14 RICHARD BORCHERDS

5. Lecture 5 The spectrum of a ring So we finally have:

Definition 5.1. The spectrum Spec(R) of a ring is a topological space whose points are the prime ideals and whose closed sets are the sets Z(I) of prime ideals containing a given ideal I. (Actually the spectrum of a ring also has the extra structure of a locally ringed space, but we temporarily ignore this.)

We should check this really is a topology: the main point is that Z(I) ∪ Z(J) = Z(IJ) so closed sets are closed under finite unions. This is another way of saying that if a prime ideal contains the ideal IJ then it contains I or J (exercise). The intersection of any collection of sets Z(Ik) is the closed set of the ideal generated by all the ideals Ik. A base for the topology is given by the sets Uf consisting of the prime ideals not containing f. It is easy to check that any homomorphism of rings from R to S induces a continuous map of Spec(S) to Spec(R); note the change in direction. Examples. The spectrum of the zero ring is empty. The spectrum of a field is just a point. Example. If R is the polynomial ring C[x] then the maximal ideals (x − a) just correspond to complex numbers a so the maximal spectrum is the complex line with a weird topology: the nonempty open sets are those with finite complement. Similarly the maximal spectrum of the ring C[x, y, z] is C3 with a strange topology, and so on. The spectrum of the integers consists of (p) for p prime or 0. Note that the point (0) is not closed: in fact its closure is the whole space. In other words you should pretty much forget everything you may have learned about topology in an analysis course. The spectrum of the ring of integers of an algebraic number field is similar. In fact the spectrum of any Dedekind domain with a countable infinite number of primes, such as polynomials over a finite or countable field, will look the same. Example. Z[i]. There is a homomorphism from Z to Z[i] inducing a map from Spec Z[i] onto Spec Z. The inverse image of a point is 1 or 2 points depending on whether p is 1 or 3 mod 4. Example: C[x]. The prime ideals are (0) and the maximal ideals (x−a) for complex numbers. So the spectrum contains the complex plane with the finite complement topology, together with a point (0) whose closure is the whole space. Example: R[x]. The spectrum contains (0) and the ideals (x − a) for real R just as for C[x]. However these are not all the maximal ideals: we also get maximal ideals (x2 + bx + c) for quadratic irreducible polynomials. These can be identified with their roots: a pair of complex conjugate complex numbers. So the spectrum has a point (0) whose closure is the whole space, and closed points corresponding to the complex plane folded in half. In general for a field k the closed points of the spectrum of k correspond to obits of points in the algebraic closure under the Galois group. Example: C[x, y]. We have closed points given by the maximal ideals (x−a, y−b) corresponding to points (a, b) of C2. (We will see later that these are all the closed points.) There is a point (0) whose closure is the whole space. There are also further MATH 250B: COMMUTATIVE ALGEBRA 15 points (f(x, y)) for f an irreducible polynomial, corresponding to irreducible curves. Exercise: What is the closure of the point (f(x, y))? Example: Z(p) The ring Z(p) is the ring of all rational numbers without p in their denominator. The only ideals are (0) and (pn), and the only prime ideals are (0) (an open point) and (p) (a closed point). So the spectrum is the 2-point space with 1 closed point (the Sierpinski space). The spectrum of any discrete valuation ring, such as the p-adic numbers or formal power series over a field, is the same. Example: Z[x]. The injection from Z to Z[x] gives a continuous map from Spec(Z[x]) to Spec(Z). The fiber over (p) is isomorphic to the spectrum of Fp([x]). More precisely, its points correspond to homomorphisms from Z[x] to fields gener- ated over Fp by one element. It has a generic point (p) (corresponding to Fp(x)) and closed points (p, f) where f is an irreducible polynomial over Fp corresponding to Fp[x]/(f). Similarly the fiber over (0) is Spec Q[x] and has a generic point (0) and further points (f) for irreducible polynomials in Z[x] that we can picture as orbits of algebraic numbers under the absolute Galois group of Q. These are not closed points. For example, the closure of (5x − 1), which we think of as the point 1/5 in Q, also contains a point over the fiber of (p) for any p 6= 5, giving the inverse of 5 mod p. So the spectrum of Z[x] is sort of 2-dimensional, and the closure of points can be thought of as ”horizontal curves” while the fibers are ”vertical curves”. For example (x2 + 1) gives another curve describing the square roots of 1 mod p. It intersects the first curve in 2 closed points: (13, x − 5) and (2, x − 1). Example. A Hecke ring. A Hecke ring is generated by the Hecke operators Tn on a space of modular forms. We will look at the case of modular forms of level 1 and weight 12. The space of such forms is 2-dimensional, spanned by the Eigenforms P n E12 = (691/65520) + σ11(n)q and ∆. The Hecke operator Tn has eigenvalue cn on any eigenform whose coefficient of q is 1. We now find the spectrum. there is a homomorphism into Z × Z given by taking Tn to (σ11(n), τ(n), the eigenvalues on the eigenforms. This gives a map from SpecZ, a union of two ”lines”, onto the spectrum of the Hecke algebra. The Hecke algebra is the set of pairs (a, b) in Z2 that are congruent mod n, where n is the largest number such that the coefficients of the normalized eigenforms are always equivalent mod n. The spectrum of this ring is 2 copies of the spectrum of Z joined at the primes dividing n. Looking at the smallest coefficients ∆ = q−24q2 +252q3 +··· and ∗+q+2049q2 + P λ2/2 ··· shows that n is 1 or 691. The theta function λ q of any even unimodular lattice is a modular form of level 1 and weight half its dimension. In particular in dimension 24 there is such a lattice with no norm 2 vectors called the Leech lattice Λ. Looking at coefficients of 1 and q we have ΘΛ = (65520/691)(E12 − ∆) giving Ramanujan’s congruence τ(n) ≡ σ11(n) mod 691. So the spectrum is a union of two copies of SpecZ corresponding to the two eigenforms. They joined at the point (691), and this intersection point occurs because of the existence of the Leech lattice. Recent news: Maryna Viazovska proved that the Leech lattice is the densest packing in 24 dimensions.) This gives an example of an ”Eisenstein prime”: roughly an intersection of the component corresponding to an Eisenstein series with something else.