Turgul said:
In reading your questions it is clear that a big picture understanding of what is going on has been elusive. I don't have a copy of Dummit & Foote on hand to remember what exactly they say, but let's see if we can't shed some light on the situation.
In algebraic geometry, we are interested in studying "algebraic sets" as objects of geometric interest. What is an algebraic set? It is the zero set of a collection of polynomials. For example all of the points $(x,y)$ satisfying $y=x^2$ form an algebraic set because they are the solutions of $y-x^2=0$. Another example of an algebraic set is $\{(0,0),(1,1)\}$ because it is the common zeroes of the polynomials $y-x=0$ and $y-x^2=0$, that is both of these points are zeroes of both polynomials at the same time, and no other points have this property.
There are a couple of things to note above:
First, in the above examples all points were ordered pairs and all polynomials were in two variables. That tacitly assumes that I'm living in an ambient two dimensional space (the space of all possible ordered pairs $(x,y)$ with no restrictions on $x$ or $y$ at all). Note that in this space, the equation $x=0$ cuts a line of points $(0,y)$ where $y$ can be anything. If we were living in one dimensional space, then $x=0$ would cut out precisely the point $x=0$. This means we should pay at least some attention to where we are looking for zeroes of our polynomials. At some point, we would like to realize that "one-space" lives inside of "two-space" lives in "three-space" and so on, but some care must be taken as to what we mean when we say that, and it's a more technical point to be puzzled later.
Second, polynomials have different solutions over different rings. Consider $x^2+1=0$. Over the real numbers, this has no zeroes at all! Over the complex numbers, we have $x = i$ and $x = -i$, and over the field of two elements, we have the single solution $x=1$. So if we are looking for solutions to polynomials, we have to pick a ring to look in! For practical purposes, we often restrict our attention to fields to make our lives easier (in practice, we actually often further assume that the field is algebraically closed, but there is something to be learned about how that matters, so we'll avoid that assumption for now), which is what D&F certainly do and we will as well, for the time being.
For now, let's restrict our attention to algebraic sets sitting inside of two-dimensional space over a fixed field $k$. Algebraic sets are just special kinds of sets, we can ask if they have nice properties in terms of typical set operations. Consider the polynomials $y-x=0$ and $y-x^2=0$. One cuts out a line, the other a parabola. If I intersect the two shapes, I get the two points from before. That is to say, if I require that points satisfy both at the same time, I get the intersection of the shapes. Since algebraic sets are allowed to be simultaneous zero sets of multiple polynomials (we don't even require finitely many!), this operation is allowed.
On the other hand, what happens if we look at $(y-x)(y-x^2)=0$, the zero set of the product of the two polynomials? Since we live over a field, if the product of two things is zero, either one or the other must be zero, so if $(a,b) \in k^2$ with $(b-a)(b-a^2)=0$, it must be that $b-a = 0$ or $b-a^2=0$. Hence the zero set of $(y-x)(y-x^2)=0$ is contained in the union of the line and the parabola. On the other hand, any point on either the line or the parabola will be a zero of the product. So by looking at the zeroes of the product polynomial, we get the union of the first two!
What's more is that if $A$ is an algebraic set defined by $f_1(x,y)=0,f_2(x,y)=0,\ldots$
and $B$ is defined by $g_1(x,y)=0,g_2(x,y)=0,\ldots$, the the intersection $A \cap B$ is defined by $f_1(x,y)=0,g_1(x,y)=0,f_2(x,y)=0,g_2(x,y)=0, \ldots$ the common zeroes of all the polynomials defining each. On the other hand, the union $A \cup B$ is defined by $f_1(x,y)g_1(x,y)=0,f_1(x,y)g_2(x,y)=0,f_2(x,y)g_1(x,y)=0,f_2(x,y)g_2(x,y)=0, \ldots$ the products of every polynomial defining $A$ by each polynomial defining $B$. You should convince yourself of this.
Now, with many polynomials running around, this gets very complicated quickly, so it is natural to wonder if there is a better way of managing the collections of polynomials. Fortunately, there is. It turns out that the ideal of $k[x,y]$ generated by all of the polynomials defining an algebraic set defines the same algebraic set! At first glance this just seems to have made things worse. After all, if we defined a set using two polynomials, the ideal generated by them has infinitely many polynomials! The great simplification comes when we realize that all that really matters are the generators of the ideal, so we can pick our favorite generators for the ideal and do all the computations with them. Instead of looking at the zero set of $xy,x^2y,xy^2,x^3y^4$, we can just look at the zero set of $xy$ since it generates the same ideal as the other four polynomials!
