's theorem 4.14 that $\bar S$ is again finitely generated (and thus free) as an $R$-module. Show that it has the same rank as $S$.

[Feel free to make use of the structure theorem for finitely generated modules over a PID.]
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3. (4.7 of Eisenbud) Show that the Jacobson radical of $R$ is
$$
J=\{ r\in R\bigm| 1+rs \hbox{ is a unit for every } s\in R\}.
$$
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4. (4.11 of Eisenbud minus the graded bit) 

(a) Use Nakayama's lemma to show that if $R$ is a commutative local ring and $M$ is a finitely generated projective module, then $M$ is free. 

[Identify the radical, consider factoring out its action, produce a map from a free module that is an isomorphism with $M$.]

(b)  Use Proposition 2.10 to show that a finitely presented module $M$ is projective if and only if  $M$ is locally free, in the sense that the localization $M_P$ is free over $R_P$ for every maximal ideal $P$ of $R$ (and then of course $M_P$ is free over $R_P$ for every prime ideal  $P$ of $R$).
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5. (4.20 of Eisenbud)
For each $n\in\ZZ$, find the integral closure of $\ZZ[\sqrt n ]$ as follows:

(a) Reduce to the case where $n$ is square-free.

(b) $\sqrt n$ is integral, so what we want is the integral closure  $R$ of $
\ZZ$ in the field $\QQ[\sqrt n]$. If $\alpha = a + b\sqrt n$ with $a,b\in\QQ$, then the minimal polynomial of $\alpha$ is $x^2 - \hbox{Trace}(\alpha)x + \hbox{Norm}(\alpha)$ where $\hbox{Trace}(\alpha) = 2a$ and $\hbox{Norm}(\alpha) = a^2-b^2n$. Thus $\alpha\in R$ if and only if  $\hbox{Trace}(\alpha)$ and $\hbox{Norm}(\alpha)$ are integers.

(c) Show that if $\alpha\in R$ then $a\in{1\over 2}\ZZ$. If $a=0$, show $\alpha\in R$ iff $b\in\ZZ$. If $a={1\over 2}$ and $\alpha\in R$, show that $b\in {1\over 2}\ZZ$. Thus, subtracting a multiple of $\sqrt n$, we may assume $b=0$ or ${1\over 2}$. Observe $b=0$ is impossible.

(d) Conclude that the integral closure is $\ZZ[\sqrt{n} ]$ if $n\not\equiv 1\hbox{ (mod 4)}$, and is $\ZZ[{1\over 2} + {1\over 2}\sqrt{n}]$ if $n\equiv 1\hbox{ (mod 4)}$.
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6. (1.3 of Matsumura plus)  Let  $A$ and $B$ be rings, and $f:A\to B$ a surjective homomorphism.

(a) Prove that  $f(\Jac A)\subseteq \Jac B$, and construct an example where the inclusion is strict.

(b) Prove that if  $A$ is a semilocal ring (a ring with only finitely many maximal ideals) then  $f(\Jac A) = \Jac B$.

(c) Continue to assume that $A$ is a semilocal ring. Show that, as an $A$-module, $A/\Jac(A)$ is a direct sum of finitely many simple $A$-modules, and that $\Jac(A)$ is  the smallest ideal with this property. (That is, if $J$ is an ideal so that $A/J$ is a direct sum of simple $A$-modules, then $J\supseteq \Jac(A)$.)

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{\bf Extra question: do not upload to Gradescope.}
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7. Show that the Jacobson radical of $k[x_1,\ldots,x_n]$ is 0.

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