Week 1, Two Classes.)

(Week 1, two classes.)

Abstract. This is the note for Math 511 that I am teaching in UIUC in 2009. The class is moved to 9-10:20 TR in room 57 Everitt.

1. Goal of the lecture.

Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is considered to be one of the greatest mathematicians of the 20th century. He is the chief designer of modern algebraic geometry. Now he is almost 81, and it is my great honor to present here an introduction to modern algebraic geometry which has been greatly influence by his work. Some textbook on introduction of algebraic geometry (or any field of mathematics) tries to minimize or erase the trace of history. This is some sense is reasonable: a science should be coherent and be independent of who have discovered it. Calculus is a typical example. We barely remember who have contributed to the development of it. Calculus has been born for more than 300 years if we count from Newton and Leibniz, and it becomes so mature and standard that it is easily accessed by a high school student. But algebraic geometry, on the other hand, is not old enough to forget about how it was created by force, and in my point of view, is not yet mature. (One evidence is that we are still stuck in curve, surface and dimension 3 manifold and barely know anything in higher dimensional; the other evidence is that the new objects kept born: manifolds, varieties, schemes, algebraic spaces, stacks, etc; third evidence, I did not successfully explained to my wife what is algebraic geometry, and I believe that algebraic geometry is far from accessible to general audience.) So in this class we follow instead the order of history, and that is roughly what Hartshorne did in his book. Start from varieties, and when we are familiar with their properties, we proceed to schemes. My suggestion for the class: be skeptical to the material, always ask me or yourself, if it is necessary, is there a better approach? On the other hand, you should be open minded, do not refuse to accept a concept simply because it is complicated or ugly. You might appreciate it when you are used to it :-) This class is not complex algebraic geometry, hence almost no transcendental method will be introduced, and I believe you have learned or will learn something like Hodge theory in other classes. Exercise are essential to understand the material. Some exercises you should do it at least once in your lifetime instead of simply believe it. Asking the following: affine varieties and projective varieties, Zariski topology, rational maps, birational equivalence, blow-up, valuation rings, Hilbert polynomials, sheaves, schemes, coherent sheaves, divisors, invertible sheaves, ample line bundles(invertible sheaves), linear system, derived functors, cohomology of a sheaf, Cech cohomology, Ext functor and Serre duality, higher direct images. CHAPTER 1

Varieties

1. Aﬃne varieties

It is simply the set of points that is defined by one or several polynomial equations. For example, let k be your favorite field, consider a parabola in k2 defined by y − x2 = 0. The polynomial y − x2 is in k[x, y]. We should expected that to study the zero locus of a polynomial is closely related to the study of the polynomial in the ring k[x, y]. Indeed, to understand the interplay between the geometry and algebra is the central goal of algebraic geometry. We will see today such a relation. n Let k be an algebraic closed field, the most common choice is C. We call it the base field. Ak (or n n n A ) is the affine n-space, or k if you like. The polynomials on A form a ring, say A = k[x1, . . . , xn], n it is called the coordinate ring of A , xi are called coordinates. Given any subset T of A, we define the zero set of T to be the common zeros of all elements in T , i.e. n Z(T ) = {P ∈ A |f(P ) = 0, ∀f ∈ T } It is obvious that if a is the ideal generated by T , then Z(a) = Z(T ). Example:(1) x2 + y2 = 0. (2) x and x2. (3) the defining equations for (t3, t4, t5). n Definition 1.1. A subset Y of A is an algebraic set if Y = Z(T ) for some subset T ⊆ A.

Remark:T = ∅, T = (1), etc. Proposition 1.2. The union of ﬁnite many algebraic set is an algebraic set. The intersection of any family of algebraic set is algebraic set.

Example:The union of a point (0, 0) and the line x = 1. Now notice each algebraic set is determined by not necessarily unique subset T . For example, (x, y) and (x, y2) and (x2, xy, y2) in k[x, y]. Can one specify a canonical subset? The answer is yes by the following. n Theorem 1.3. There is a one-one correspondence between algebraic sets in A and radical ideals in A, given by Y 7→ I(Y ) := {f ∈ A|f(P ) = 0, ∀P ∈ Y } and a 7→ Z(a). √ Proof. Enough to show that (1) for any ideal a, I(Z(a) = a and (2) for any algebraic set Y , Z(I(Y )) = Y . For (2), if not equal, then ∃f ∈ I(Y ), z ∈ Z(I(Y )), s.t. f(z) 6= 0, this is impossible. (1) is the famous Hilbert’s Nullstellensatz. Theorem 1.4 (Hilbert’s Nullstellensatz (German: “theorem of zeros”)). Let k be an algebraically closed ﬁeld, a be an ideal in A = k[x1, . . . , xn], then √ I(Z(a)) = a.

