Functoriality of Chow Groups Over Regular Base Schemes

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 polynomial 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 codimension 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 multiplicities – 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 considering 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 representatives, 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 similar 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 notion 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-projective 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 homomorphism 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 purpose 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 intersection 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 technical 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 intersection 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 functorial 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 essential 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 isomorphism 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 homomorphism 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 homomorphisms. 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 obtain 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

dim(B) = dim(A) + dim B/mAB .

Proof: Let d := dim(A) and e := dim (B/mAB). So by Lemma 2.1.6 there are elements x1, . . . , xd ∈ A and y1, . . . , ye ∈ B such that

n m mA ⊆ (x1, . . . , xd) ⊆ mA and mB ⊆ (y1,..., ye) ⊆ mB

for some n, m 0. The latter chain yields

m mB ⊆ mAB + (y1, . . . , ye) ⊆ mB.

So putting this together we obtain

mn n n mB ⊆ (mAB + (y1, . . . , ye)) ⊆ mAB + (y1, . . . , ye)

⊆ (x1, . . . , xd, y1, . . . , ye)B ⊆ mB

Applying Lemma 2.1.6 we get

dim(B) ≤ d + e = dim(A) + dim (B/mAB) .

To show the other direction, let q ⊆ B be a prime ideal that is minimal over

mAB. This means dim (B/mAB) = dim(B/q). Let mA ) p1 ) ··· ) p`

be a maximal chain of prime ideals with ` = ht(mA) = dim(A). Now by deﬁnition −1 of q we have φ (q) = mA, so we can apply ﬂat Going-Down (Proposition 2.1.4) and obtain a chain of prime ideals

q ) q1 ) ··· ) q`

in B yielding ht(q) ≥ ` = dim(A). We conclude the proof as

dim(B) ≥ dim (B/q) + ht(q) = dim (B/mAB) + ht(q)

≥ dim (B/mAB) + dim(A).

– 15 – 2 Preparation

Eventually we collected everything we need to elaborate the dimension formula by Grothendieck, which we mentioned in the prologue:

Theorem 2.1.8. Let A, B be domains and φ: A,→ B be an injective ring homomorphism of ﬁnite type, where A is Noetherian and universally catenary (cf. Deﬁnition 3.1.1). Let further q ⊆ B be a prime ideal and p := φ−1(q). Then we have

dim (Bq) + TrDeg (κ(q)/κ(p)) = dim (Ap) + TrDeg Quot(B)/ Quot(A) .

Proof: We will do the proof in several cases.

(Case 1) B =∼ A[X] is a polynomial ring. In this case clearly

TrDeg Quot(B)/ Quot(A) = 1.

As A is Noetherian, also Ap and Bq are Noetherian and local rings. Further B is ﬂat over A (as it is a free A-module), so the induced homomorphism φp : Ap → Bq is a ﬂat (cf. [AM, Prop. 3.10]) and local ring homomorphism, to which we can apply Proposition 2.1.7 and obtain

dim (Bq) = dim (Ap) + dim (Bq/pBq) = dim (Ap) + dim ((B/pB)q) . (*)

Now there are two sub-cases:

(Case 1.1) If we have pB = q, then

∼ Bq/pBq = (B/pB)q = κ(q)

is a ﬁeld, hence dim (Bq/pBq) = 0. Furthermore, we see

κ(q) =∼ Quot(B/pB) =∼ Quot (A[X]/pA[X]) =∼ Quot ((A/p)[X]) = κ(p)(X)

so TrDeg(κ(q)/κ(p)) = 1. Putting everything together we conclude

dim (Bq) + TrDeg (κ(q)/κ(p)) = dim (Ap) + 0 + TrDeg (κ(q)/κ(p))

= dim (Ap) + 1

= dim (Ap) + TrDeg (Quot(B)/ Quot(A)) .

(Case 1.2) In this case we assume pB ( q. Consider a chain of prime ideals pB ⊆ ℘ ⊆ q in B. This corresponds to a chain of primes

(0) ⊆ ℘ ⊆ q (†)

in B/pB =∼ (A/p)[X]. By deﬁnition we have A ∩ q = p, hence (A/p) ∩ q = (0).

– 16 – 2.1 Dimension

So all the primes in (†) lie over (0) and therefore one of the inclusions in (†) has to be an equality (e.g. shown in [Hei, Thm. 4.4]). We obtain that there are no primes properly between pB and q, yielding

dim (Bq/pBq) = dim ((B/pB)q) = 1.

Summing up the elaborated facts, we get:

(*) dim (Bq) + TrDeg (κ(q)/κ(p)) = dim (Ap) + 1 + TrDeg (κ(q)/κ(p))

= dim (Ap) + TrDeg (Quot(B)/ Quot(A)) + TrDeg (κ(q)/κ(p)) .

So the only thing left to show is TrDeg(κ(q)/κ(p)) = 0, but as we have seen above q ∩ (A/p) = (0), hence q corresponds to some non-zero prime ideal in Quot(A/p)[X] =∼ κ(p)[X], so we get a ﬁnite ﬁeld extension

κ(p) ,→ κ(p)[X]/q =∼ Quot (A/p)[X]/q =∼ Quot (A[X]/q) =∼ κ(q)

and we win.

(Case 2) B =∼ A[X]/I for some non-zero prime ideal I ⊆ A[X]. As φ is injective, we get A ∩ I = ker(φ) = (0), hence I corresponds to some non-zero prime ideal I0 ⊆ Quot(A)[X] (cf. [Hei, Cor. 4.2]). Now as above we conclude that the extension

Quot(A) ,→ Quot(A)[X]/I0 =∼ Quot A[X]/I =∼ Quot(B)

is ﬁnite, hence the assertion we have to show becomes

dim (Bq) = dim (Ap) − TrDeg (κ(q)/κ(p)) .

The prime ideal q ⊆ B corresponds to a prime ideal q0 ⊆ A[X] containing I, so we get ∼ ∼ Bq = (A[X]/I)q = A[X]q0 /IA[X]q0 and κ(q) =∼ κ(q0). Furthermore with the same arguments as in (Case 1.2) we get ht(I) = 1. Now let ` := dim (Bq) = dim A[X]q0 /IA[X]q0 and let

(0) ( p1 ( ··· ( p`

be a maximal chain of prime ideals in A[X]q0 /IA[X]q0 . This yields a chain of prime ideals (0) ( I ( p1 ( ··· ( p` of A[X] contained in q0. This chain is maximal as ht(I) = 1. Since A is universally

– 17 – 2 Preparation

catenary, all maximal chains of primes in A[X] between (0) and q0 have the same length, in this case ` + 1. So we conclude

0 dim A[X]q0 = ht(q ) = ` + 1 = dim (Bq) + 1. (+)

We now apply (Case 1) to the situation B = A[X] with prime ideal q0 lying over p and we get

0 dim A[X]q0 = dim (Ap) + TrDeg (Quot(A[X]/ Quot(A)) − TrDeg κ(q )/κ(p)

= dim (Ap) + 1 − TrDeg (κ(q)/κ(p))

where we used κ(q0) =∼ κ(q). Putting this equation into (+) we obtain

dim (Bq) + 1 = dim (Ap) + 1 − TrDeg (κ(q)/κ(p))

and we win.

(General Case) In the general case B is ﬁnitely generated over A. We will reduce this case to the situation where B is generated by one element. Then the assertion follows from (Case 1) and (Case 2). χ ψ For the reduction let A ,→ C ,→ B be a factorization of φ where C is a ﬁnitely generated A-subalgebra of B and let q0 := ψ−1(q). By means of induction we assume we know the assertion for χ and ψ, i.e.

