TOPICS IN COMMUTATIVE ALGEBRA: MULTIPLICITIES Mel Hochster Math 615: Lecture of January 7, 2015 In these lectures, all rings are assumed to be commutative, associative, with multiplica- tive identity denoted 1, which may be subscripted with the letter denoting the ring if precision is needed. Ring homomorphisms R ! S are assumed to map 1R 2 R to 1S 2 S. Modules M over a ring R are assumed to be unital, i.e., 1 · u = u for all u 2 M.A local ring is a Noetherian ring with a unique maximal ideal. The statement that (R; m) is local means that R is a local ring with maximal ideal m. The statement that (R; m; K) is local means that R is local with maximal ideal m and residue class field K = R=m. We use N ⊆ Z ⊆ Q ⊆ R ⊆ C for the nonnegative integers, the integers, the rational numbers, the real numbers, and the complex numbers, respectively. We give an overview of some the material that will be covered near the beginning of this set of lectures. We will study various notions of multiplicity that are currently objects of study within commutative algebra and the relationships among them. These include the usual Hilbert- Samuel notion, intersection multiplicities in the sense of Serre, and Hilbert-Kunz multi- plicities. Consider a local ring (R; m; K) of Krull dimension d and an m-primary ideal I. The condition on I simply means that for some N, mN ⊆ I ⊆ m. The function `(R=In), where ` is length, is called the Hilbert function of I, and agrees with a polynomial in n of degree d for all n 0. The polynomial is called the Hilbert polynomial. Its leading term will have e d the form d! n where e is a positive integer. The integer e is called the multiplicity of I. Note that e can be obtained as a limit: `(R=In) e = d! lim : n!1 nd The multiplicity of m is called the multiplicity of R. Under mild assumptions, a local ring of multiplicity 1 is regular. (One can also study `(M=InM) for any finitely generated R- module M. This is gives the Hilbert function and Hilbert polynomial of M: one difference is that in the module case, the degree of the Hilbert polynomial is the dimension of M.) These multiplicities can be obtained in other ways. Given a sequence of elements x = x1; : : : ; xd of ring R, and an R-module M, we shall define a homology theory called Koszul homology: the Koszul homology modules are denoted Hi(x; M). (They vanish if i < 0 or if i > d.) We won't give the definition at this point. Koszul homology has many uses, including the proofs of the fundamental facts about behavior of cohomology of coherent 1 2 sheaves on projective space. The connection with the multiplicities defined in the preceding paragraph is this. If x1; : : : ; xd is a system of parameters for the local ring R, which simply means that d = dim (R) and I = (x1; : : : ; xd)R is m-primary, then the multiplicity of the Pd i ideal I is the same as i=0(−1) ` Hi(x; R) , the alternating sum of the lengths of the Koszul homology modules, which do turn out to have finite length in this situation. Recently, other asymptotic invariants have been studied intensively, including several that are defined in prime characteristic p > 0 using the Frobenious endomorphism. If R is a ring of prime characteristic p > 0 the Frobenius endomorphism F : R ! R is defined n by F (r) = rp. Thus, its n th iterate F n sends r 7! rp . If q := pn and I is an ideal of R, the expanded ideal ISn, where Sn = R is thought of as an R-algebra using the structural homomorphism F n : R ! R, is denoted I[q] when thought of as an ideal of R. Said differently, I[q] is the ideal of R generated by all q th powers of elements of I. (It does not consist only of the q th powers of elements of I: typically, that set is not closed under multiplication by elements of R.) I[q] is generated by the q th powers of any set of generators of I. If m is a maximal ideal of the Noetherian ring R, and I is m-primary, by analogy with the Hilbert-Samuel function one is led to study the Hilbert-Kunz function [pn] `(R=I ). (This does not change if we replace R by Rm. When R is an N-graded ring L1 such that R0 = K, a field, we take m to be the homogeneous maximal ideal i=1 Ri unless otherwise specified.) Monsky proved the first substantial results about this function. It is asymptotic to cqd, where d is the Krull dimension of R and c is a positive real constant. More precisely, it is the sum of cqd and a function bounded in absolute value by a positive constant times qd−1, i.e., it is cqd + O(qd−1). Thus, `(R=I[q]) c = lim : q!1 qd Early examples suggested that c is rational, but this turns out not to be true in general. It is now believed that c can even be transcendental. c is called the Hilbert-Kunz multi- plicity of R with respect to I. If I = m, it is called the Hilbert-Kunz multiplicity of R, and denoted eHK(R). It is very hard to calculate. E.g., for n ≥ 3, it is not known what its value is if R is the ring K[Xij : 1 ≤ i; j ≤ d]m=(D) where the numerator is a polynomial ring over a field K of positive characteristic generated by the entries of an n×n matrix Xij of indeterminates, and the denominator is the principal ideal generated by the determinant D of that matrix. (The maximal ideal we use is generated by the images of the Xij.) Again, under mild conditions, R is regular if and only if eHK(R) = 1. We shall also discuss a notion of intersection multiplicity for two varieties intersecting in an isolated point (in the Zariski topology). The simplest situation of some interest is when 2 2 2 one intersects V (y) and V (x − y ) in the plane K = AK over an algebraically closed field, which we assume for simplicity has characteristic different from 2. The coordinate ring of the plane is K[x; y]. For the purpose of this example it is perfectly fine to take K = C. There are several points of view from which one wants to think of the point of intersection 3 at the origin as having multiplicity 2. Note that if one replaces y by y − (think of as \small" scalar) there really are two points of intersection unless = 0. Another point of view is that the the curves V (y) and V (y − x2) share more than a point of intersection at the origin: they have a common tangent. Yet another way to think about this is that, in the theory of schemes, the scheme-theoretic intersection of V (I) and V (J) corresponds to V (I + J). When I + J is not radical, taking the radical discards useful information. The nilpotents correspond, heuristically, to the fact that there is more to the intersection than one perceives when only thinking about the underlying sets. In the example above, K[x; y]=(I + J) =∼ K[x; y]=(y; y − x2) =∼ K[x]=(x2) =∼ K + Kx, a two-dimensional vector space over K. The dimension of this vector space gives the intersection multiplicity. To exclude other points of intersection, one may work with the local ring K[x; y]m, where m is the maximal ideal corresponding to the point of intersection. In this example, m = (x; y). Quite generally, if f, g are irreducible in K[x; y] and do not generate the same ideal (i.e., neither is a scalar multiple of the other), then the intersection multiplicity of V (f) and V (g) at an isolated point of intersection is the vector space dimension of K[x; y]m=(f; g). Note ∼ that if R = K[x; y]m, then R=(f; g) = R=fR ⊗R R=gR. That is, intersection corresponds to tensor product. This notion of intersection multiplicity at an isolated point in the affine plane also gives, 2 a notion for irreducible curves in the projective plane PK , since the isolated point will be 2 contained in an affine open set isomorphic with AK , and we may study the intersection there. In fact, as in the affine case, we wind up working in the local ring at the point of intersection. With this notion of intersection multiplicity one gets Bezout's theorem: the number of points of intersection of V (F ) and V (G), counting multiplicities, where F; G are homogeneous irreducible polynomials in three variables, so that V (F );V (G) are irreducible 2 curves in PK , is deg(F ) deg(G). In higher dimension, the dimension of the tensor product is not the right notion to use. One needs to take account of higher Tors, and we shall use this fact to provide motivation for the study of Tor. Thus, in developing the material on multiplicities we shall introduce and use a number of powerful homological techniques, including derived functors such as Tor and Ext, and the theory of spectral sequences. Spectral sequences take some getting used to, but often provide the right way to look at a problem.