Let $X$ be an algebraic subset of affine $n$-space over a field $k$. If $X$ is a point then $I(X)$ is a maximal ideal of $k[x_1,\ldots,x_n]$.
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 $a,b\in \mathbb{R}$.
Let $I$ and $J$ be ideals in $k[x_1,\ldots,x_n]$ such that $Z(J)\subset Z(I)$. Prove or disprove $I\subset J$.
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 $C=Z(I):=\{p\in X: f(p)=0 \forall f\in I\}$.
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?
Let $X$ and $Y$ be algebraic subsets of $\mathbb{A}^m$ and $\mathbb{A}^n$ respectively. Show that $I(X\times Y)$ in $k[x_1,\ldots,x_m, y_1,\ldots,y_n]$ is generated by $I(X)\subset k[x_1,\ldots,x_m]$ and $I(Y)\subset k[y_1,\ldots y_n]$. (Hint: For $f\in k[x_1,\ldots,x_m,y_1,\ldots, y_n]$ show that $f=\sum_1^l f_ig_i$ with $f_i\in k[x_1,\ldots, x_m]$ and $g_i\in k[y_1,\ldots, y_n]$. If possible choose an $f\in I(X\times Y)\setminus (I(X)\cup I(Y))$ with the smallest possible $l$ to get some contradiction.)
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.
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 $k$ be an algebraically closed field, $R$ be a finitely generated $k$-algebra and $I$ be an ideal of $R$. Show that $Z(I)=\emptyset$ iff $I=R$.
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 $\{D_f:=X\setminus Z(f)| f\in k[X] \}$ is a basis of Zariski topology on $X$.
Let $X$ be an affine variety and $f$ a regular function on $X$. Show that the bijections between $D_f$ and $Y=$mspec$(\mathcal{O}_X[f^{-1}])$ defined in the class are homeomorphisms.
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 $k$ be a field and $A$ be a finite $k$-algebra. Show that $A$ has finitely many maximal ideals. Conclude that if $A\subset B$ be is a finite ring extension (i.e. $B$ is a finite $A$-module) then 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.)
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 $I(X)$.
Let $S\subset \mathbb{P}^n$, show that $Z(I(S))$ is the Zariski closure of $S$.
Show that the map $\Phi_0:\mathbb{A}^n\to \mathbb{P}^n$ is homeomorphism 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.
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: $x=X_0/X_2, y=X_1/X_2, u=X_1/X_0, v=X_2/X_0$).