0 dim Cq0 + TrDeg κ(q )/κ(p) = dim (Ap) + TrDeg (Quot(C)/ Quot(A)) 0 dim (Bq) + TrDeg κ(q)/κ(q ) = dim Cq0 + TrDeg (Quot(B)/ Quot(C)) .

Summing both equations and using additivity of the degree of transcendence yields

dim (Bq) + TrDeg (κ(q)/κ(p)) = dim (Ap) + TrDeg (Quot(B)/ Quot(A))

and this ﬁnishes the proof.

Translating this dimension formula into algebraic geometry we obtain the dimension formula for schemes, which we will need in the thesis:

Corollary 2.1.9. Let X, Y be integral schemes and f : X → Y be a morphism locally of ﬁnite type. Assume Y is locally Noetherian and universally catenary. Further let x ∈ X and y := f(x), then

dim (OX,x) + TrDeg (κ(x)/κ(y)) = dim (OY,y) + TrDeg (K(X)/K(Y )) .

Proof: Follows directly from Theorem 2.1.8.

– 18 – 2.1 Dimension

A nice little corollary of the algebraic dimension formula is the following ﬁniteness- condition for some ﬁbers:

Corollary 2.1.10. Let φ: A,→ B be a finite type extension of Noetherian domains such that the induced field extension Quot(A) ,→ Quot(B) is finite. Further assume A is universally catenary. Then for any prime ideal p ⊆ A with ht(p) = 1 there are only finitely many primes qi ⊆ B lying over p (necessarily of height one).

Proof: Let p ⊆ A be a prime ideal of height one and let q ⊆ B be a prime lying over p. Then by the dimension formula (Theorem 2.1.8) we get

dim(Bq) + TrDeg κ(q)/κ(p) = dim(Ap) = ht(p) = 1.

But dim(Bq) ≥ 1, thus the extension κ(p) ,→ κ(q) is algebraic (and ht(q) = 1).

0 So the prime q corresponds to q in Bp/pBp = B ⊗A κ(p) =: C. Observe the diagram Ap / C

κ(p) / C/q0 / κ(q0). We know that κ(p) ,→ κ(q0) is algebraic, hence any intermediate ring is a ﬁeld. In particular q0 ⊆ C is a maximal ideal.

So we have shown that any prime ideal lying over p deﬁnes a closed point in the

ﬁber Spec (B ⊗A κ(p)), but by assumptions this is a Noetherian space and we just have explained that it consists of closed points only, hence it is ﬁnite.

2.1.3 Local dimension

This subsection elaborates two results about local dimensions. Both are needed for proving that base-changes and compositions of morphisms of relative dimension, are again of relative dimension; and to calculate the dimension of pre-images under those morphisms.

The nice thing about fibers of morphisms locally of finite type is, that they always are a scheme of finite type over a field (namely the residue field of the point whose fiber is taken). This allows us to use all the nice results we have for schemes of finite type over a field.

The following lemma is due to [GW, Lemma 14.94] and uses well-known facts about dimension of schemes of ﬁnite type over a ﬁeld which we freely use without a proof.

– 19 – 2 Preparation

Lemma 2.1.11. Let f : X → Y be a morphism locally of ﬁnite type, x ∈ X and y := f(x) ∈ Y . Then

dimx(Xy) = dim OXy,x + TrDeg (κ(x)/κ(y)) . = dim OX,x ⊗OY,y κ(y) + TrDeg (κ(x)/κ(y)) .

Xy Proof: Let Z := {x} be the closure of x in the fiber Xy = X ×Y Spec(κ(y)). As a base-change of morphisms locally of finite type are again locally of finite type

([GW, Prop. 10.7]), we see that Xy is a κ(y)-scheme locally of ﬁnite type, hence

dimx(Xy) = sup dim(W ) = sup dim(Z) + codim(Z,W ) x∈W ⊆Xy W irred. comp.

= dim(Z) + codim(Z,Xy). (*)

Now Z is an irreducible κ(y)-scheme, hence

dim(Z) = TrDeg (κ(ηZ )/κ(y)) = TrDeg (κ(x)/κ(y)) .

Further from the description of the ﬁber we see that

∼ OXy,ηZ = OXy,x = OX,x ⊗OY,f(x) κ(f(x)) = OX,x ⊗OY,y κ(y).

Putting both into (*) and using Proposition 2.1.2 we obtain

dimx(Xy) = dim(Z) + codim(Z,Xy) = dim(Z) + dim OXy,ηZ = TrDeg (κ(x)/κ(y)) + dim OX,x ⊗OY,y κ(y) .

Corollary 2.1.12. Let A B be a surjective ring homomorphism and assume A, B are of ﬁnite type over some ﬁeld k. Let q ⊆ B be a prime ideal. This corresponds to a prime p ⊆ A containing ker(A → B). Denote by x, y the corresponding points in Spec(A), Spec(B) respectively. Then we have

dimx(Spec(A)) − dimy(Spec(B)) = ht(p) − ht(q).

Proof: Applying Lemma 2.1.11 to Spec(A) → Spec(k) and Spec(B) → Spec(k) we obtain

dimx(Spec(A)) = dim(Ap) + TrDeg (κ(p)/k)

dimy(Spec(B)) = dim(Bq) + TrDeg (κ(q)/k) .

Note that κ(p) = κ(q), hence subtracting both equations above the claim.

– 20 – 2.2 Homology

2.2 Homology

In this section we first collect some technical details concerning flatness. We will see that a flat local ring homomorphism is faithfully flat, which will be needed for calculating the length of tensor-products. Then we proceed to coherent OX - modules, where we show that their support is closed and that “being coherent” is stable under some push-forward and pull-back. These are necessary assertions for associating cycles to coherent modules and working with those in the context of push-forward or pull-back. We conclude with two results on quasi-coherent modules under base-change.

Lemma 2.2.1. Let φ: A → B be a ﬂat ring homomorphism and

f g M1 −→ M2 −→ M3 a complex of A-modules with H := ker(g)/ im(f). Assume that

M1 ⊗A B −→ M2 ⊗A B −→ M3 ⊗ B is an exact sequence, then H ⊗A B = 0.

Proof: As B is a ﬂat A-module, tensoring with B preserves kernels. That means for example ∼ ker(g) ⊗A B = ker (g ⊗ idB) . Further

∼ ∼ . im (f ⊗ idB) = (M1 ⊗A B)/ ker (f ⊗ idB) = (M1 ⊗A B) (ker(f) ⊗A B) ∼ ∼ = (M1/ ker(f)) ⊗A B = im(f) ⊗A B,

where the third isomorphism holds because B is ﬂat. It follows

∼ H ⊗A B = ker(g)/ im(f) ⊗A B = (ker(g) ⊗A B) (im(f) ⊗A B) ∼ = ker (g ⊗ idB) im (f ⊗ idB) = 0.

Proposition 2.2.2. Let A, B be local rings and φ: A → B be a ﬂat local ring homomorphism. Then φ is faithfully ﬂat.

f g Proof: Consider a complex of A-modules N1 → N2 → N3 and suppose

N1 ⊗A B −→ N2 ⊗A B −→ N3 ⊗A B

is exact. We have to show that M := ker(g)/ im(f) is zero.

For this let m ∈ M be arbitrarily chosen and set I := AnnA(m) as its annihilator.