3 4 1. VARIETIES

A special case: when a is maximal, hence proper, Z(a) is not empty, suppose (a1, . . . , an) ∈ Z(a), then (x1 − a1, . . . , xn − an) ⊇ a, by the maximality of a, equality holds. Theorem 1.5 (Weak Nullstellensatz). Let k be an algebraically closed ﬁeld, the maximal ideal in A = k[x1, . . . , xn] are the ideals (x1 − a1, . . . , xn − an).

Remark:Note it is not true when k is not algebraically closed, say R, the nullstellensatz is not 2 true. Eg. (x + 1) is maximal in R[x].

On the other hand, the√ weak Nullstellensatz implies the strong one, by the following proof. It is easy to see I(Z(a)) ⊇ a, so we need to show the inclusion in the other direction. If we take a ﬁnite set of generators f1, . . . , fm of a, then we need to show that, if g ∈ k[x1, . . . , xn] vanishes at ` P the zero locus f1 = ··· = fm = 0, then g = hifi.

Here is the lifting trick of Zariski: consider k[x1, . . . , xn+1]. Deﬁne ideal

b = ak[x1, . . . , xn+1] + (1 − gxn+1). We ﬁrst show that b must be (1). Otherwise it is proper, by weak Nullstellensatz it is in a maximal ideal m = (x1 − a1, . . . , xn+1 − an+1). Then 1 − gxn+1 is in the maximal ideal m, so are f1, . . . , fm. But this is impossible since f1 = ··· = fm = 0 at (a1, . . . , an) implies g(a1, . . . , an) = 0.

Now since 1 ∈ b, 1 = h1f1 +···+hmfm +hm+1(1−gxn+1). Plug in xn+1 = 1/g, then eliminate ` P 0 the denominator, we will obtain g = hifi. Now we need to prove Theorem 1.5, we need Noether’s Normalization Lemma: let k be any field. R be an integral domain finitely generated over k. There are algebraically independent elements x1, . . . , xn in R, such that R is an integral extension of k[x1, . . . , xn]. (I am not suppose to say it now, but geometrically, the Lemma says that any variety is a branched covering of an affine space.)

Proof of Theorem 1.5. Let m be a maximal ideal. Then R = k[x1, . . . , xn]/m is a field. Let d be its transcendental degree over k. Then S = k[y1, . . . , yd] ⊂ R such that R is integral over S and is a field. This easily implies S itself is a field (check it!), which is only possible if d = 0.

Then R is integral over k, but k is algebraically closed (!), so R = k, hence k+m = k[x1, . . . , xn], in particular ai + mi = xi for each i = 1, . . . , n. So m ⊇ (m1, . . . , mn) = (x1 − a1, . . . , xn − an), but the right hand side is already maximal, so equality holds.

Now we introduce Zariski topology and irreducibility, and deﬁne variety.

n n Definition 1.6. The Zariski topology on A is deﬁned as follows: a closed set in A is to be n a algebraic set. The Zariski topology on an algebraic set in A is induced from the Zariski topology n of A .

(Check it is a topology.) Definition 1.7. A nonempty subset Y of a topological space X is irreducible if it cannot be expressed as the union Y = Y1 ∪ Y2 where Y1 and Y2 are closed proper subsets of Y . The empty set is not considered to be irreducible. 1. AFFINE VARIETIES 5

Example:Open subset of an irreducible space. Closure of a irreducible subset. n Definition 1.8. An affine (algebraic) variety is an irreducible closed subset of A . An open subset of an affine variety is a quasi-affine variety. n Proposition 1.9. An algebraic set Y on A is a variety iff I(Y ) is prime.

Now is more about topology: Definition 1.10. A topological space X is noetherian if it satisﬁes descending chain condition for closed subsets.

(Note that it is dual to the deﬁnition of noetherian rings or modules.) Homework: any algebraic set is noetherian. Proposition 1.11. In a noetherian topological space X, any nonempty closed subset Y can be uniquely written as a ﬁnite union Y = Y1 ∪ · · · ∪ Yr of maximal irreducible closed subsets. They are called the irreducible components of Y .

We will be very sketch about dimension: Definition 1.12. Let X be a topological space, the dimension of X is the supremum of all integer n such that there is a chain Z0 ⊂ · · · ⊂ Zn of distinct irreducible closed subsets of X.

Algebraically, we need to wait until we define the function field k(X) of X. Then it can be shown that dim X = tr.degk(X). For affine variety, k(X) is the fraction field of the coordinate ring of X. Homework: 1.1, 1.2*, 1.3, 1.6, 1.7, 1.9*.