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<title>Homework Commutative Algebra MMath and BMath</title>
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<h1><center>Homework for Commutative Algebra I, M.Math and B.Math</center></h1>

<br>Unless otherwise stated a <b>ring</b> would mean a <b>commutative ring with identity</b>.<br>
<ol>
<h3>Week 1</h3>

<li><p> Let $R$ be a commutative ring with unity. Let $J \subset I$ be ideals of $R$. Let $\bar I$ denote the ideal $I / J$ of $\bar R=R/J$. Show that $\bar R/\bar I$ is isomorphic to $R/I$.</p>

<li><p>An element $x$ in a ring $R$ is called idempotent if $x^2=x$. Show that if a ring $R$ has an idempotent different from 0 and 1 then $R$ is not an integral domain. Show that the only idempotents in $\mathbb{R}[X,Y]/(XY)$ are 0 and 1.

<li><p> Show that every short exact sequence of $k$-module splits. Give an example of a short exact sequence of $R$-modules for some ring $R$ which does not split.</p>

<li><p> Let $M$ be an $R$-module and $\mu:R\to End(M)$ the resulting structure map. Show that $End_R(M)$ is the centralizer of $Image(\mu)$.</p>

<li><p> Show that $R$ is a local ring iff for any $x\in R$, $x$ or $1+x$ is a unit.</p>

<li><p> Let $p$ be a prime number. Show that the subring  $\{r\in \mathbb{Q}: r=\frac{m}{n}, m,n \in \mathbb{Z} \text{ and } p \nmid n \}$ of $\mathbb{Q}$ is a local ring.</p>

<h3>Week 2</h3>

<li><p>Let $R=k[x,y]$ and $M=k[x,y]/(x)$. Compute $S^{-1}M$ for <br>
 (a) $S=R\setminus (x)$.<br>
 (b) $S=\{1, y, y^2, \ldots \}$.
</p>

<li><p> Show that $Spec(R)$ is connected iff only idempotent in $R$ are 0 and 1.</p>

<li><p>Let $R$ be a ring. For a subset $X$ of $Spec(R)$, define $I(X)=\cap_{P\in X} P$. <br> 
 (a) Show that $Y\subset X \subset Spec(R)$ implies $\mathscr{I}(X)\subset \mathscr{I}(Y)$.<br>
 (b) $\mathscr{I}(V(I))=\sqrt{I}$.<br>
</p>

<li><p> Show that $I$ is a prime ideal of $R$ then $V(I)$ is irreducible. </p>

<li><p> Show that $S^{-1}R=0$ iff $0\in S$. Show that the natural map $R\to S^{-1}R$ is injective iff $S$ does not contain a zero divisor. The map is an isomorphism iff every element of $S$ is a unit.</p>

<li><p> Let $R$ be a ring and $S$ a multiplicative subset. Show that $S^{-1}R$ is naturally an $R$-module. Show that it is not finitely generated if $S$ consists of nonzero divisors and at least one non-unit.</p>





<h3>Week 3 and 4</h3>

<li> <p> Let $R=\mathbb{C}[X,Y]$. Show that the ideal $I=(X,Y)$ is not a free $R$-module.</p>

<li><p> Let $R$ be a ring, $M$ a finitely generated $R$-module and $S$ amultiplicative subset of $R$. Show that $S^{-1}M$ is a finitely genrated $S^{-1}R$-module and if $M$ is a free $R$-module then $S^{-1}M$ is a free $S^{-1}R$-module. Is $S^{-1}M$ a finitely generated $R$-module?</p>

<li><p> Let $0\to A\to B \to C\to 0$ be a short exact sequence of $R$-modules. Show that if $A$ and $C$ are finitely generated then so is $B$. But the converse fails ($B$ is finitely generated does not imply $A$ is finitely generated).</p>

<li><p> Let $R$ be a ring and $M$, $N$ and $K$ be $R$-modules. Show that $Hom(M,N\oplus K)$ is isomorphic to $Hom(M,N)\oplus Hom(M,K)$ as $R$-modules.</p>

<li>Prove the Five Lemma.