– 21 – 2 Preparation

Then we obtain an injective homomorphism A/I ,→ M which – as B is a ﬂat A-module by assumption – yields the injectivity of

∼ B/IB = A/I ⊗A B,→ M ⊗A B.

But M ⊗A B = 0 by Lemma 2.2.1, hence B/IB = 0. So we have I = A since otherwise we would have I ⊆ mA, hence 0 = B/IB ⊇ B/mAB ⊇ B/mB, a contradiction (note that mAB ⊆ mB because φ is local). As I = A we have 1 · m = 0, hence m = 0 as B is ﬂat and it follows M = 0.

Lemma 2.2.3. Let A be a ring and

0 −→ M1 −→ M2 −→ M3 −→ 0 be an exact sequence of A-modules where M3 is ﬂat. Then for all A-modules N we obtain an exact sequence

0 −→ M1 ⊗A N −→ M2 ⊗A N −→ M3 ⊗A N −→ 0. L Proof: Let φ: I A N be a surjection. Then we obtain an exact sequence

0 −→ ker(φ) −→ M A −→ N −→ 0. I

Tensoring this sequence with Mi yields

M ∼ M Mi ⊗A ker(φ) −→ Mi ⊗A A = Mi −→ Mi ⊗A N −→ 0, I I

where, for i = 3 this sequence is short exact as M3 is ﬂat by assumption. Further we can tensor the short exact sequence in the assumption with ker(φ), L with I A and with N respectively. Putting everything together we get a diagram

M1 ⊗A ker(φ) / M2 ⊗A ker(φ) / M3 ⊗A ker(φ) / 0

α β γ L L L 0 / I M1 / I M2 / I M3 / 0

M1 ⊗A N / M2 ⊗A N / M3 ⊗A N / 0

0 0 0.

– 22 – 2.2 Homology

L with exact rows and columns. Note that the second row is short exact as I A is free, hence ﬂat. The snake lemma gives an exact sequence

ker(γ) −→ coker(α) = M1 ⊗A N −→ coker(β) = M2 ⊗A N,

where ker(γ) = 0 as mentioned above (M3 is ﬂat). The claim follows.

Now we proceed to coherent OX -modules and their support.

Lemma 2.2.4. Let A be a Noetherian ring and M be a ﬁnitely generated A- module. Then the support of M is closed in Spec(A). More precisely, we have

suppA(M) = V (AnnA(M)) .

Proof: Assume we have a prime ideal p ∈ suppA(M), then Mp 6= 0 implies ∼ 0 6= Mp/pMp = M ⊗A κ(p)

by Nakayama-lemma, so we necessarily have AnnA(M) ⊆ p yielding

p ∈ V (AnnA(M))

and as p was arbitrarily chosen we have one direction.

On the other hand assume p ∈/ suppA(M) and take a generating system

{m1, . . . , mn} ⊆ M.

As Mp = 0 there are ti ∈ A \ p such that timi = 0. Now as p is prime we have Q t := i ti ∈ A \ p and clearly tM = 0 as {mi}i was a generating system. Hence t ∈ AnnA(M) \ p yielding AnnA(M) 6⊆ p and the claim follows.

Proposition 2.2.5. Let X be a locally Noetherian scheme and F a coherent OX - module. Then the support supp(F) is closed in X.

Proof: Let x ∈ X with Fx = 0, then we have to show that there exists an open

neighborhood U ⊆ X of x with F U = 0. For this, choose an aﬃne open neighbor-

hood U = Spec(A) ⊆ X of x. If F U = 0 we are through. Otherwise by assump-

tion we have F U = Mf for some ﬁnitely generated A-module M and x corresponds to a prime ideal px ⊆ A. By Lemma 2.2.4 the support suppA(M) ⊆ Spec(A) is closed and by assumption px ∈/ suppA(M), hence there is an open neighborhood V of x in Spec(A) \ suppA(M), which does the job.

We later deﬁne a cycle on X associated to a coherent OX -module F. During the thesis we will come to the point where we push this cycle forward (or pull it back).

– 23 – 2 Preparation

In that situations we want to look at the cycle associated to the push-forward (or pull-back) of F and need that those modules are coherent again. The proof is quite elaborate, so we only give a reference, as it would go beyond the scope of this thesis.

Theorem 2.2.6. Let f : X → Y be a morphism of locally Noetherian schemes.

∗ (i) Let F be a coherent OY -module, then f (F) is a coherent OX -module.

(ii) Assume f is proper and F is a coherent OX -module, then f∗(F) is a coherent OY -module.

Proof: To show (i), ﬁrst observe that the pull-back f ∗(F) is quasi-coherent (cf. [GW, Rem. 7.23]), so by [GW, Prop. 7.45] it is enough to verify that it is also of

ﬁnite type over OX . This can be checked locally and follows immediately from commutative algebra. For (ii) see [GW, Thm 12.68] or [EGA.III, Cor. 3.2.2].

We conclude this section with two basic results from the theory of cohomology of sheafs and base-change. For both see also [GW, Prop. 12.6]. First we do the aﬃne base-change.

Proposition 2.2.7. Consider the following ﬁber diagram of schemes

f X / Y O O j i

W g / Z.

Let further F be a quasi-coherent OX -module and assume f is aﬃne. Then

∗ ∼ ∗ i (f∗(F)) = g∗(j (F)).

Proof: The statement is local on both Z and Y , hence assume Y = Spec(A) and Z = Spec(A0). Since f is aﬃne we further can assume X = Spec(B) and F = Mf for some B-module M by quasi-coherence of F. Thus we have

0 W = Spec B ⊗A A ∼ ∗ 0 j (F) = (B ⊗A A ) ⊗B M .

So the assertion follows as we have an isomorphism of A0-modules

0 ∼ 0 (B ⊗A A ) ⊗B M = A ⊗A M.

– 24 – 2.3 Morphisms

The following theorem about ﬂat base-change is due to the seminary on cohomology of schemes by Walter Gubler in 2014. We also refer to [Ha, Prop. 9.3] or better [GW, Prop. 12.6] for the direct statement.

Theorem 2.2.8. Consider a ﬁber square as in Proposition 2.2.7. Let further F be a quasi-coherent OX -module and assume i is ﬂat and f is proper. Then we get an isomorphism ∗ ∼ ∗ i (f∗(F)) = g∗(j (F)).

2.3 Morphisms

In this section we collect some assertions on morphisms that we need for this thesis.

We start with a subsection on open and dominant morphisms. Here we use Cheval- leys theorem and some basic facts to show that a ﬂat morphism (under some weak assumptions) is open. Further, we give some results on the image of generic points, namely we show that open maps send generic points to generic points and we elaborate that a morphism is dominant iﬀ all generic points are in the image (as we would expect by intuition). These assertions will be used repeatedly within this thesis.

Next, the second subsection basically shows that a quasi-compact morphism locally of finite type with a finite generic fiber can be restricted to a finite morphism. This technical detail will be important in the proof that proper push-forward commutes with taking associated cycles.

In the last subsection we have a look at finitely generated locally free OX -modules and the notion of finite locally free morphisms. The theory we build up there, will be used in the thesis to calculate the composition of push-forward and pull-back along a finite morphism.

2.3.1 Open and dominant morphisms

In this subsection we will elaborate that a ﬂat morphism locally of ﬁnite type of locally Noetherian schemes is open. The proof bases on Chevalley’s Theorem that images are constructible sets.

