Universität Regensburg Fakultät für Mathematik
Functoriality of Chow groups over regular base schemes
Masterarbeit im Studiengang Mathematik
Eingereicht von: Daniel Heiß Eingereicht bei: Prof. Dr. Moritz Kerz
Zweitgutachter: Prof. Dr. Walter Gubler
Ausgabetermin: 29.11.2016 Abgabetermin: 31.03.2017
Functoriality of Chow groups over regular base schemes
Daniel Heiß
Contents
1 Introduction 3
2 Preparation 11 2.1 Dimension ...... 12 2.1.1 Codimension ...... 12 2.1.2 The dimension formula ...... 13 2.1.3 Local dimension ...... 19 2.2 Homology ...... 21 2.3 Morphisms ...... 25 2.3.1 Open and dominant morphisms ...... 25 2.3.2 Finite morphisms ...... 31 2.3.3 Finitely generated locally free modules ...... 33 2.4 Length ...... 35 2.5 Order function ...... 44 2.6 Lattices ...... 45
3 S-schemes and cycles 51 3.1 S-schemes ...... 52 3.2 Relative S-dimension ...... 54 3.2.1 S-dimension and its properties ...... 54 3.2.2 Morphisms of relative dimension ...... 57 3.3 Cycles ...... 60 3.4 Proper push-forward ...... 65 3.5 Flat pull-back ...... 69 3.6 Pull-back and push-forward ...... 75
4 Rational equivalence and functoriality of Chow groups 81 4.1 Principal divisors and rational equivalence ...... 82 4.2 Proper push-forward ...... 84 4.3 Flat pull-back ...... 90 4.4 Properties of the Chow group ...... 96
References 103
1 Introduction
«From the ancient origins of algebraic geometry in the solution of poly- nomial equations, through the triumphs of algebraic geometry during the last two centuries, intersection theory has played a central role.» William Fulton
Although intersection theory has rich applications, its basic interest is – as the name suggests – to study the intersection of subschemes in a given scheme and to define an intersection product; a problem that has a long history in algebraic geometry. Possibly the first theorem in this area was Bézout’s theorem:
Theorem (Bézout). Two projective plane curves of degrees d and e intersecting transversally and having no common component meet in d · e points.
Figure 1.1: Two curves of degree 2 and 3 intersecting transversally in 6 points.
In the special case, where one of the curves is a line, this theorem boils down to the fundamental theorem of algebra for polynomials without any multiple roots. To obtain the complete fundamental theorem, we have to introduce the notion of multiplicity of a root, or – more generally in the setting of Bézout’s theorem – the multiplicity of an intersection point. Moreover, we want to intersect algebraic objects of higher dimension than curves. This leads to the following generalization of Bézout’s theorem:
– 3 – Introduction
Theorem. Let X be a smooth variety and V , W be two sub-varieties of codimen- sion p, q respectively such that their intersection is proper (i.e. every irreducible component of V ∩ W is of codimension p + q). Then there are integers – called mul- tiplicities – such that we can write the intersection product of V and W as a formal sum of the irreducible components of V ∩ W with their multiplicities as coefficients.
Finding the right definition for those multiplicities was an open problem for almost the entire first half of the 20th century until J.P. Serre suggested
∞ X(−1)i len Tori (O /I, O /J) , OX,z OX,z X,z X,z i=0 where I, J are the ideals representing V , W respectively and z is the generic point of the irreducible component. In particular, Serre showed that the intuitive guess for the multiplicity len(OZ,z) is not sufficient (in fact, it is the first summand above). Further he proved non-obvious properties like for example that this integer is strictly positive in case of a proper intersection and that it vanishes otherwise. This leaves the problem how to deal with non-proper intersection. To motivate this, we make a little journey to homology.
Consider a torus T as in Figure 1.2 and two cycle classes α, β ∈ H1(T, Z) represented by the cycles A and B. We want to define an intersection index of α and β. By intuition it should be 1, as A and B have exactly one intersection point. But con- sidering Figure 1.3, we see that the number of intersection points is not independent of the choice of the cycles representing the cycle class.
