There is an exercise in Hartshorne's Algebraic Geometry with says to show that $Y=\{(t, t^2, t^3) \mid t \in k\}$ is an affine variety of dimension $1$. But I actually don't even know how to show that it is algebraic set. The easiest polynomials I thought of that vanish on $Y$ are: $x^2-y, x^3-z, y^3-z^2$. How do we show that $Y$ is affine algebraic set? How do we come up with the polynomials to define $Y$? How would we go about showing that $Y$ is the zero set of these polynomials?
121570
I am trying to work through Hartshorne's book and while working through one of the exercises, I need to show the following:  Let $k$ be an algebraically closed field. Let $Y \subseteq A^3$ be the set $Y=\{(t,t^2,t^3)\;|\;t \in k\}$. Show that $Y$ is an affine variety.  The rest of the problem I have completed. I'm just not sure how to show that this set is closed and irreducible. I assume that they intend for $A^3$ to have the Zariski topology.
26230
Suppose that $k$ is an algebraically closed field. Then what do the prime ideals in the polynomial ring $k[x,y]$ look like?   As far as I know, the maximal ideals of $k[x,y]$ are of the form $(x-a,y-b)$ where $a,b\in k$. What can we say about the prime ideals? Are there similar results? And what about $k[x,y,z], k[x,y,z,w]$ and so on. Would someone be kind enough to give me some hints or referrence on this topic? Thank you very much!
67143
I find difficulty to understand the proof of this theorem :  Theorem : Let $Y\subseteq\mathbb A^n$ be an affine variety. Let $Ρ\in Y$ be a point. Then $Y$ is nonsingular at $Ρ$ if and only if the local ring $\mathcal O_P$ is a regular local ring.  Let $P$ be the point $(a_1,\dots,a_n)$ in $\mathbb A^n,$ and let $\frak {a}$$=(x_1-a_1,\dots,x_n-a_n)$  be the corresponding maximal ideal in $A = k[x_1,\dots,x_n].$ We define a  $\theta : A\longrightarrow k^n$ by $$\theta(f)=\left(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n}\right).$$ So $\theta$ induce an isomorphism $\theta':\frak{a}/\frak{a}^2\longrightarrow$$k.$   Now let $\frak b$ be the ideal of $Y$ in $A,$ and let $f_1,\dots,f_n,$ be a set of generators of $\frak b.$ Then the rank of the Jacobian matrix $\left(\frac{\partial f_i}{\partial x_j}\right)_{i,j}$ is just the dimension of $\theta(\frak b)$ as a subspace of $k^n.$ Using the isomorphism $\theta'$ this is the same as the  dimension of the subspace $\frak a^2+b/a^2$ of $\frak a/a^2,$ if $\frak m$ is the maximal ideal of $\mathcal{O}_Р,$ we have $$\frak m/m^2\cong a/(b+a^2).$$   I don't understand how I can use $\theta'$ to get $\dim_k \theta(\frak b)$$=\dim_k(\frak a^2+b/a^2)$ and why  $$\frak m/m^2\cong a/(b+a^2).$$   Thank you for any help.
32240
How can we prove that (the set of all polynomials in infinitely non commutative indeterminants over $k$) is a ring, where $k$ is a ring with unity?   Adding two polynomials is a polynomial, but for commutativity  How can we show the following: $ax_1+bx_2=bx_2+ax_1$ (commutativity of addition) If we have finite number of variable it is easy to prove it, (we take the $n-1$ variables...), but what about the infinite number of variables?  Can someone give me a hint?
123992
I am having some trouble understanding Proposition 5.15 in Introduction to Commutative Algebra by Atiyah and Macdonald.      Let $A\subset B$ be integral domains, $A$ integrally closed, and $x\in B$ be integral over an ideal $\mathfrak{a}$ of $A$. Then $x$ is algebraic over the field of fractions $K$ of $A$, and if its minimal polynomial over $K$ is $t^n+a_1t^{n-1} + \cdots + a_n$, then the $a_1, a_2, \ldots , a_n$ all lie in the radical of $\mathfrak{a}$.   The proof of the proposition states that the coefficients of the minimal polynomial of $x$ are polynomials in the $x_i$ (the conjugates of $x$). I don't understand this, and am struggling with the rest of the proof as well. Thanks for any help.
35633
I've got to show that any irreducible conic in $\mathbb{P^2(\mathbb{K})}$ where $\mathbb{K}$ is algebraically closed is isomorphic as a quasiprojective variety to the projective line. I wrote the conic as equivalent to the zero locus of $ x^2 +y^2+z^2$ in $\mathbb{P^2(\mathbb{K})}$. I tried to use the standard affine charts to cover this simpler variety and parametrize it but i didn't manage to do that. What am i missing?