I suspect this gets overwhelming, so instead of rambling on, I will leave you with the following question:
The first thing to consider before you attack your problem is why is the union of algebraic sets given by the common zeroes of the products of the defining polynomials of each? That is, instead of considering $m_1$ and $m_2$, consider just the pairs of polynomials $x,y$ and $x-1, y-1$ defining the two points $(0,0)$ and $(1,1)$ seperately. Why are the two points together defined by the four polynomials $xy,x(y-1),(x-1)y,(x-1)(y-1)$?
Your problem is just a supped up version of this in terms of ideals instead of polynomials.
Thanks Turgil, you certainly have helped my understanding of affine algebraic sets, but I still have a way to go in trying to get an understanding of the basics of algebraic geometry ,,,
In this post, I will try to address your question:
" ... ... why is the union of algebraic sets given by the common zeroes of the products of the defining polynomials of each?"
Since the exercise deals in 2 dimensions let us consider your question in $$ k[x_1, x_2] $$.
In Dummit and Foote's terminology (see attachment)
In $$ k[x_1, x_2] $$ we have
$$ V_1 = \mathcal{Z} (S) $$
$$ = \mathcal{Z} ( \{ f_1, f_2, ... \ ... f_s \} )$$
$$ = \{ (a_1, a_2) \in \mathbb{A}^2 \ | \ f_i (a_1, a_2) = 0 \ \forall \ f_i \in S \} $$
and
$$ V_2 = \mathcal{Z} (T) $$
$$ = \mathcal{Z} ( \{ g_1, g_2, ... \ ... g_s \} )$$
$$ = \{ (a_1, a_2) \in \mathbb{A}^2 \ | \ g_i (a_1, a_2) = 0 \ \forall \ g_i \in T \} $$
Then for $$ V_1 \cup V_2 $$ we have
$$ f_i (a_1, a_2) \times g_j (a_1, a_2) = 0 \ \forall \ i, j $$ and $$ \forall \ (a_1, a_2) \in V_1 \cup V_2 $$ since if $$ (a_1, a_2) \in V_1 $$ then $$ f_i (a_1, a_2) = 0 $$ and if $$ (a_1, a_2) \in V_2 $$ then $$ g_j (a_1, a_2) = 0 $$.
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I am still puzzling a bit over your next suggestions since to apply the above re affine algebraic sets $$ V_1, V_2 $$ I would be looking for two affine algebraic sets (again in D&F's terminology - see attachment) , say, $$ V_1 = \mathcal{Z} (S_1) , V_2 = \mathcal{Z} (S_2) $$ where $$ S_1 = \{ f_1, f_2, ... \ ... f_{s_1} \} , S_2 = \{ g_1, g_2, ... \ ... g_{s_2} \} $$.
But when we come to $$ V= \{ (0,0) , (1,1) \} $$, although it looks like we could express this as $$ V = V_1 \cup V_2 $$ where $$ V_1 = \{ (0,0) \} $$ and $$ V_2 = \{ (1,1) \} $$ it does not appear intuitive to do this as the polynomial $$ f_1 = x-y $$ defines an affine algebraic set $$ V_1 \cup V_2 = \{ (0,0) , (1,1) \} $$ as does $$ f_2 = y - x^2 $$ as do some other polynomials like $$ f_3 = y^2 - x $$ and $$ f_4 = x^2 - y $$.
However, I am still thinking over this ...
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You advise me to consider the four polynomials $$ g_1 = xy, g_2 = x(y-1), g_3 = (x-1)y , g_4 = (x-1)(y-1) $$
Inspection reveals that in $$ \mathbb{F}_2 $$:
$$ g_1 = xy = 0 $$ at the points $$ (0,0), (0,1), (1,0) $$
$$ g_2 = x(y-1) = 0 $$ at the points $$ (0,0), (0,1), (1,1) $$
$$ g_3 = (x-1)y = 0 $$ at the points $$ (0,0), (1,0), (1,1) $$
$$ g_4 = (x-1)(y-1) = 0 $$ at the points $$ (0,1), (1,0), (1,1) $$
Thus in D&F's terminology (see attachment)
$$ V_1 = \mathcal{Z} (g_1) = \{ (0,0), (0,1), (1,0) \} $$
$$ V_2 = \mathcal{Z} (g_2) = \{ (0,0), (0,1), (1,1) \} $$
$$ V_3 = \mathcal{Z} (g_3) = \{ (0,0), (1,0), (1,1) \} $$
$$ V_4 = \mathcal{Z} (g_4) = \{ (0,1), (1,0), (1,1) \} $$
SO just seeing how $$ g_1 = xy, g_2 = x(y-1), g_3 = (x-1)y , g_4 = (x-1)(y-1) $$ might define $$ \{ (0,0), (1,1) \} $$
Any comments on the above?
Thanks again for your post, it was most helpful!
Peter