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 $(X,\mathcal{O}_X)$ 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 $S=(k[x,y]/(xy))\setminus (\bar x,\bar y-1)$.
Show that the set of ideals of $R$ not intersecting a multiplicative subset $S$ is not in bijection with ideals of $S^{-1}R$ under the extension and contraction maps between the two sets.
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 $S^{-1}R/\text{ker}(q)S^{-1}R$.
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 $B$ be a finitely generated $A$-algebra and $C$ be a finitely generated $B$ algebra then $C$ is a finitely generated $A$-algebra.
Let $R$ be a finitely generated $\mathbb{Z}$-algebra and $I$ be an ideal in $R$. Show that $\cap\{m\in mspec(R)| I\subset m\}=\sqrt{I}$.
Let $\mathbb{Z} \subset A$ be integral domains. If $A$ is a finite $\mathbb{Z}$-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 $A$.
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.
(Complete the proof of Noether normalization)
a) Let $S$ be a finite collection of $d+1$-tuple of non negative integers and $r$ be greater than every integer that appears in $S$. Show that if $i_n+i_1r+i_2r^2+\ldots+i_dr^d=j_n+j_1r+j_2r^2+\ldots+j_dr^d$ for $(i_n,i_1,\ldots,i_d),(j_n,j_1,\ldots,j_d)\in S$ then $(i_n,i_1,\ldots,i_d)=(j_n,j_1,\ldots,j_d)$.
b) Let $f(X_1,\ldots X_d)$ be a nonzero polynomial over an infinite field $k$ then show that $F(a_1,\ldots,a_d)\ne 0$ for some $a_1,\ldots, a_d\in k$.
a) 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 $R[X]/I[X]\cong (R/I)[X]$ and zerodivisors in $(R/I)[X]$ are nilpotents.)
b) 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 (a))
(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 $\bar x\bar y \in P^2$)
Let $R$ be a noetherian ring and $I$ an $R$-ideal. Show that $Z(I)$ as a subset of $spec(R)$ or $mspec(R)$ (with Zariski topology) has finitely many irreducible components. Show that the ideal of definition of the irreducible components $Z(I)$ are the minimal primes of $I$. Show that if $P$ is a minimal prime of $I$ then the $P$-primary component of $I$ is the same ideal in every primary decomposition of $I$.
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 $R=k[x,y]/(x^2-y^2-y^3)$ where $k=\mathbb{C}$. Let $K$ be its fraction field and $\alpha$ denote $\bar x/\bar y\in K$. Show that $R\subset k[\alpha]$ and it is an integral extension. Determine two maximal ideals $m_1$ and $m_2$ of $k[\alpha]$ lying above the maximal ideal $(\bar x, \bar y)$ of $R$. Find a prime ideal of $k[\alpha][z]$ contained in $(m_1,z)$ but not in $(m_2,z)$.
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 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 $U$ be a nonempty open subset of an affine variety $X$. Note that $k[X]\subset \mathcal{O}_X(U) \subset k(X)$. Show that if $\Phi:X\dashrightarrow Y$ be a rational map and $\Psi:Y\to X$ be a morphism such that image of $\Psi$ is $U$, $\Phi\circ \Psi=id_Y$ and $\Psi\circ\Phi_{|U}=id_U$. Show that $\Phi^{\#}:\mathcal{O}_Y\to k(X)$ is injective and surjects onto $\mathcal{O}_X(U))$
Let $S\subset \mathbb{P}^n$, show that $Z(I(S))$ is the Zariski closure of $S$.
Let $X\subset \mathbb{P}^n$ be such that $X\cap U_i$ is closed in $U_i$ for $0\le i \le n$. Show that $X$ is closed in $\mathbb{P}^n$ in Zariski topology.
Let $U_0\subset \mathbb P^n$ be the open subset $X_0\ne 0$. Let $V$ be a closed subset of $U_0$ viewed as the affine space $\mathbb A^n$ and $I$ be the ideal of definition of $V$. Let $J$ be the homogenization of $I$ as defined in the class. Show that the ideal of definition $V \subset \mathbb P^n$ is $J$. Conclude that $Z_{\mathbb P}(J)$ is the closure of $V$ in $\mathbb P^n$
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.
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.
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 $X\subset \mathbb{P}^n$ be a projective variety. Show that a rational function on $X$ which is regular on all points of $X$ is a constant. (Hint: If $f$ is such a function then show that there exist $N$ such that $X_i^Nf$ is in homogeneous coordinate ring $A$ of $X$. Use this to show that $f$ is integral over $A$ and then conclude that $f$ is in the base field.)
Let $f:X\to Y$ be a rational map of varieties with $Y\subset \mathbb P^n$ projective variety. Show that there exist rational functions $F_0,\ldots,F_n$ on $X$ and a nonempty open subset $U$ of $X$ such that for all $x\in U$, $f(x)=[F_0(x),\ldots,F_n(x)]$.
In the above setup, if $X$ is smooth curve then show that $f$ extends to a morphism (i.e. f is regular at all points of $X$).
(Prime avoidance lemma) Let $I_1, I_2, \ldots, I_n$ be ideals with all but two of them prime ideals and $J$ be an ideal of $R$, such that $J \subset \bigcup_j I_j$. Then $J$ is contained $I_j$ for some $j$.
Given finitely many points in $\mathbb P^n$, show that there exist a hyperplane in $\mathbb P^n$ which avoids these points.(Hint: Observe that the set of hyperplanes in $\mathbb{P}^n$ is also $\mathbb{P}^n$). Use this to conclude that given a collection $S$ of finitely many points on a projective variety $X$, there exist an affine open subset $U$ of $X$ containing $S$.
Given a smooth projective $X$, a finite set of points $S$ on $X$ and integers $n_P$ for $P\in S$, show that there exist a rational function $f\in k(X)$ such that $ord_P(f)=n_P$. (Hint: Use the previous problem to reduce to $X$ affine. For a fixed $P\in S$, use prime avoidance to obtain functions $t_P$ such that $ord_P(t_P)=1$ and $ord_Q(t_P)=0$ for $Q\in S$ different from $P$.)
Let $D$ be a divisor. Show that $l(D) > 0$ if and only if $D$ is linearly equivalent to an effective divisor.
Show that $deg(D) = 0$ and $l(D) > 0$ are true if and only if $D$ is linearly equivalent to the zero divisor.
Let $D = n_PP$ be an effective divisor on X, $S = \{P \in X | n_P > 0\}$, $U = X \setminus S$. Show that elements of $L(rD)$ are regular on $U$ for all $r>0$.
Let $X$ be the plane curve given by $Y^2Z=X(X-Z)(X-2Z)$. Let $P$ be any point on $X$. Show that there is a nonconstant rational function $f$ regular away from $P$ with $ord_P(f)>-3$.