Figure 1.2: Torus with two 1-cycles Figure 1.3: Torus with other 1-cycles (Pictures by [GH])
To solve this problem, we choose an orientation on T and then define for any point of transversal intersection p the intersection index of A and B at p to be +1 if the tangent vectors to A and B in p form a positive oriented basis and −1 otherwise. Then the total intersection index of α and β is defined as the sum of all these local intersection indices. This integer depends only on the homology class of the cycles involved, as one easily verifies. Now assume given two cycles A and B do not intersect transversally, then we just “vary them a little bit”, i.e. we choose other representatives of their cycle class that
– 4 – Introduction do intersect transversally. One can show that this is always possible. This provides a bilinear pairing H1(T, Z) × H1(T, Z) −→ Z. The whole construction can be generalized to any n-dimensional compact oriented manifold M: For cycle classes α ∈ Hk(M, Z) and β ∈ H`(M, Z) we find piecewise smooth representatives intersecting transversally almost everywhere and then endow their intersection with a suitable orientation. This induces a bilinear pairing
Hk(M, Z) × H`(M, Z) −→ Hk+`−n(M, Z).
By Poincaré duality this becomes (for i = n − k and j = n − `)
i j i+j H (M, Z) ⊗ H (M, Z) −→ H (M, Z).
For details we refer to [GH, Chap. 0].
Summarizing, the idea in this journey was that, if cycles A and B do not intersect transversally, then we pass to their classes modulo homology and choose other repre- sentatives, whose intersection is transversal. This is well-defined as the intersection index depends on the homology class only. Now back to intersection theory, the challenge was to establish an analogue to the notion of homology on sub-varieties (or algebraic cycles) such that we obtain a simi- lar situation as described above. For that purpose, mathematicians searched for the right notion of equivalence of algebraic cycles. Severi was the first to propose a no- tion of rational equivalence in 1932 and Todd introduced the notion of the subgroup of cycles rationally equivalent to zero, which led to significant clarification. It took until 1956 that Chow published a paper proving that rational equivalence classes determine a well-defined intersection class using Chow’s “Moving-lemma”:
Theorem (Chow’s Moving-lemma). Let X be a non-singular quasi-projec- tive variety and V , W be two algebraic cycles on X. Then there is an algebraic cycle V 0 on X that is rationally equivalent to V and intersects W properly.
Further, it was shown that the intersection product up to rational equivalence does not depend on the choice of the cycle. However, one inconvenient consequence using the Moving-lemma is that the intersection product is not constructed as a cycle on V ∩ W (note for example using cohomology with support gives a cohomology class supported on V ∩ W ). This problem was solved later by Fulton when he replaced the Moving-lemma by the “deformation to the normal cone”.
In 1984 Fulton published a textbook about intersection theory, which became the first precise rigorous foundation of this area and is still the most important and influential textbook in intersection theory.
– 5 – Introduction
In the very first part of the book Fulton defines the notion of algebraic cycles on a variety X and the notion of rational equivalence on them. Passing to the quotient yields the Chow group CH(X). This group is graded by dimension of the cycles, L i.e. CH(X) = k∈Z CHk(X), where CHk(X) is the group of cycles of X having dimension k (the so-called k-cycles) modulo rational equivalence. Moreover, CH(X) L k can also be graded by codimension CH(X) = k∈Z CH (X). In case of X being smooth over the ground field, the Chow group is not only a graded group, but also a graded ring with multiplication
CHk(X) × CH`(X) −→ CHk+`(X) given by the intersection product, which was discussed above. It is the counter- part to the multiplication in homology on a smooth compact manifold imported by Poincaré duality from the natural ring structure on cohomology, which we have discussed during the journey to homology.
Further, the Chow group is functorial in the following sense: Let f : X → Y be a morphism of varieties. Assume f is proper, then there is an induced group homomor- phism f∗ : CH(X) → CH(Y ), called the (proper) push-forward along f. Moreover, if f is flat, it induces a homomorphism f ∗ : CH(Y ) → CH(X), called the (flat) pull-back.
In [Ful, Chap. 20], Fulton states that his theory would hold in a greater generality, namely for schemes over some fixed regular base-scheme S. This is exactly the pur- pose of this thesis. We will generalize the definitions and statements in the beginning of Fultons landmark book and prove them in the greatest generality possible.
Leitfaden
The second chapter is somehow the “algebraic footing” this thesis relies on. It has several sections and provides calculations and other facts required for the intersec- tion theory we build up from the third chapter onwards. To motivate the theory developed in this chapter, emphasis was put on the prologues to explain what the respective theory is good for and why it is important for the topics of this thesis. The fact that this “off-topic chapter” is the longest is because we outsourced every tech- nical detail into there. More concretely, we elaborate some assertions on dimension like the dimension formula given by Grothendieck in [EGA.IV, Prop. 5.6.5]. Fur- ther, we collect facts about coherent OX -modules, including their push-forward and pull-back, as well as flat and affine base-change. A longer section will be about Mor- phisms, where we provide, among other facts, some connections between dominance
– 6 – Leitfaden and images of generic points, the fact that flat morphisms (in the given setting) are open and some assertions on finite locally free morphisms. The most basic section in this chapter will be about lengths of modules. Nevertheless, it will be essential for the thesis as most proofs boil down to comparing certain lengths. This chapter concludes with the “determinant lemma”, a formula on which the proofs of both main results will heavily depend on.
