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Here is my favourite way to motivate the Zariski topology: it is the coarsest topology which makes the functions defined (below) by ring elements "continuous" in the following sense:

Given a classical variety $V$ over $\mathbb{C}$ and a "regular function" $f:V\to\mathbb{C}$, one can identify the value
$f(x)$ with the the image of $f$ in $A_V/m_x$, where $A_V$ is the coordinate ring of $V$ and $m_x$ is the maximal ideal at $x$. This perpsective has the advantage of generalizing to any ring, if you allow the target field to vary from point to point:

First, motivate working with primes instead of maximal ideals because primes pull back under ring maps, and because non-maximal primes act like "generic points" in classical algebraic geometry.

Next, at each prime ideal of a ring $p\triangleleft A$, you get a domain $A/p$ (which people often like to think of as living inside a residue field, $k(p):=Frac(A/p)$). Then an element of the ring $a\in A$ defines a function $f_a$ on $Spec(A)$ taking values in various domains (or fields): $f_a(p):=image_{A/p}(a)$.

All domains/fields have the element $0$ in common, so it makes sense to talk about the vanishing set
$f_a^{-1}(0)=V(a):=$ {$p\in Spec(A) | a\in p$}, and these sets form a base for the closed sets of the Zariski topology.

Moreover, the finite unions and arbitrary intersections we need turn out to be extremely manageable because of the definition of primes, in a way that is intuitively meaningful in the context above: For any collection of basic closed sets $V(a)$ with $a$ ranging over a set $E\subset Spec(A)$, we get

  • $\bigcap_{a\in E} V(a) = V(E) := $ {$p\in Spec(A) | E \subset A\setminus P $}. These are the primes where
    "not all of $E$ vanishes" in the residue domain/field.

  • $V(a)\cup V(b) = V(ab)$, the primes where $a$ and $b$ "both vanish".
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Here is my favourite way to motivate the Zariski topology: it is the coarsest topology which makes the functions defined (below) by ring elements "continuous" in the following sense:

Given a classical variety $V$ over $\mathbb{C}$ and a "regular function" $f:V\to\mathbb{C}$, one can identify the value
$f(x)$ with the the image of $f$ in $A_V/m_x$, where $A_V$ is the coordinate ring of $V$ and $m_x$ is the maximal ideal at $x$. This perpsective has the advantage of generalizing to any ring, if you allow the target field to vary from point to point:

First, motivate working with primes instead of maximal ideals because primes pull back under ring maps, or motivate and because non-maximal primes as act like "generic points"points" in classical algebraic geometry. Now

Next, at each prime ideal of a ring $p\triangleleft A$, you get a domain $A/p$ (which people often like to think of as living inside a residue field, $k(p):=Frac(A/p)$). Then an element of the ring $a\in A$ defines a function $f_a$ on $Spec(A)$ taking values in various domains (or fields): $f_a(p):=image_{A/p}(a)$, which can be thought of inside various fields $k(p)$, whatever you prefer.f_a(p):=image_{A/p}(a)$.

All domains/fields have the element $0$ in common, so it makes sense to talk about the zero vanishing set
$f_a^{-1}(0)=V(a):={p\in f_a^{-1}(0)=V(a):=$ {$p\in Spec(A) | a\in p }$, p$}, and these sets form a base for the closed sets of the Zariski topology.

Moreover, the finite unions and arbitrary intersections we need turn out to be extremely manageable because of the definition of primes, in a way that is intuitively meaningful in the context above: For any collection of basic closed sets $V(a)$ with $a$ ranging over a set $E\subset Spec(A)$, we get

  • $\bigcap_{a\in E} V(a) = V(E) := $ {$p\in Spec(A) | E \subset A\setminus P $}. These are the primes where
    "not all of $E$ vanishes" in the residue domain/field.

  • $V(a)\cup V(b) = V(ab)$, the primes where $a$ and $b$ "both vanish".
show/hide this revision's text 2 fixed some errors; added 9 characters in body

Here is my favourite way to motivate the Zariski topology: it is the coarsest topology which makes the functions defined (below) by ring elements "continuous" in the following sense:

Given a classical variety $V$ over $C$ \mathbb{C}$ and a "regular function" $f:V\to\mathbb{C}$, one can identify the value
$f(x)$ with the the image of $f$ in $A_V/m_x$, where $A_V$ is the coordinate ring of $V$ and $m_x$ is the maximal ideal at $X$. x$. This perpsective has the advantage of geralizing generalizing to any ring, if you allow the codomain target field to vary from point to point:

At

First, motivate working with primes instead of maximal ideals because primes pull back under ring maps, or motivate non-maximal primes as "generic points". Now, at each prime ideal of a ring $p\triangleleft A$, you get a domain $A/p$ (which people often like to think of as living inside a residue field, $k(p)=Frac(A/p)$. k(p):=Frac(A/p)$). Then an element of the ring $a\in A$ defines a function $f_a$ on $Spec(A)$ taking values in various fieldsdomains (or fields): $f_a(p):=image_{k(p)}(a)$.

All f_a(p):=image_{A/p}(a)$, which can be thought of inside various fields $k(p)$, whatever you prefer.

All domains/fields have the element $0$ in common, so it makes sense to talk about $f_a^{-1}(0)$, which should be closed if we want $f_a$ to look like continuous fuction. But the zero set
$f_a^{-1}(0)=V(a):={p\in Spec(A) | a\in p }$, and these sets form a base for the closed sets of the Zariski topology.
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