The basic idea of the proof is to show that the image is both constructible and generically stable, as both together imply that it is open. So ﬁrst we elaborate needed facts concerning constructible subsets.

Deﬁnition 2.3.1. Let X be a Noetherian topological space. A subset of X is called locally closed, if it is a closed subset of an open subset of X. A ﬁnite union of locally closed subsets is called a constructible subset.

– 25 – 2 Preparation

Theorem 2.3.2 (Chevalley). Let A be a Noetherian ring and φ: A → B be a ring homomorphism of ﬁnite type. Denote by f : Spec(B) → Spec(A) the induced morphism on the aﬃne schemes, then the image of f is a constructible subset of Spec(A).

Proof: See [Ei, Corollary 14.7].

Now we need two very technical lemmas.

Lemma 2.3.3. Let X be a topological space, A ⊆ X be a constructible subset and Y ⊆ X be a closed irreducible subset. Assume that A ∩ Y ⊆ Y is dense, then A ∩ Y contains a non-empty open subset of Y .

Proof: By assumption A is constructible, hence

n [ A = (Ui ∩ Vi) i=1

for open subsets Ui ⊆ X and closed subsets Vi ⊆ X. By density of A ∩ Y in Y we get Y [ Y [ Y Y = A ∩ Y = (Ui ∩ Vi ∩ Y ) = Ui ∩ Vi ∩ Y , i i where the latter equality holds because the union is ﬁnite. But Y is irreducible,

so there is an index i0 with

Y Y Y = Ui0 ∩ Vi0 ∩ Y ⊆ Vi0 ∩ Y = Vi0 ∩ Y.

Hence Y ⊆ Vi0 which provides a non-empty open subset

U := Ui0 ∩ Vi0 ∩ Y = Ui0 ∩ Y ⊆ Y

which is contained in A ∩ Y (note U ⊆ Ui0 ∩ Vi0 ⊆ A).

Lemma 2.3.4. Let X be a Noetherian topological space and A ⊆ X be a constructible subset. Let further x ∈ A and assume that for any irreducible closed subset Y ⊆ X with x ∈ Y the set Y ∩ A ⊆ Y is dense. Then A is a (not necessarily open) neighborhood of x.

Proof: We deﬁne a set

M := Y ⊆ X closed x ∈ Y,Y ∩ A ⊆ Y is not a neighborhood of x .

Our aim is to show that M = ∅ as then in particular X/∈ M, hence A ⊆ X is a neighborhood of x and we are done. So assume for a contradiction that M 6= ∅. As X is Noetherian, the set M contains a minimal element Y ∈ M.

– 26 – 2.3 Morphisms

To show that Y is irreducible we assume Y = Y1 ∪ Y2 for proper closed subsets Yi ⊆ Y . Since x ∈ Y , we may assume w.l.o.g. x ∈ Y1. There are two cases:

(Case 1) x∈ / Y2. By the minimality of Y ∈ M and x ∈ Y1 we get that Y1 ∩ A is a neighborhood of x in Y1. That means there exists an open subset U˜ ⊆ Y1 with x ∈ U˜ and U˜ ⊆ Y1 ∩ A. So there is an open subset U ⊆ Y with U ∩ Y1 = U˜. Now the set U ∩ (Y \ Y2) is open in Y and it contains x. Further

U ∩ (Y \ Y2) ⊆ U ∩ Y1 = U˜ ⊆ Y1 ∩ A ⊆ Y ∩ A

so Y ∩ A is a neighborhood of x in Y . Contradiction to Y ∈ M.

(Case 2) x ∈ Y2. Again by minimality of Y ∈ M we get that Yi ∩ A is a neighborhood of x in Yi (for both i = 1 and i = 2). So as above there exist open subsets Ui ⊆ Y with x ∈ Ui ∩ Yi ⊆ Yi ∩ A. Thus clearly x ∈ U1 ∩ U2 and U1 ∩ U2 is open in Y . Further as U1 ∩ U2 = (U1 ∩ U2) ∩ Y1 ∪ (U1 ∩ U2) ∩ Y2 ⊆ (Y1 ∩ A ∩ U2) ∪ (U1 ∩ Y2 ∩ A) ⊆ A

we have U1 ∩ U2 ⊆ Y ∩ A, so Y ∩ A is a neighborhood of x in Y , contradiction.

Both cases show that we cannot write Y = Y1 ∪ Y2 for proper closed subsets, hence Y is irreducible. Therefore, by assumption we have that Y ∩ A ⊆ A is dense, hence Lemma 2.3.3 yields some non-empty open subset U ⊆ Y that is contained in Y ∩ A. There again are two cases:

(Case A) x ∈ U. In this case we immediately get that Y ∩ A is a neighborhood of x in Y , a contradiction.

0 0 (Case B) x∈ / U. In this case we deﬁne Y := Y \ U. Then Y ( Y is a closed subset with x ∈ Y 0. Again by minimality of Y ∈ M we get that Y 0 ∩ A is a neighborhood of x in Y 0. Thus – as above – there exists an open subset V ⊆ Y such that x ∈ V ∩ Y 0 ⊆ Y 0 ∩ A. Now x ∈ U ∪ V and U ∪ V is open in Y . Further since V = (V ∩ Y 0) ∪ (V ∩ (Y \ Y 0)) = (V ∩ Y 0) ∪ (V ∩ U) ⊆ A

we have U ∪ V ⊆ Y ∩ A, hence again Y ∩ A would be a neighborhood of x in Y . Contradiction and we win.

Next we deﬁne the notion of generically stable subsets and stabilizing maps.

Notation. Let X be a topological space and x ∈ X. If x0 ∈ {x} then we write 0 x C x.

Deﬁnition 2.3.5. (i) Let X be a topological space and let A ⊆ X be a subset. 0 0 Then A is generically stable, if for any x ∈ A and x C x we have x ∈ A.

– 27 – 2 Preparation

(ii) Let f : X → Y be a map of topological spaces. Then f is generically stabilizing, 0 −1 0 0 if for all y C y and for all x ∈ f (y) there is some x ∈ X with x C x and f(x0) = y0.

Lemma 2.3.6. Let f : X → Y be a generically stabilizing map of topological spaces and let A ⊆ X be generically stable. Then f(A) ⊆ Y is generically stable.

0 0 Proof: Let y ∈ f(A) and y ∈ Y with yCy . By deﬁnition there is some x ∈ A with 0 0 0 0 f(x) = y and by assumption on f we have some x ∈ X with xCx and f(x ) = y . 0 0 0 But as A is generically stable, x ∈ A induces x ∈ A, hence y = f(x ) ∈ f(A).

As mentioned above, the basic idea of the main proof of this subsection is that a constructible generically stable subset is open:

Proposition 2.3.7. Let X be a Noetherian topological space and let U ⊆ X be a constructible subset that is generically stable. Then U ⊆ X is open.

Proof: Let x ∈ U be arbitrarily chosen and let Y ⊆ X be any irreducible closed subset with x ∈ Y . As Y is irreducible we have x C ηY and as x ∈ U we obtain ηY ∈ U by assumption on U. Thus U ∩ Y ⊆ Y is dense and so we can apply Lemma 2.3.4 and get that U is a neighborhood of x. As x was chosen arbitrarily we win.

We now conclude the “local analogue” of our main result.

Proposition 2.3.8. Let A and B be Noetherian rings and φ: A → B be a ﬂat homomorphism of ﬁnite type. Then the induced morphism f : Spec(B) → Spec(A) on the spectra is open.

