The map I: from points back to polynomials
Guide 1 sent ideals to point-sets via V. Now reverse the arrow. For any subset X ⊆ A^n, define I(X) = { f in k[x_1, …, x_n] : f(p) = 0 for all p in X }. This is the ideal of the variety X — the collection of all polynomial relations the points of X satisfy. It is genuinely an ideal: if f and g vanish on X so does f + g, and so does h·f for any polynomial h. See ideal of a variety.
The two operations almost invert each other. Always V(I(X)) ⊇ X, and equality holds exactly when X is itself a variety — so V(I(X)) is the smallest variety containing X, its closure. And always I ⊆ I(V(I)). The exact gap in that second inclusion is the whole content of the Nullstellensatz, deferred to guide 4.
The coordinate ring: functions on a variety
Two polynomials restrict to the same function on X exactly when their difference lies in I(X). So polynomial functions on X form the quotient ring k[X] = k[x_1, …, x_n] / I(X), the coordinate ring of X. Its elements are the regular functions — the polynomial-valued observables you can measure on the variety. This ring is the algebraic avatar of the geometric object: everything about X is encoded in k[X].
Example: the parabola X = V(y - x^2) in A^2.
k[X] = k[x, y] / (y - x^2).
In the quotient, y == x^2, so every class has a unique
representative that is a polynomial in x ALONE:
a0 + a1*x + a2*x^2 + ... (replace each y by x^2)
Hence the map k[X] -> k[t], x |-> t, y |-> t^2, is an
ISOMORPHISM. Geometrically: the projection (x,y) |-> x
identifies the parabola with the line A^1.
Contrast: the node Y = V(y^2 - x^3) in A^2.
k[Y] = k[x, y] / (y^2 - x^3).
Now x and y are tangled: y^2 = x^3 cannot be solved away,
and t |-> (t^2, t^3) maps A^1 ONTO Y but is NOT an
isomorphism of rings -- k[Y] sits inside k[t] as the
subring k[t^2, t^3] (missing the element t).
The ring SEES the cusp at the origin that the picture shows.The Zariski topology
The closure properties of V from guide 1 are exactly the axioms for closed sets of a topology: A^n and ∅ are varieties, finite unions of varieties are varieties, and arbitrary intersections of varieties are varieties. Declaring the varieties to be the closed sets defines the Zariski topology on A^n (and on any variety, by restriction). Read Zariski topology.