In the third chapter we first define a new notion of dimension, as the usual Krull dimension is not well-behaved in the situations of relative schemes. For example there are dense subsets having a dimension strictly smaller than the surrounding space. Then we generalize Fultons definition of cycles to the situation of S-schemes. We try to do the whole theory in the greatest generality possible. In particular, we do not assume that our schemes are regular; regularity will be needed for the intersec- tion product, which is not part of this thesis. Nor do we assume our schemes are quasi-compact or separated. Taking this into account, we also allow cycles to be infinite sums (which are locally finite though).
In addition to Fultons theory, we also introduce cycles associated to coherent OX - modules, which will link this theory to K-theory, but we will not elaborate this connection in detail. Having defined the cycle group, we proceed with the two basic constructions: The proper push-forward and the flat pull-back on cycles. Both constructions are func- torial and compatible with the theory of coherent modules in the following sense: In the theory of cohomology of sheafs there is a notion of push-forward and pull-back of sheafs (such as coherent modules). We will show that taking the cycle associated to the push-forward of a coherent module yields the same result as pushing forward the cycle associated to that module and the analogue result holds for the pull-back. In the end of the third chapter we will combine both, proper push-forward and flat pull-back and elaborate some assertions on the level of cycles which will carry on to properties of the Chow group in the following chapter.
Our fourth and final chapter generalizes the definitions of principal divisors and rational equivalence, which leads to the definition of the Chow group. Then the two main results of this thesis follow: We show that rational equivalence is compatible with both proper push-forward and flat pull-back, hence there are induced such homomorphisms on the Chow group. We want to note that the main idea for the proof of the proper push-forward is the same as in [Ful], but for the proof of the flat pull-back we have chosen a completely different way. In the conclusive section, we summarize properties of the Chow group. Beyond the functoriality for proper and flat maps, we elaborate the localization sequence, which is somehow similar to excision in homology. Further we prove an analogue to the Mayer-Vietoris sequence (again showing a connection to homology). Moreover, for
– 7 – Introduction a fiber square f X / Y O O 0 g g W / Z. f 0 where f is proper and g is flat of some relative dimension we prove
∗ 0 0 ∗ g ◦ f∗ = (f )∗ ◦ (g ) .
Finally we calculate that the composition of push-forward and pull-back along a finite morphism of degree d is just the multiplication by d.
– 8 – Leitfaden
Notation and Conventions
Rings Any ring A is assumed to be unitary and commutative. Its subgroup of units will be denoted by A∗. For a prime ideal p ⊆ A, by κ(p) we mean the residue field. If A is a local ring, we denote its maximal ideal by m , if not stated otherwise. √ A Further, for any ideal I ⊆ A the radical is denoted by I. When passing from A to a quotient ring, we usually write I for the image of I in that quotient. Moreover if p ⊆ A is a prime ideal, we write ht(p) for its height.
Topology If A ⊆ B are two sets, the complement of A in B will be denoted by B \ A, or sometimes A{, if it is clear where we take the complement in. We assume that every irreducible space has a unique generic point (i.e. all spaces are sober) and if X is such an irreducible space, we write ηX for its generic point. For a subspace A ⊆ X the closure of A in X will be denoted by A, or more precisely AX , if it is not clear from the context, where we take the closure in. If x ∈ {y}, we write x C y.
Schemes For a scheme X and a point x ∈ X we write OX,x for its local ring at x and κ(x) for its residue field at x. Further, if X is integral, K(X) denotes its function field. By Xred we mean the reduced subscheme of X. Further, if we have a morphism of schemes f : X → Y , we denote by f(X) or im(f) the set-theoretic image of f and by Im(f) its scheme-theoretic image. If we have a closed subset Z ⊆ X, then – if not specified otherwise – without further mentioning we always view it as a closed subscheme endowed with the reduced structure. Finally, for immersions we sometimes will be sloppy in notions. For example, if i: Z → X is a closed immersion, by the OX -module OZ we mean i∗(OZ ).
