Let $R$ be a nonzero commutative ring with 1. Show that $R[x,y]/(x-f(y))$ is isomorphic to $R[y]$, where $R[x,y]$ is a polynomial ring over $R$ in two variables and $f(y)\in R[y]\subset R[x,y]$.
Show that $I=(x^2+y^2-1)$ is a prime ideal in $\mathbb{R}[x,y]$. Find a maximal ideal containing $I$ different from $(x-a,y-b)$ for some $a,b\in \mathbb{R}$.
Give an example of a reduced ring which is not an integral domain.
Let $(X,\mathcal{O}_X)$ be an affine variety. Show that $C\subset X$ is closed iff there exist an ideal $I$ of $\mathcal{O}_X$ such that $C=Z(I):=\{p\in X: f(p)=0 \forall f\in I\}$.
Let $R$ be a ring and $S$ a multiplicative subset. Let $J$ be an ideal of $S^{-1}R$. The subset $\phi^{-1}(J)$ of $R$ is an ideal of $R$ and is (sometimes) denoted by $J\cap R$, where $\phi:R\to S^{1-}R$ is the natural localization map. Show that $J=(J\cap R)S^{-1}R$.
List all the ideals of $\mathbb{Z}_{(p)}$ and the prime ideals of $\mathbb{Z}[\frac{1}{p}]$, where $\mathbb{Z}[\frac{1}{p}]=S^{-1}\mathbb{Z}$ for $S=\{1, p, p^2, \dots \}$.
For a field $k$ show that $S^{-1}[k[x,y]/(xy)]$ is isomorphic to $k[x]_{(x)}$ where $S=(k[x,y]/(xy))\setminus (\bar x,\bar y-1)$.
Let $q:R\to \bar R$ be surjective ring homomorphism and $S$ be a multiplicative subset of $R$ then $q(S)$ is a multiplicative subset of $\bar R$ and $q(S)^{-1}\bar R$ is isomorphic to $S^{-1}R/\text{ker}(q)S^{-1}R$.
Show that every short exact sequence of (finitely generated) modules over the ring $F\times F'$ splits where $F$ and $F'$ are fields.
Let $R$ be a commuatative ring with unity and $S$ be a multiplicative set. Let $0\to A\to B \to C\to 0$ be a short exact sequence of $R$-modules. Show that $0\to S^{-1}A\to S^{-1}B\to S^{-1}C\to 0$ is exact.
Let $R$ be a ring and $A$, $B$ and $C$ be $R$-modules. Show that $A\otimes_R (B \otimes_R C)$ is isomorphic to $(A \otimes_R B)\otimes_R C$ as $R$-modules.
Let $R$ be a ring, $I$ an $R$-ideal and $M$ an R-module. Show that $R/I \otimes_R M \cong M/IM$.
Let $k$ be a ring, $A$ and $B$ be $k$-algebra then show that $A\otimes_k B$ has a natural $k$-algebra structure.
Show that if $0\to A\to B\to C\to 0$ be a short exact sequence of $R$-modules and $P$ is a projective $R$-module then the induced exact sequence $0\to A\otimes P\to B\otimes P\to C\otimes P\to 0$ is exact.
Let $R$ be a ring, $M$ a finitely generated $R$-module and $S$ a multiplicative subset of $R$. Show that $S^{-1}M$ is a finitely generated $S^{-1}R$-module. Is $S^{-1}M$ necessarily a finitely generated $R$-module as well?
State Hom-Tensor duality. Use it and left exactness of $Hom$ to show that $\otimes$ is right exact. (Hint: First show that if $A\to B\to C\to 0$ is a complex of $R$-modules such that $0\to Hom(C,N)\to Hom(B,N)\to Hom(A,N)$ is exact for all $R$-module $N$ then the complex $A\to B\to C\to 0$ is exact.)
Let $B$ be a fnitely generated $A$-algebra and $C$ be a finitely generated $B$ algebra then $C$ is a finitely generated $A$-algebra.
Let $\mathbb{Z} \subset A$ be integral domains. If $A$ is a finite $Z$-module then show that there exist $\alpha_1,\ldots,\alpha_n\in \bar{\mathbb{Q}}$such that $A$ is isomorphic to the subring $\mathbb{Z}[\alpha_1,\ldots, \alpha_n]$ of $\bar{\mathbb{Q}}$.
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 $A$.
Show that $\mathbb{Z}[i]$ is the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$.
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 $S^{-1}B$.
For the following $\mathbb{C}$-algebra homomorphisms. Describe the corresponding varieties and the morphism between them.
(i) The map $\phi^{\#}:\mathbb{C}[y]\to \mathbb{C}[x,y]/(y^2-x^3-x)$ which sends $y$ to $\bar y$ the image of $y$ in the co-domain.
(ii) The inclusion $\mathbb{C}[y]\to \mathbb{C}[x,y]$.
(iii) The inclusion $\mathbb{C}[x] \to \mathbb{C}[x,x^{-1}, (x-1)^{-1}]$.
Consider the space curve $C\subset \mathbb{A}^3$ given by the image of the morphism $\phi: \mathbb{A}^1\to \mathbb{A}^3$ which send $t\to (t,t^2,t^3)$. Compute the co-ordinate ring $\mathcal{O}_C$ by computing the ideal of definition of $C$ in $\mathbb{A}^3$ and the ring homomorphism $\phi^{\#}:\mathcal{O}_C\to \mathcal{O}_{\mathbb{A}^1}$.