44262
My question is about  about solving Hartshorne's Exercise I-1.9 by using exercise I-1.8:  Hartshorne's Chapter 1, exercise 1.9: Irreducible components of $Z(\mathfrak a)$ have dimension $\geq n-r$ if $\mathfrak a$ is an ideal generated by $r$ elements. Hartshorne's Chapter 1, exercise 1.8: Let $Y$ be an affine variety of dimension $r$ in $\mathbf A^n$. Let $H$ be a hypersurface in $\mathbf A^n$, and assume that $Y \nsubseteq H$. Then every irreducible component of $Y \cap H$ has dimension $r-1$.  Suppose $\mathfrak{a}$ can be generated by $f_1, \dots, f_r$ and no one is in the ideal generated by the rest of them (note they are not necessarily irreducible. But it seems to me that I don't need to assue $H$ in Ex. 1.8 is the zero set of an irreducible polynomial. Let me know if I am wrong). We'll induct on $r$. I can prove that any irreducible component $Y$ of $Z(\mathfrak{a})$ is contained in an irreducible component $Y'$ of $Z(f_1, \dots, f_{r-1})$. But, to use 1.8, I have to show that $Y'$ is not contained in $Z(f_r)$ and that $Y$ is an irreducible component of $Y' \cap Z(f_r)$ Question 1: does the assumption that $f_r\notin (f_1,\dots, f_{r-1})$ imply that $Y'$ is not contained in $Z(f_r)$? If not, how do we remedy this? Question 2: how to prove that $Y$ is an irreducible component of $Y' \cap Z(f_r)$ (something might be missing here and I can't prove it at least)? It is irreducible but why it appears in the irreducible decomposition of $Y' \cap Z(f_r)$?
53629
An algebraic set (not necessairly a variety) $X \subseteq \mathbb{A}^2$ defined by a polynomial of degree $2$ is called a conic.  The problem is:     Show that any irreducible conic is isomorphic either to $Z(y-x^2)$ or to $Z(xy-1)$ after an affine change of coordinates in $\mathbb{A}^2$.   However, with this definition of conic, I only can conclude that if the polynomial is irreducible, then the conic is irreducible (because $f$ irreducible $\Rightarrow Z(f)=X$ is irreducible)     What can I do to prove that if a conic is irreducible, then the polynomial that defines the conic is irreducible?   And for the main problem, how can I do the affine change of coordinates?  Thanks
60661
I was trying to proof the following without any luck. I would appreciate good hints.  A projective variety that is isomorphic to an affine variety is a finite number of points.
109500
I am trying to figure out the normalisation of $k[x,y]/(y^2-x^2(x-1))$, for an algebraically closed field $k$.    I can show that it is not normal and I have the information that the normalisation is $k[t]$, where $t=\frac{y}{x}$. I can't figure out how to prove that this is the normalisation i.e. that the fraction fields of $k[x,y]/(y^2-x^2(x-1))$ and $k[t]$ are the same, that $k[t]$ is integrally closed and that $k[t]$ is an integral extension of $k[x,y]/(y^2-x^2(x-1))$.   Also, how does one get to the idea of $t=\frac{y}{x}$? Algebraically if possible, since this is from a commutative algebra class, not algebraic geometry.
64970
In my recent reading of various books and notes on algebraic geometry and scheme theory, I have come across three definitions of affine $n$-space over a field $k$:   $\mathbb{A}_k^n$ is $k^n$ 'without an origin'; $\mathbb{A}_k^n$ is simply $k^n$ with the Zariski topology (i.e. zero loci of sets of polynomials are closed); $\mathbb{A}_k^n=\operatorname{Spec}k[x_1,\ldots,x_n]$.   I know that things like the Nullstellensatz give us a nice way of going from one point of view to another when it comes to polynomial rings and affine varieties, but here I'm having some confusion. For $k=\mathbb{C}$ (which is algebraically closed, so things should all act reasonably nicely) we have that the points of $\operatorname{Spec}\mathbb{C}[x]$ are in bijective correspondence with points of $\mathbb{C}$, apart from the general point $[(0)]$. So it seems to me that these definitions can't all be the same, but searching for affine space on Google or in books leads to any one of the above definitions, depending on the level of the text.  Am I just making a silly mistake, or is there a difference in these definitions? If so, which one is the 'right' one?
103216
Suppose $R$ is a field and $f$ is a polynomial of degree $d$ in $R[x]$.  How do you show that each coset in $R[x]/\langle f\rangle$ may be represented by a unique polynomial of degree less than $d$?  Secondly, if $R$ is finite with $n$ elements, how do you show that $R[x]/I$ has exactly $n^d$ cosets?