Proof: It clearly suﬃces to show the assertion for the standard open subsets. So

let g ∈ B and show that f(D(g)) ⊆ Spec(A) is open. Thus replacing B by Bg we only have to show that the image im(f) ⊆ Spec(A) is open. By Theorem 2.3.2 the image is a constructible subset. We show now that the image also is generically stable. Then Proposition 2.3.7 finishes the proof. To show this, we just have to argue that f is generically stabilizing, the rest follows 0 from Lemma 2.3.6. So let pCp be elements in Spec(A) and let q ∈ Spec(B) with f(q) = p. This means that p0 ⊆ p ⊆ A is a chain of prime ideals in A and q ⊆ B is a prime ideal with φ−1(q) = p. As φ is flat (and localizations are flat) it fulfills flat Going-Down (Proposition 2.1.4), hence there is some prime ideal q0 ⊆ q such −1 0 0 0 0 that φ (q ) = p meaning f(q ) = p . Thus we are through.

Now the main result follows as a corollary:

– 28 – 2.3 Morphisms

Proposition 2.3.9. Let X and Y be locally Noetherian schemes and let f : X → Y be a ﬂat morphism locally of ﬁnite type. Then f is open.

Proof: By assumption on f, we can cover X with open affine subsets Ui that map into open affine subsets Vi ⊆ Y such that the induced ring homomorphisms φi : OY (Vi) → OX (Ui) are of finite type and flat. Thus, by Proposition 2.3.8 the

morphisms f U : Ui → Vi are open, hence f : X → Y is open because: i Let U ⊆ X be an open subset, then f (Ui ∩ U) ⊆ Y is open for all i. Therefore Ui

[ [ [ f(U) = f Ui ∩ U = f(Ui ∩ U) = f (Ui ∩ U) i Ui i i

is open in Y and we are through.

Now we proceed to results about the image of generic points. Those results will be used repeatedly in this thesis.

Lemma 2.3.10. Let f : X → Y be a morphism of schemes and assume X has only ﬁnitely many irreducible components. Then f is dominant if and only if for every irreducible component W of Y the generic point ηW lies in f(X). In this case the generic point of some irreducible component of X is mapped to ηW .

Proof: If the generic points of all irreducible components of Y are contained in f(X), then clearly f(X) = Y , hence f is dominant.

Now let η1, . . . , ηn ∈ X be the generic points of the irreducible components of X. If f is dominant, then

[ [ cont. [ Y = f(X) = f(Xi) = f {ηi} = {f(ηi)}, i i i

where clearly {f(ηi)} are irreducible subsets of Y whose generic points are in the image of f.

The next purpose is to show that open maps send generic points to generic points.

Proposition 2.3.11. Let f : X → Y be an open morphism of locally Noetherian schemes. Then f is generically stabilizing.

In particular, if Y is irreducible, then f(ηZ ) = ηY for any irreducible component Z ⊆ X.

0 Proof: Let x ∈ X, y := f(x) and y C y in Y . Then choose some aﬃne open neighborhood V ⊆ X of x. By openness of f, the set f(V ) ⊆ Y is open and contains y. Thus it contains y0 since otherwise y0 was contained in the closed

– 29 – 2 Preparation

subset Y \ f(V ) yielding y ∈ {y0} ⊆ Y \ f(V ),

a contradiction. So we can replace X by V and Y by f(V ) and therefore assume that X is Noetherian.

Denote W := f −1({y0}). First we show x ∈ W . If not, then U := X \ W is an open neighborhood of x and thus – by the openness of f – the set f(U) ⊆ Y is an open neighborhood of y. By the same argument as above this implies y0 ∈ f(U), which contradicts the deﬁnitions (note that U ∩ W = ∅).

Now, as X is Noetherian, also W is Noetherian, hence the latter is a ﬁnite union of irreducible components

W = W1 ∪ · · · ∪ Wn

and as x ∈ W we may assume w.l.o.g. x ∈ W1, thus x C η := ηW1 . We want to show that f(η) = y0 (this will ﬁnish the proof).

First observe that by continuity of f we get

0 0 cont. y ∈ {y } = f(W1) ⊆ f W1 = f {η} ⊆ f {η} = f ({η}) = {f(η)}

and on the other hand

0 f(η) ∈ f W1 ⊆ f(W1) = {y }.

Taking the closures yields {f(η)} = {y0}

and therefore f(η) = y0 is the unique(!) generic point of this set.

The second assertion now immediately follows: Let Y be irreducible and let ηZ be the generic point of some irreducible component Z ⊆ X. Then f(ηZ ) is contained in Y = {ηY }, hence f(ηZ ) C ηY . By the ﬁrst assertion we obtain a generalization ηZ C x in X with f(x) = ηY , but ηZ is a generic point of X, hence ηZ = x.

Corollary 2.3.12. Let f : X → Y be a morphism of locally Noetherian schemes, where Y is irreducible. Assume f is ﬂat and locally of ﬁnite type, then f(ηZ ) = ηY for any generic point ηZ of any irreducible component Z ⊆ X.

Proof: Combine Proposition 2.3.9 and Proposition 2.3.11.

– 30 – 2.3 Morphisms

2.3.2 Finite morphisms

The purpose of this subsection is to prove the following:

Proposition 2.3.13. Let X and Y be locally Noetherian schemes and f : X → Y be a quasi-compact morphism locally of ﬁnite type. Let further η be a generic point of an irreducible component of Y . Further assume that

−1 f ({η}) = {ζ1, . . . , ζn} ⊆ X is a ﬁnite subset and that all ζi are generic points of irreducible components of X. Then there is an aﬃne open neighborhood V ⊆ Y of η such that

−1 f f −1(V ) : f (V ) −→ V is a ﬁnite morphism.

To ease the proof, we ﬁrst state and prove the “local analogue”:

Lemma 2.3.14. Let φ: A → B be a ring homomorphism and assume B is of finite type over A. Let p ⊆ A be a minimal prime ideal such that only finitely many prime ideals q1,..., qr ⊆ B lie over p. Then there is an element g ∈ A \ p such that the induced ring homomorphism φg : Ag → Bg is finite.

Proof: By assumption the κ(p)-algebra Bp/pBp is of finite type and has only finitely many prime ideals (namely those coming from qi). So Spec(Bp/pBp) is Jacobson and finite, hence discrete and therefore dim(Bp/pBp) = 0. So Noether normalization yields that κ(p) ,→ Bp/pBp is finite.

Now let {b1, . . . , b`} be a generating system of B over A. By ﬁniteness of Bp/pBp over κ(p) there are monic polynomials fi ∈ Ap[X] such that fi(bi) = 0 in Bp/pBp (meaning fi(bi) ∈ pBp). As p ⊆ A is minimal, every element in pAp is nilpotent, hence also every element in pBp is nilpotent, so for all i we ﬁnd an exponent n ni such that fi(bi) i = 0 in Bp. That means there are hi ∈ A \ p such that n hifi(bi) i = 0 in B and thus

Y h := hi ∈ A \ p i

n fulﬁlls hfi(bi) i = 0 in B and hence

ni fi(bi) = 0 in Bh.

Further as we only have ﬁnitely many fi ∈ Ap[X] we can ﬁnd a common denom- 0 h 1 i inator h ∈ A \ p such that fi ∈ A h0 [X] for all i.

– 31 – 2 Preparation

Putting both together we get that for g := h · h0 ∈ A \ p the homomorphism Ag → Bg is ﬁnite.