Let the following be a commutative diagram of $R$-modules with exact rows:

$$
\require{AMScd}
\begin{CD}
M_1 @>{f_1}>> M_2 @>{f_2}>> M_3 @>{f_3}>> M_4 @>{f_4}>> M_5 \\
@V{h_1}VV        @V{h_2}VV        @V{h_3}VV        @V{h_4}VV        @V{h_5}VV \\
N_1 @>>{g_1}> N_2 @>>{g_2}> N_3 @>>{g_3}> N_4 @>>{g_4}> N_5
\end{CD}
$$

1. If $h_1$ is surjective, and $h_2, h_4$ are injective, then $h_3$ is injective.<br>
2. If $h_5$ is injective, and $h_2, h_4$ are surjective, then $h_3$ is surjective.<br>
3. If $h_1, h_2, h_4, h_5$ are isomorphisms, then $h_3$ is an isomorphism.

<li><p>Let $k$ be a field and $R=k[[x,y]]/(x^2+xy)$. Let $M=R/(x)$. Show that $M$ is not a projective $R$-module. Also show that $M/xM$ is a free $R/(x)$-module.</p>   

<li><p> In the proof of construction of tensor product, show that the map from $\tilde \psi:T \to Q$ satisfying $\tilde\psi \circ \phi=\psi$ is unique.</p>
  
<li><p>Let $R$ be a ring and $A, B, C$ be $R$-modules. Show that $(A\otimes B)\otimes C \cong A\otimes (B\otimes C)$.</p>



<h3>Week 5</h3>

<li><p> Let $R$, $A$ and $B$ be rings. Let $i:R\to A$ and $j:R\to B$ be ring homomorphisms. Show that there exist an $R$-bilinear map $\mu:(A\otimes_R B) \times (A\otimes_R B) \to A\otimes_R B$ which sends $(a\otimes b, a'\otimes b')$ to $aa'\otimes bb'$. Show that this makes $A\otimes_R B$ into a ring.</p>

<li><p> Show that for a field $k$, $k[X]\otimes_k k[Y]\cong k[X,Y]$.</p>

<li><p>Let $A$ be a ring, $B$ be an $A$-algebra and $M$ a finitely generated $B$-module. Show that $M$ is a finitely generated $A$-module if $B$ is a finitely generated $A$-module.</p>

<li><p> Let $0\to A\to B\to C\to 0$ be an exact sequence of $R$-modules. Show that if $A$ and $C$ are noetherian $R$-module then so is $B$.</p>

<li><p>Show that if $M$ is a flat $R$-module and $S$ is a multiplicative subset of $R$ then $S^{-1}M$ is a flat $S^{-1}R$-module.</p>


<li><p> Let $(M_i)_{i\in \Omega}$ be $R$-modules and $M=\oplus_{i\in \Omega}M_i$. Show that $M$ is flat iff $M_i$ is flat for all $i\in \Omega$.</p>


<h3>Week 6</h3>

<li><p>Let $R$ be a ring, $M$ and $N$ be $R$-modules and $A$ be an $R$-algebra. Show that $A\otimes_R(M\otimes_R N)\cong (A\otimes_R M)\otimes_A (A\otimes_R N)$ as $A$-modules.</p>

<li><p>Let $R$ be a ring, $M$ and $N$ be finitely generated $R$-modules. Show that $M\otimes_R N$ is also finitely generated.</p>

<li><p>Let $M$ be flat $R$-module where $R$ is an integral domain. Show that $M$ is torsion free.</p>

<li><p> Let $R=\mathbb C[x,y]$. Show that the ideal $I=(x,y)$ is not flat. (Hint: Show that the natural map $I\otimes I\to I$ is not injective by showing that $x\otimes y-y\otimes x$ is a non zero element in $I\otimes I$ and is in the kernel of the map.) </p>


<h3>Week 7</h3>

<li><p> Give an example of a nonzero projective module which is not faithfully flat.</p>

<li><p>Let $R$ be a local ring and $M$ a nonzero finitely generated flat $R$-module then $M$ is faithfully flat.</p>

<h3>Week 8 and 9</h3>

<li><p>Let $A\subset B$ be an integral extension. Show that $A[X]\subset B[X]$ is also an integral extension.</p>

<li><p>Let $A\subset B$ be rings. Let $C$ be the integral closure of $A$ in $B$. Show that $C$ is the integral closure of $C$ in $B$.</p>