Let $\phi:X\to Y$ be a surjective morphism of varieties. Show that the induced map of $k$-algebra $\phi^{\#}:\mathcal{O}_Y\to \mathcal{O}_X$ is injective. Also show that the converse is false.
(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 $\bar x\bar y \in P^2$)
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 $Q_1\cap A=Q_2\cap A=P$.
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.
Let $A$ be a ring and $S$ a multiplicative subset. Show that going down theorem holds for the following ring homomorphisms.
(i) $A\subset A[X]$ where $A[X]$ is the polynomial ring over $A$.
(ii) $A \to S^{-1}A$ where $S$ is a multiplicative subset of $A$.
Let $X$ be an irreducible affine variety and $p\in X$ a point. Show that the local ring R_p=$\{f \in k(X): f \text{ is regular at } p\}$ is isomorphic to $k[X]_P$ where $k[X]=\mathcal{O}_X$ is the co-ordinate ring of $X$ and $P$ is the maximal ideal $\{f\in k[X]: f(p)=0\}$.
Show that Zariski topology on an irreducible affine curve is same as the cofinite topology. Deduce that any two affine curves are homeomorphic.
Let $X$ be an affine variety. Show that $\{U_f:=X\setminus Z(f)| f\in k[X] \}$ is a basis of Zariski topology on $X$.
Show that every nonempty open subset of an irreducible affine curve, whose coordinate ring is a PID, is an affine curve. (The result is true without PID hypothesis on the coordinate ring).
Let $X$ be an algebraic subset of $\mathbb{P}^n$. Show that $I=\{ f \in k[X_0,\ldots, X_n] : f(a_0,\ldots,a_n)=0 \forall (a_0,\dots,a_n)\in \mathbb{A}^{n+1} \text{ with } [a_0,\ldots, a_n]\in X\}$ is a homogenous ideal and equals $I(X)$.
Consider the algebraic set $C$ in $\mathbb{P}^2$ defined by the homogenous polynomial $X_0^2+X_1^2-X_2^2$. As we saw in the class, $C\cap U_2= \{[\frac{X_0}{X_2},\frac{X_1}{X_2},1]: (\frac{X_0}{X_2})^2+(\frac{X_1}{X_2})^2-1=0\}$ and $C\cap U_0=\{[1,\frac{X_1}{X_0},\frac{X_2}{X_0}]: 1+(\frac{X_1}{X_0})^2-(\frac{X_2}{X_0})^2=0\}$.
Find the points of $C$ not in $U_0$ and $U_2$ respectively.
Show that $Y=C\setminus \{[1,\pm i,0], [0,\pm 1,1]\}$ is (1) circle minus two points and (2) hyperbola minus two points.
Show that the affine co-ordinate ring of $Y$ viewed as in (1) is $A=k[x,y,x^{-1}]/(x^2+y^2-1)$; and as in (2) is $B=k[u,v,v^{-1}]/(1+u^2-v^2)$. Show that $A$ and $B$ are isomorphic directly by giving explicit ring homomorphism. (Hint: $x=X_0/X_2, y=X_1/X_2, u=X_1/X_0, v=X_2/X_0$).
Let $S\subset \mathbb{P}^n$, show that $V(I(S))$ is the Zariski closure of $S$.
Show that the map $\Phi_0:\mathbb{A}^n\to \mathbb{P}^n$ is homemorphism onto its image. The topology on a subset of $\mathbb{P}^n$ is the subspace topology inherited from the Zariski topology and the topology on $\mathbb{A}^n$ is the Zariski topology.
Let $X\subset \mathbb{P}^n$ be a projective algebraic subset. Recall that the cone of $X$, $CX=\{(a_0,a_1,\ldots,a_n): [a_0,\ldots,a_n]\in X\} \cup \{origin\}\subset \mathbb{A}^{n+1}$. Show that $X$ is irreducible iff $CX$ is irreducible. Conclude that the ideal of definition of an algebraic subset $X$ of $\mathbb{P}^n$ is a prime ideal iff $X$ is irreducible.
Let $X\subset \mathbb{P}^n$ with homogeneous coordinate $X_0,\ldots, X_n$ be a projective variety and $U_i\subset\mathbb{P}^n$ be the open set given by $X_i\ne 0$. Show that if $U_i\cap X$ is nonempty then $k(X)$ is the fraction field of the coordinate ring of $k(U_i\cap X)$.
Let $A\subset B$ be a finite ring extension (i.e. $B$ is a finite $A$-module). Show that the map $\phi:mspec(B) \to mspec(A)$ has finite fibres. (i.e. for any maximal ideal $m$ of $A$, $\phi^{-1}(m)$ is a finite set.
Show that a variety (i.e. which for us means an open subset of a projective variety) is quasi-compact, i.e. every open cover has a finite subcover.
Show that the map $s:\mathbb{P}^m\times \mathbb{P}^n\to \mathbb{P}^{(m+1)(n+1)-1}$ given by $([a_0,\ldots, a_m], [b_0, \ldots, b_m])$ goes to $[a_0b_0, a_0b_1,\ldots, a_mb_n]$ is a well defined map and a morphism of varieties.