55318
Let $T$ be a linear transformation of an $n$-dimensional vector space $V$ over a field $k$. It's pretty easy to define the minimum polynomial of $T$ and make sure its degree is between $1$ and $n^2$, inclusive.     Observe $ I = \{ p(x) \in k[x] : p(T) =0\}$ is an ideal in $k[x]$. Indeed, $I$ is the kernel of the evaluation homomorphism $\mathrm{eval}_T: k[x] \to \mathrm{End}(V)$.  Notice also that:         $\mathrm{eval}_T$ is unital homomorphism, so $I$ is a proper ideal.    The $n^2 + 1$ transformations $I, T,T^2,T^3,\ldots, T^{n^2}$ must be linearly dependent, since $\mathrm{dim}(\mathrm{End}(V)) = n^2$, so there exist scalars $a_0,\ldots,a_{n^2}$, not all zero, such that $a_0I + a_1 T + \ldots + a_{n^2}T^{n^2} = 0$, whence the nonzero polynomial $p(x) = a_0 + a_1 x + \ldots + a_{n^2}x^{n^2}$ belongs to $I$.           Since $k[x]$ is a p.i.d., we may define the minimum polynomial $m(x)$ of $T$ to be the monic generator of the ideal $I$. By the preceding two observations, we have $1 \leq \mathrm{deg}(m(x)) \leq n^2$.    Now, of course, we know that the degree of $m(x)$ actually satisfies $1 \leq \mathrm{deg}(m(x)) \leq n$. One way to see this is to use the Cayley-Hamilton theorem which shows that the characteristic polynomial $c(x)=\det(xI - T)$, whose degree is $n$, annihilates $T$, whence $m(x)$ divides $c(x)$.     Question: Is there another way to see that $T$ is annihilated by a polynomial of degree $\leq n$ which does not require use of the characteristic polynomial?
106986
I'm working through some Hartshorne problems and have noticed that in order to do certain problems properly one must prove a given polynomial $f\in k[x,y]$ is irreducible.    For example, in problem I.5.1(b) we are studying the polynomial $f=xy-x^6-y^6$. How can I show this is irreducible?    The standard tricks (Eisenstein, linearity in one variable, etc...) don't seem to work here.  Edit: Some of the other polynomials in the problem are also giving me trouble. They are $g=x^3-y^2-x^4-y^4$ and $h=x^2y+xy^2-x^4-y^4$. Any ideas for these?
114160
I'm trying to do the following exercise from the book An Invitation to Algebraic Geometry:     Show that if $X \to Y$ is a surjective morphism of affine algebraic varieties, then the dimension of $X$ is at least as large as the dimension of $Y$.   Could anyone give me a hint? I'm looking for a proof that avoids commutative algebra since it is not mentionned in the book at this point.
48195
It is known that   Now consider a ring $R$ with exactly three ideals. Take an element $x$ such that $(x)\neq (1)$. For $y\in R$ and $y\notin (x)$, there are two possible cases:   $(y)=(1)$ $(y)\neq(1)$. Then $(y)=(x)$, that is $x=ys=xts$ for some $s,t\in R$. It follows that $$x(1-ts)=0$$ which implies that $x$ is a zero divisor.   However, if there is no such a $y$ of the second case, how to prove $x$ is a zero divisor?
52148
For each $m \ge 1$, how do I construct a homeomorphism $(\mathbb{CP}^1)^m/S_m \overset{\sim}{\to} \mathbb{CP}^m$?  My thoughts so far: I probably want to identify $\mathbb{C}^{m+1}$ with the vector space $\mathbb{C}^m[z, w]$ of degree $m$ homogeneous polynomials in two variables and use that a polynomial in one variable is determined by its roots up to a nonzero constant factor somehow... but from here, I'm kinda of stuck. Could someone help?  EDIT: Progress so far. After reading Eric Wofsey's response, I have the following. Since a projective variety is determined by its homogeneous coordinate ring, it suffices to determine the homogeneous coordinate ring of $(\mathbb{CP}^1)^m/S_m$.  For $(\mathbb{CP}^1)^m$, the coordinate ring is the graded ring $\mathcal{S}$ whose $d$th degree part is the $m$-fold tensor product (over $\mathbb{C}$) of $\mathbb{C}[s, t]_{d}$. That is, $S_d$ consists of polynomials in $s^1, t^1, s^2, t^2, \dots, s^m, t^m$ of multidegree ($d, d, \dots, d$). And $S_m$ acts in the obvious way, i.e. if $\sigma(i) = j$, then $\sigma$ sends $s^i$ to $s^j$ and $t^i$ to $t^j$. The homogeneous coordinate ring of $(\mathbb{CP}^1)^m/S_m$ is just the ring of invariants of the $S_m$ action on $\mathcal{S}$. But I still have two questions.   Which polynomials are invariant under this action? I still don't see how to conclude that Eric Wofsey's map is a homeomorphism. How do I see that Eric Wofsey's map is a homeomorphism?