For the reduction of Proposition 2.3.13 to the local situation in Lemma 2.3.14 we need an auxiliary lemma:

Lemma 2.3.15. In the situation of Proposition 2.3.13 there exists an aﬃne open U ⊆ X with ζi ∈ U for all i.

Proof: Let Mi be the closure of the set {ζ1,..., ζˆi, . . . , ζn}. As all the ζi are generic points of distinct irreducible components, we have ζi ∈/ Mi, hence there is an aﬃne open neighborhood Vi = Spec(Ai) ⊆ X \Mi of ζi. Thus Vi are Noetherian spaces, hence the open subsets n [ Wi := (Vi ∩ Vj) ⊆ Vi j=1 j6=i

are also Noetherian and therefore quasi-compact. Further we clearly have ζi ∈/ Wi as ζi ∈/ Vj for any j 6= i.

To ease notation ﬁx one i and write W := Wi, ζ := ζi and A := Ai. Let p ⊆ A be the (minimal) prime ideal belonging to ζ. As W ⊆ Spec(A) is quasi-compact, we can cover W by ﬁnitely many standard-

open subsets, i.e. there are elements g1, . . . , gr ∈ A such that

W = D(g1) ∪ · · · ∪ D(gr).

But p ∈/ W yields gj ∈ p for all j. And as p ⊆ A is minimal, the images of gj in Ap are nilpotent, hence there are some fj ∈ A \ p with

nj fjgj = 0 in A for nj 0.

Thus for n := max{nj} we have for all j that

n Y fgj = 0 for f := fj ∈ A \ p. j

So p ∈ D(f) and D(f) ∩ W = ∅ as for all q ∈ W there is some j with gj ∈/ q, but n fgj = 0 ∈ q yields f ∈ q, hence q ∈/ D(f).

Therefore Ui := D(f) ⊆ Vi is an aﬃne open neighborhood of ζi with the property that Ui ∩ Wi = ∅. By deﬁnition of Wi this implies Ui ∩ Uj = ∅ for all j 6= i.

So all-in-all we found an aﬃne open subset

n a Y U := U1 ∪˙ ... ∪˙ Un = Spec(OX (Ui)) = Spec OX (Ui) i i=1

– 32 – 2.3 Morphisms

containing all ζi (for the latter equality see [GW, Example 3.12]).

So we now prove the main result of this subsection: Proof (Proposition 2.3.13): Replacing Y by an aﬃne open neighborhood of η we can assume Y = Spec(A) for some Noetherian ring A. By Lemma 2.3.15 there is an aﬃne open U = Spec(B) ⊆ X with f −1({η}) ⊆ U. So the morphism

f U : U → Y induces a ring homomorphism φ: A → B which is of ﬁnite type.

Let p ⊆ A be the minimal prime belonging to η and let qi ⊆ B be the minimal −1 primes belonging to ζi. Then φ (qi) = p for all i by deﬁnition. So by the local case (Lemma 2.3.14) there is an element g ∈ A \ p such that Ag → Bg is ﬁnite, hence if we replace Y by D(g) (note η ∈ D(g)) and U by U ∩ f −1(D(g)) we see that

f U : U −→ Y is ﬁnite.

As Y is aﬃne, it is quasi-compact and by assumption on f, we get that X is quasi-compact. Therefore the set Z := X \ U and thus the morphism

0 f := f Z : Z −→ Y

are quasi-compact. So the morphism f˜: Z → Im(f 0) into the scheme-theoretic image is dominant (cf. [GW, Prop 10.30]). Now if η ∈ Im(f 0), then clearly it is a generic point of some irreducible component of Im(f 0), hence because f˜ is dominant it would be in the set-theoretic image f 0(Z) = f(Z) (cf. Lemma 2.3.10), but then −1 ζi ∈ f (f(Z)) ⊆ Z,

0 0 a contradiction to the deﬁnition of Z as ζi ∈ U . So we have shown η∈ / Im(f ) which is closed, thus there is some aﬃne open neighborhood V ⊆ Y \ Im(f 0) of η. This means f −1(V ) ∩ Z = ∅, hence f −1(V ) ⊆ U and thus the morphism

−1 f f −1(V ) : f (V ) −→ V

is ﬁnite.

2.3.3 Finitely generated locally free modules

This subsection defines the notion of finitely generated locally free modules and finite locally free morphisms. The aim will be to show that a flat finite morphism (of locally Noetherian schemes) is finite locally free. In fact both assumptions are equivalent, as we will mention at the end of this subsection. The theory around finite locally free morphisms will give an easy proof for the fact that the composition of

– 33 – 2 Preparation pull-back and push-forward along a ﬁnite morphism of degree d will just be given by multiplication by d.

Deﬁnition 2.3.16. (i) Let A be a ring and M an ﬁnitely generated A-module, then M is locally free, if for any prime ideal p ⊆ A there exists an f ∈ A \ p

such that Mf is a ﬁnitely generated free Af -module. We say M is locally free of rank d, if for all prime ideals p above the free

Af -modules Mf are of the same rank d.

(ii) Let X be a locally Noetherian scheme and F be a coherent OX -module. We say that F is locally free (of rank d), if every x ∈ X has an open neighborhood ∼ L U ⊆ X such that F U = I OX U for a ﬁnite index set I (of cardinality d).

(iii) Let f : X → Y be a morphism of schemes. Then f is ﬁnite locally free (of

degree d), if it is aﬃne and f∗(OX ) is a locally free OY -module (of degree d).

Lemma 2.3.17. Let A be a Noetherian ring and M be a ﬂat and ﬁnitely generated A-module. Then M is locally free.

Proof: Let p ⊆ A be a prime ideal. By assumption on M, the κ(p)-vector space 0 M := M ⊗A κ(p) is ﬁnitely generated, so we can choose {m1, . . . , mn} ⊆ M that map onto a basis of M 0. By Nakayama-lemma (and ﬁniteness) there is an

element g ∈ A \ p such that {mi}i generate the Ag-module Mg. So we get an exact sequence n φ 0 −→ ker(φ) −→ Ag −→ Mg −→ 0.

Now Mg is ﬂat, because M is ﬂat, hence Lemma 2.2.3 yields that

n 0 −→ ker(φ) ⊗A κ(p) −→ Ag ⊗A κ(p) −→ Mg ⊗A κ(p) −→ 0

is an exact sequence of κ(p)-vector spaces, so ker(φ) ⊗A κ(p) = 0. But ker(φ) is ﬁnitely generated as A is Noetherian, hence we can apply Nakayama again and 0 obtain an element g ∈ A \ p such that ker(φ)g0 = 0. 0 Putting things together we see that f := gg ∈ A \ p does the job.

Corollary 2.3.18. Let f : X → Y be a finite and flat morphism (of degree d, cf. Definition 3.6.3) of locally Noetherian schemes. Then f is finite locally free (of degree d).

Proof: Let y ∈ Y be arbitrarily chosen and V = Spec(A) be some affine open neighborhood of y. As f is finite (hence affine) and flat, there is some finite and flat A-algebra B such that f −1(V ) = Spec(B). Note that A is Noetherian by assumption. Let p ⊆ A be the prime ideal belonging to y. By Lemma 2.3.17

there is an element g ∈ A \ p such that Bg is a ﬁnitely generated free Ag-module.