<li><p>Let $A\subset B$ be rings and $S\subset A$ be a multiplicative set. Let $C$ be the integral closure of $A$ in $B$. Show that $S^{-1}C$ is the integral closure of $S^{-1}A$ in $S^{-1}B$.</p>

<li><p>Show that the ring $k[X,Y]/(Y^2-X^2-X^3)$ is not a normal domain.</p>

<li><p>Show that $\mathbb{Z}[i]$ is the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$.</p>

<li><p>Let $A\subset B$ be a finitely generated $k$-algebras. Show that for any maximal ideal $n$ of $B$, $B/n$ is a finite extension of $A/(A\cap n)$.</p>

<li><p> (Power of prime ideals need not be primary): Let $R=k[x,y,z]/(z^2-xy)$. Show that $P=(\bar x, \bar z)$ is a prime ideal of $R$. Show that $P^2$ is not primary. Also show $\sqrt{P^2}=P$. (Hint: First show $\bar x\bar y \in P^2$)</p>

<li><p> Let $R$ be a ring and $I$ an ideal in $R$. Let $R[X]$ be a polynomial ring over $R$ and $I[X]$ the ideal generated by $I$ in $R[X]$. Show that if $I$ is primary then $I[X]$ is primary. (Hint: Show that zero divisors in $R[X]/I[X]\cong (R/I)[X]$ are nilpotents.)</p>

<li><p> Let $k$ be a field and $k[X_1,\ldots, X_n]$ be a polynomial ring. Show that every power of prime ideals $(X_1,\ldots, X_i)$ is a primary ideal for $1\le i \le n$. (Hint: Use above)</p>

<h3>Week 10</h3>

<li><p> Let $R$ be a ring and $S$ a multiplicative subset of $R$. Let $Q$ be a primary ideal of $R$ disjoint with $S$ then show that $S^{-1}Q$ is a primary ideal of $S^{-1}R$. </p>  

<li><p>Let $\phi:A\to B$ be a surjective ring homomorphism and $I$ be an ideal of $B$. Let $J=\phi^{1}(I)$. Show that $J$ is primary iff $I$ is primary. If $I$ admits a minimal primary decomposition $I=\cap_{i=1}^n Q_i$. Compute the minimal primary decomoposition of $J$.</p>   

<li><p> Let $R=\mathbb{C}[X,Y,Z]$ and $I=(X^2,XY,YZ,ZX)$. Determine the minimal prime ideals of $I$. Show that $(X,Y,Z)$ is an embedded prime of $I$.</p>

<li><p> Let $R=\mathbb{C}[X,Y]$ be polynomial ring and $I=((X^2+Y^2-1)(X-1)Y)$. Compute a minimal primary decomposition of $I$.</p>  

<li><p> In the previous two problems find a subring $S$ of $R/I$ isomorphic to the polynomial ring such that $R/I$ is integral over $S$.</p> 
<!-----------------------

<li><p> Let $\mathscr{C}$ be an abelian category and let $f\in Mor(A,B)$ for some objects $A$ and $B$ of $\mathscr{C}$. Write down the universal property an object $I$ together with $i\in Mor(I,B)$ should satisfy for it to be image of $f$. Show that kernel(cokernel($F$)) satisfy this universal property.</p> 

<li><p>Let $B$ be an integral domain and $A$ a subring of $B$. Let $K$ be the field of fractions of $A$. Show that if $\alpha \in B$ is integral over $A$ then every $K$-Galois conjugate of $\alpha$ (which lies in an algebraic closure of the quotient field of $B$) is integral over $A$. </p>



<li><p>Let $k$ be a field and $R$ be a finitely generated $k$-algebra. Show that $Jac(R)=nil(R)$.</p>


<li><p>Let $A\subset B$ be integral extension. Show that dimension of $A$ and $B$ are same. </p>

<li><p> Find an example of a ring extension $A \subset B$, a prime ideal $P$ in $A$ and prime ideals $Q_1\subset Q_2$ in $B$ such that $Q_1\cap A=Q_2\cap A=P$.</p>

<li><p> Find an example of a ring extension $A \subset B$ such that the induced map from Spec$(B)$ to Spec$(A)$ is surjective (i.e. lying over holds) but going up property for the extension fails.</p>

<li><p>Let $A\subset B$ be integral extension and $P$ a prime ideal of $A$. Show that $PB\cap A=P$.</p>