Acknowledgements
First of all, I would like to thank my supervisor Prof. Dr. Moritz Kerz very much for providing me with the topic for this thesis, his uncomplicated guidance and for always being understanding and supportive in tough situations. Moreover, I am very thankful to Prof. Dr. Walter Gubler for his unprompted helpful hints and his interest on the progress of my thesis, as well as for teaching me Algebraic Geometry during my studies at University of Regensburg. Furthermore, a special thanks goes to Prof. Dr. Niko Naumann – without whom I never would have decided to change my major into mathematics – for his outstand- ing, well-wrought and humorous lectures during my undergraduate study. Finally, I want to express my deepest gratitude to my parents Albert and Luise Heiß and to my brothers Markus, Andreas and Florian for always being there for me and especially for being so extremely supportive in the current hard times.
– 9 –
2 Preparation
This chapter is somehow the technical framework providing various algebraic facts we need within this thesis. Some of the statements in this chapter are well-known, nevertheless for convenience of the reader we give a proof.
The first section is about dimension. The big result will be the dimension formula due to [EGA.IV, Prop. 5.6.5] stating that – under some assumptions on a morphism f : X → Y of schemes – we have
dim (OX,x) + TrDeg (κ(x)/κ(y)) = dim (OY,y) + TrDeg (K(X)/K(Y )) .
This is needed to ensure, that S-dimension behaves well with codimension. In addition we have some results on local dimension of fibers, which are needed for pulling back cycles.
The second section collects results that – in the widest sense – have to do with homology. Besides some technical lemmas from homological algebra we have a look at (quasi-)coherent OX -modules, their support, pull-back, push-forward and some base-changes. These results will be quite important, as many proofs will heavily depend on the connection between push-forward (or pull-back) of cycles associated to coherent modules and the cycles associated to the push-forward (or pull-back) of these modules.
The next section will deal with morphisms. There we will elaborate some connections between dominance of a morphism and images of generic points. Further we will use Chevalleys theorem to show that a flat morphism is open. These facts are needed in the context of the pull-back of cycles. Moreover, we will prove that we can restrict a quasi-compact morphism with a finite generic fiber to a finite morphism. This technical detail will be of importance, when we show the compatibility of push- forward with cycles associated to coherent modules, which we already mentioned a few times. The subsection on finitely generated locally free modules will define the notion of finite locally free morphisms and we will see that a finite flat morphism is finite locally free. This theory will be used for calculating the composition of push-forward and pull-back along a finite morphism.
The basic objects of study within this thesis are k-cycles. These will be formal sums of subschemes with some integer coefficients. Defining cycles associated to closed subschemes or associated to coherent modules, the coefficients will be given by some
– 11 – 2 Preparation lengths of modules. Therefore many of the central proofs in this thesis will boil down to showing some equality of lengths. All of these calculation are collected here in section four.
In the next section we use length to generalize the notion of valuation over a DVR to any Noetherian domain A of Krull dimension one. The so obtained group ho- ∗ momorphism Quot(A) → Z will be called the order function. This translates into geometry and gives an order functions w.r.t. subschemes of codimension one. These order functions will come into play when defining principal divisors, which are es- sential for the notion of rational equivalence.
We conclude this chapter with a section on lattices. There we define the notion of a lattice and the distance of two lattices. This section ends with the proof of the “determinant lemma”. This result is extremely important, as the proofs of both main theorems of this thesis – namely showing that push-forward and pull-back are compatible with rational equivalence, yielding the functoriality of Chow groups – strongly depend on the equation supplied by the determinant lemma.
2.1 Dimension
The main result will be the dimension formula given by Grothendieck, which we will need for proving that the notion of S-dimension (to be defined in Definition 3.2.1) behaves well with codimension. That fact will be essential for the whole thesis. We conclude this section with some results on fiber dimension, which are needed later in the context of pulling back cycles.
2.1.1 Codimension
In this subsection we just give two basic facts about codimension which we will use repeatedly.
Lemma 2.1.1. Let X be a topological space, Y ⊆ X an irreducible subset and U ⊆ X be an open subset such that Y ∩ U 6= ∅. Then
codim(Y,X) = codim(Y ∩ U, U).
Proof: By taking the closure in X we have an bijective inclusion-preserving cor- respondence of closed irreducible subsets of U and those of X. This yields the lemma.
– 12 – 2.1 Dimension
Proposition 2.1.2. Let X be a scheme and Y ⊆ X be an irreducible closed subset with generic point ηY . Then
codim(Y,X) = dim (OX,ηY ) .