– 34 – 2.4 Length

Thus the set U := D(g) is an open neighborhood of p and

∼ ∼ M ∼ M f∗(OX ) U = Bfg = Afg = OY U . I I

For the last assertion just observe that by definition of f being finite of degree d we have that B has rank d over A (for any choice of the affine opens), so in the proof of Lemma 2.3.17 we can choose n = d generators and thus the claim follows.

Remark 2.3.19. Clearly the converse in Corollary 2.3.18 is also true: A finite locally free morphism f : X → Y is both flat and finite, hence over locally Noetherian schemes these two notions are equivalent. For details (e.g. for the case of non-locally Noetherian schemes) see [GW, Prop. 12.19].

Lemma 2.3.20. Let f : X → Y be a morphism of locally Noetherian schemes and assume f is ﬁnite locally free (of degree d). Then for any Y -scheme g : Z → Y the 0 base-change f : Z ×Y X → Z of f is ﬁnite locally free (of degree d).

Proof: Follows from the stability of ﬁniteness ([GW, Prop. 12.11]) and ﬂatness under base-change and Remark 2.3.19. For details see [GW, p. 329].

2.4 Length

In some sense this section can be seen as the “technical core” this thesis bases on:

Throughout the thesis we will define cycles on X associated to coherent OX -modules as well as cycles on X associated to closed subschemes Z ⊆ X. In both definitions the coefficients will be lengths of certain modules. Therefore, many important proofs boil down to the equality of lengths of certain modules.

In this section we introduce the length of a module and we elaborate formulas for their lengths in various situations.

Deﬁnition 2.4.1. Let A be a ring and M an A-module. We deﬁne its length to be

lenA(M) := sup n ∈ ∃ 0 = M0 M1 ··· Mn = M ∈ ∪ {∞}. N ( ( ( N

In case A is a ﬁeld, the notion of length clearly coincides with the notion of dimension for A-vector spaces. The additivity of dimension carries on to – its generalization – the length:

– 35 – 2 Preparation

Proposition 2.4.2. Let A be a ring and

0 −→ M 0 −→ι M −→π M 00 −→ 0 be a short exact sequence of A-modules of ﬁnite length, then

0 0 lenA(M) = lenA(M ) + lenA(M ).

More generally if we have an exact sequence

f0 f1 f2 fn 0 −→ M1 −→ M2 −→ · · · −→ Mn −→ 0 of A-modules of ﬁnite length, then

n X n (−1) lenA(Mi) = 0. i=1 Proof: For the ﬁrst assertion consider ﬁltrations

0 0 0 00 00 00 0 = M0 ( ··· ( Mn = M and 0 = M0 ( ··· ( Mm = M

0 00 0 00 of M and M with n = lenA(M ) and m = lenA(M ). Then

0 0 −1 00 −1 00 0 = ι(M0) ( ··· ( ι(Mn) = ker(π) = π (M0 ) ( ··· ( π (Mm) = M

is a ﬁltration of M of length m + n showing

0 00 lenA(M) ≥ lenA(M ) + lenA(M ).

On the other hand let 0 = M0 ( ··· ( Mk = M be a ﬁltration of M with k = lenA(M), then we obtain chains of A-submodules

−1 0 00 (0) ⊆ ι (M1) ⊆ · · · ⊆ M and (0) ⊆ π(M1) ⊆ · · · ⊆ M

of M 0 and M 00. If there is an index 1 ≤ ` ≤ k with

−1 −1 ι (M`) = ι (M`+1) and π(M`) = π(M`+1),

then we obtain a commutative diagram

−1 0 / ι (M`) / M` / π(M`) / 0

−1 0 / ι (M`+1) / M`+1 / π(M`+1) / 0.

– 36 – 2.4 Length

The Five-Lemma yields M` = M`+1, a contradiction. Hence no such ` exists yielding 0 00 lenA(M) ≤ lenA(M ) + lenA(M ). For the last assertion observe the short exact sequences

0 −→ ker(fi) −→ Mi −→ im(fi) −→ 0

yielding lenA(Mi) = lenA ker(fi) + lenA im(fi) . Together with the exactness we obtain:

n X i X i X i (−1) lenA(Mi) = (−1) lenA(ker(fi)) + (−1) lenA(im(fi)) i=1 i i X i X i = (−1) lenA(im(fi−1)) + (−1) lenA(im(fi)) i i n = − lenA(im(f0)) + (−1) lenA(im(fn)) = 0.

Note that – if we assume the usual rules for calculating with infinity – the first assertion of this proposition remains valid, if we drop the assumption that the A- modules in the sequence are of finite length.

The following lemma is an easy fact that will often be used. For example to argue that lenA(M) < ∞, if lenA/I (M) < ∞.

Lemma 2.4.3. Let φ: A → B be a ring homomorphism and let M be an B- module. Then M becomes an A-module via φ and we have lenB(M) ≤ lenA(M) with equality, if φ is surjective.

Proof: Clearly every B-submodule of M is an A-submodule, hence the inequality follows. If in addition φ is surjective, then every A-submodule of M is also an B-submodule of M yielding the claim.

Corollary 2.4.4. Let A be a ring with maximal ideal m and let M be an A-module with mM = 0. Then M is an A/m-module and

lenA(M) = lenA/m(M) = dimA/m(M).

The following theorem is extremely important and will be used throughout this section. It shows in particular that an A-module, which has a finite filtration that cannot be refined, is of finite length and further that every maximal filtration is of the same length (i.e. Theorem of Jordan-Hölder).

– 37 – 2 Preparation

Theorem 2.4.5. Let A be a ring and M an A-module with ﬁltration

0 = M0 ( M1 ( ··· ( M` = M that cannot be reﬁned. Then

(i) For all 1 ≤ i ≤ ` the quotient Mi/Mi−1 is a simple A-module.

∼ (ii) For each 1 ≤ i ≤ ` there is a maximal ideal mi ⊆ A with Mi/Mi−1 = A/mi.

(iii) We have lenA(M) = `. In particular M is of ﬁnite length and every maximal ﬁltration has the same length.

(iv) The quotients appearing in (ii) are unique up to permutation. More precisely for every maximal ideal m ⊆ A we have

# 1 ≤ i ≤ ` mi = m = lenA (Mm). m

Proof: If Mi/Mi−1 is not simple then there is a non-trivial A-submodule yielding a reﬁnement of the ﬁltration between Mi−1 and Mi. Thus (i) follows.

Now let 0 6= m ∈ Mi/Mi−1. Then hmi = Mi/Mi−1, as the module is simple by ∼ (i). So we obtain an A-isomorphism A/ AnnA(m) −→ Mi/Mi−1. As the quotient is non-zero, there is a maximal ideal mi ⊆ A with AnnA(m) ⊆ mi, but then mi/ AnnA(m) ⊆ A/ AnnA(m) is an A-submodule, hence – as the latter is simple ∼ – we have mi = AnnA(m), so Mi/Mi−1 = A/mi.

Next we show (iii) by induction on `.

The assertion is clear for ` = 0, so assume we know lenA(N) = n for any A- module N having a maximal ﬁltration of length n < `.