<h3>Week 9</h3>

<li><p>Let $A=k[X,Y,Z]/(Y^2-X^2-X^3)$ and $x,y,Z$ be the images of $X,Y,Z$ in $R$. Let $B=A[y/x]$. Show that $B=k[y/x,Z]$ is isomorphic to the polynomial ring in two variables and hence $Q_1=(y/x+1,Z)$ is a prime ideal of $B$. Show that $Q_1\cap A=(x,y,Z)$ (=$P_1$ say). Show that for any nonzero $\alpha\in k$, $(y/x-1+\alpha Z)\cap A=(y-x+\alpha xZ)A$ (=$P_0$ say). Show that $P_0B=(y/x-1+\alpha Z)$ and hence is not contained in $Q_1$. Deduce that going down does not hold for the integral extension $B/A$.</p>

<li><p>Let $A\subset B$ be an integral extension. Let $P\in spec(A)$ and $Q\in spec(B)$. Show that $Q$ is a minimal prime of $PB$ iff $Q\cap A=P$.</p>

<li><p> Let $A$ be a ring and $S$ a multiplicative subset. Show that going down theorem holds for the following ring homomorphisms. <br>
(i) $A\subset A[X]$ where $A[X]$ is the polynomial ring over $A$.<br>
(ii) $A \to S^{-1}A$ where $S$ is a multiplicative subset of $A$.<br>
(More generally, going down holds for the extension $A\subset B$ if $B$ is a flat $A$-module.) 
</p>



<li><p> Let $R=k[X,Y,Z]$ and $I=(X^2,XY,YZ,ZX)$. Determine the minimal prime ideals of $I$. Show that $(X,Y,Z)$ is radical of a primary ideal which appear in a minimal primary decomposition of $I$.</p>

<h3>Week 10</h3>


<li><p> Let $R$ be a ring and $M$ a simple $R$-module then show that $M$ is isomorphic to $R/m$ for some maximal ideal $m$ of $R$.</p>

<li><p> Let $A$ be a noetherian ring. Prove that the following are equivalent.<br>
   (i) $A$ is Artinian.<br>
  (ii) Spec($A$) is finite and discrete.<br>
 (iii) Spec($A$) is discrete.</p>

<li><p> Let $k$ be a field and $A$ be a finitely generated $k$-algebra. Show that $A$ is artinian iff $dim_k(A)< \infty$.</p>

<li><p>Let $v:K^*\to G$ be a valuation where $K$ is a field and $G$ an ordered abelian group. Show that the valuation ring $R_v$ is noetherian iff $Im(v)$ is isomorphic to $\mathbb{Z}$.</p>

<h3>Week 11 and 12</h3>

<li><p> Let $A$ be a Dedekind domain and $I$ a nozero ideal in $A$. Show that every ideal in $A/I$ is principal. Deduce that every ideal in $A4 can be generated by two elements.</p>

<li><p> Let $M$ be a finitely generated module over a dedekind domain. Show that $M$ is flat iff $M$ is torsion free.</p>

<li><p> Let $(R,m)$ be a noetherian local ring such that $m$ is a principal ideal and $R$ is not artinian. Show that $R$ is a discrete valuation ring.</p>

<li><p>Let $\hat{\mathbb{Z}}_p$ be the inverse limit of $\mathbb{Z}/p^n\mathbb{Z}$ as $n$ varies. Show that every element of $\hat{\mathbb{Z}}_p$ can be uniquely written as a power series:<br>
 $ a_0 + a_1p + a_2p^2 + \ldots $ where $a_i\in \{0, 1, \ldots, p-1\}$ and addition happens with carry over, i.e. $3p+(p-2)p = 1p + 1p^2$. 
</p>

<li><p>Let $A$ be noetherian ring $I$ an ideal and $\hat A$ the $I$-adic completion of $A$. Show that if $x\in A$ is a non-zero divisor then its image in $\hat A$ is a no0n-zero divisor. Consider the local ring $A=\mathbb{C}[x,y]_{(x,y)}/(x^2-y^2-y^3)$ and let $m$ denote its maximal ideal. Show that the $m$-adic completion of $A$ is not an integral domain.</p>

<li><p> Let $(R,m)$ be a DVR. Show that its $m$-adic completion $\hat R$ is also a DVR.</p> 
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