Proof: Let U ⊆ X be an affine open subscheme with ηY ∈ U, say U = Spec(A). Then Y ∩U ⊆ U is irreducible closed and corresponds to some prime ideal p ⊆ A. The chains of closed irreducible subsets ascending from Y ∩U correspond to prime ideals in A contained in p which in turn correspond to prime ideals in Ap, hence codim(Y ∩ U, U) = dim (Ap). Now the claim follows using Lemma 2.1.1
codim(Y,X) = codim(Y ∩ U, U) = dim (A ) = dim (O ) = dim (O ) . p U,ηY X,ηY
2.1.2 The dimension formula
In this subsection we elaborate the dimension formula given in [EGA.IV, Prop. 5.6.5] with methods like those in [Ei, Sec. 10.1] and [Mat, Thm 15.5, 15.6].
As a preparatory work, we also need facts like flat Going-Down and Krulls Principal Ideal Theorem and its inverse. First lets have an easy result about flatness:
Lemma 2.1.3. Let φ: A → B be a ring homomorphism and let M be a flat A- 0 module. Then M := M ⊗A B is a flat B-module.
Proof: For an exact sequence
0 −→ M1 −→ M2
we have to show that
0 0 0 −→ M1 ⊗B M −→ M2 ⊗B M
is also exact. But this follows from flatness of M as an A-module and the iso- morphism 0 ∼ ∼ Mi ⊗B M = Mi ⊗B B ⊗A M = Mi ⊗A M.
Proposition 2.1.4 (Flat Going-Down). Let φ: A → B be a flat ring homomor- phism of Noetherian rings. Let further p0 ⊆ p ⊆ A be prime ideals in A and let q ⊆ B be a prime ideal with φ−1(q) = p. Then there exists a prime ideal q0 ⊆ q such that φ−1(q0) = p0.
Proof: Since p0B ⊆ pB ⊆ q, there is a prime ideal q0 ⊆ B that is minimal over p0B and contained in q. We will show that φ−1(q0) = p0 which yields the claim.
– 13 – 2 Preparation
0 0 0 ∼ 0 By passing from A to A/p and from B to B/p B (note that B/p B = B ⊗A A/p is flat over A/p0 by Lemma 2.1.3) we may assume that p0 = 0, that A is a domain and that q0 ⊆ B is a minimal prime. So q0 consists of zero-divisors. But any non-zero element in A is a non-zero-divisor on B because B is flat over A, hence −1 0 φ (q) = (0) = p .
Next we need an auxiliary assertion about Krull dimension and flat local ring homo- morphisms. Before we can elaborate this we need a little bit of preparatory work. We start with a recollection of Krulls Principal Ideal Theorem (in short: Krulls PIT) and its inverse:
Theorem 2.1.5 (Krulls PIT and inverse PIT). Let A be a Noetherian ring.
(i) Let I ⊆ A be an ideal which is generated by r elements. Then for any minimal prime p ⊆ A over I we have ht(p) ≤ r.
(ii) Let p ⊆ A be a prime ideal with ht(p) = r, then there exists an ideal I generated by r elements such that p is minimal over I.
Proof: Those are very well-known facts. See for example the bachelor thesis [Hei, Theorems 3.12, 3.14].
With this we can give another description of the Krull dimension of a local Noethe- rian ring, namely:
Lemma 2.1.6. Let A be a local Noetherian ring and let d denote the minimum integer such that there exist elements x1, . . . , xd ∈ mA such that
n mA ⊆ (x1, . . . , xd) ⊆ mA for some n 0. Then d = ht(mA) = dim(A).
n Proof: First let I := (x1, . . . , xd) be an ideal such that mA ⊆ I ⊆ mA. Then mA is minimal over I, as for any prime I ⊆ p we have √ q n √ mA = mA ⊆ I ⊆ p = p ⊆ mA
hence p = mA. Now Krulls PIT (Theorem 2.1.5) yields ht(mA) ≤ d.
On the other hand we apply the inverse PIT (Theorem 2.1.5) to mA and ob- tain elements y1, . . . , ye ∈ A (with e := ht(mA)) such that mA is minimal over J := (y1, . . . , ye). It follows that R/J has only one prime ideal (namely mA), n hence it is Artinian. In particular there is some n 0 such that mA = 0. This yields n mA ⊆ J ⊆ mA
– 14 – 2.1 Dimension
and thus – by definition of d – we get d ≤ e = dim(A).
Equipped with that, we can proceed to the formula about Krull dimension and flat local ring homomorphisms, which we will need in several situations:
Proposition 2.1.7. Let φ: A → B be a flat local homomorphism of Noetherian local rings. Then we have