We have to show lenA(M) ≤ ` (as we already have a filtration of length `), so assume for a contradiction that lenA(M) > `, hence we have a proper A- submodule N ( M with lenA(N) = ` (e.g. the `-th module in a filtration of length ` + 1). Define Ni := N ∩ Mi and obtain a sequence

0 = N0 ⊆ N1 ⊆ · · · ⊆ N` = N (*)

of A-submodules of N that need not be strict. Omitting those Ni for which we have Ni−1 = Ni, we obtain a ﬁltration of length n ≤ `. Next we show that this ﬁltration is maximal. For that observe that the kernel of the map

φi : Ni −→ Mi/Mi−1

∼ is given by ker(φi) = Mi−1 ∩ Ni = Ni−1, hence Ni/Ni−1 = im(φi) ⊆ Mi/Mi−1, but the latter A-module is simple by (i) and Ni/Ni−1 6= 0 as we omitted those

– 38 – 2.4 Length

Ni, hence ∼ Ni/Ni−1 = Mi/Mi−1 is simple. So (*) is a maximal ﬁltration of length n ≤ ` and in fact n < ` as

we will show now by proving inductively that Ni = Mi for all i ≤ n (note that N` = N ( M = M`): Firstly N0 = M0 is clear, so now assume we already know that Ni−1 = Mi−1. As we omitted those Ni with Ni/Ni−1 = 0 the maps φi are surjective, hence

Mi ⊆ Ni + Mi−1 = Ni + Ni−1 = Ni ⊆ Mi

yielding Mi = Ni and we are through. So we have elaborated that (*) is a maximal ﬁltration of N of length n < `,

hence by induction hypothesis we have lenR(N) = n < `, a contradiction to lenR(N) = ` by deﬁnition.

To show (iv) ﬁrst observe that – as localization is exact – we get

∼ (Mi/Mi−1)m = (Mi)m/(Mi−1)m.

Further, we see that for mi 6= m we have (A/mi)m = 0, because then there exists ∼ x ∈ mi \m and thus in (A/mi)m = A/mi ⊗A Am we have for an arbitrary generator a ⊗ b that xb b a ⊗ b = a ⊗ = ax ⊗ = 0. x x So localizing the ﬁltration of M w.r.t. m we get a sequence M˜ i := (Mi)m of i Am-submodules of Mm with quotients  ∼ ∼ ∼ 0 , mi 6= m M˜ i/M˜ i−1 = (Mi/Mi−1)m = (A/mi)m = A/m , m = mi.

Omitting those M˜ i with M˜ i/M˜ i−1 = 0 we obtain a ﬁltration of Mm of length

# i mi = m

that cannot be reﬁned as the successive quotients are simple. By (iii) this length

is equal to lenAm (Mm).

Corollary 2.4.6. Let A be a ring, M an A-module and

0 = M0 ( M1 ( ··· ( Mk = M ∼ a sequence of A-submodules. Assume that for all i we have Mi/Mi−1 = A/mi for some maximal ideals mi ⊆ A, then lenA(M) = k < ∞.

– 39 – 2 Preparation

Corollary 2.4.7. In the situation of Theorem 2.4.5 we have Mp 6= 0 for only ﬁnitely many prime ideals p ⊆ A and all of them are maximal. Further we have

X X lenA(M) = lenAp (Mp) = lenAm (Mm). p∈Spec(A) m⊆A max

Proof: The same argument as in Theorem 2.4.5 (iv) shows that lenAp (Mp) = 0 for all prime ideals p ∈/ {m1,..., mn}. Hence Mp = 0 for all prime ideals p ⊆ A except for m1,..., mn. The second assertion now follows from Theorem 2.4.5 (iv) by summing up the lengths len (M ). Ami mi

Often we will be in the situation where we have a local ring homomorphism A → B and a module M of ﬁnite length over B and we are interested in the length of M over A. The following proposition answers this question even more generally for a semi-local ring homomorphism.

Proposition 2.4.8. Let A be a local ring with maximal ideal m = mA and let B be a semi-local ring with maximal ideals M1,..., Mn. Further let φ: A → B be a −1 ring homomorphism such that φ (Mi) = m for all 1 ≤ i ≤ n. Further, assume that [κ(Mi): κ(m)] < ∞ for all i and let M be an B-module of ﬁnite length, then

n len (M) = X [κ(M ): κ(m)] · len (M ) < ∞. A i BMi Mi i=1

Proof: As lenB(M) < ∞ we have a maximal ﬁltration

0 = M0 ( ··· ( Mm = M

by B-submodules Mi. By Theorem 2.4.5 we have

∼ Mi/Mi−1 = B/Mθ(i) = κ(Mθ(i)) for some θ(i) ∈ {1, . . . , n}.

Applying additivity of length (Proposition 2.4.2) repeatedly to the sequences

0 −→ Mi−1 −→ Mi −→ Mi/Mi−1 −→ 0

yields lenA(M) = lenA κ(Mθ(m)) + lenA(Mm−1) = lenA κ(Mθ(m)) + lenA κ(Mθ(m−1)) + lenA(Mm−2) = ... m n X X = lenA κ(Mθ(i)) = lenA (κ(Mj)) · (#{1 ≤ i ≤ m | θ(i) = j}) i=1 j=1 n 2.4.5= X len (κ(M )) · len (M ) (†) A j BMj Mj j=1

– 40 – 2.4 Length

−1 Now by assumption we have φ (Mi) = m, so we can apply Corollary 2.4.4 to obtain

lenA (κ(Mj)) = dimA/m (κ(Mj)) = [κ(Mj): κ(m)] .

Plugging this into (†) yields the claim.

Clearly any module of ﬁnite length is both Noetherian and Artinian. In fact, also the converse is true. However we only need a weaker assertion, namely that an Artinian ring has ﬁnite length over itself.

Proposition 2.4.9. Let A be an Artinian ring and M a ﬁnitely generated A- module, then lenA(M) < ∞. In particular `A(A) < ∞.

Proof: By assumption M is both Noetherian and Artinian. We now construct a

sequence of A-submodules of M as follows: Start with M0 = {0} and given Mi−1 we consider the set of all A-submodules of M that strictly contain Mi−1. If this set is non-empty, it contains a minimal element as M is Artinian. Choose such

a minimal element to be Mi. Now this process terminates, as M is Noetherian and gives us a ﬁltration

0 = M0 ( M1 ( ··· ( Mn = M

which is maximal by construction, hence lenA(M) < ∞ by Theorem 2.4.5.

Corollary 2.4.10. Let A be a Noetherian domain with dim(A) = 1 and let a ∈ A be a non-zero element. Then lenA A/(a) < ∞.

Proof: Using Krulls PIT (Theorem 2.1.5) one can show that the ring A/(a) is zero- dimensional, hence Artinian. Combining Proposition 2.4.9 and Lemma 2.4.3 the claim follows.

In the next section we will deﬁne the notion of an order function. This will be done in terms of lengths and we want order functions to be homomorphisms. The following lemma ensures that.

Proposition 2.4.11. Let A be a Noetherian domain with dim(A) = 1 and let a, b ∈ A be non-zero elements. Then

lenA A/(ab) = lenA A/(a) + lenA A/(b) .

Proof: All lengths are ﬁnite by Corollary 2.4.10. The claim follows from additivity of lengths (Proposition 2.4.2) applied to the exact sequence

·b 0 −→ A/(a) −→ A/(ab) −→ A/(b) −→ 0.

– 41 – 2 Preparation

Lemma 2.4.12. Let φ: A → B be a ﬂat local ring homomorphism of local rings and let M be an A-module of ﬁnite length. Then we have

lenA(M) · lenB(B/mAB) = lenB(M ⊗A B).

Proof: Consider the following composition series of length ` := lenA(M)

0 = M0 ( M1 ( ··· ( M` = M ∼ and successive quotients Mi/Mi−1 = A/mA (cf. Theorem 2.4.5). We obtain a series 0 = M0 ⊗A B ( ··· ( M` ⊗A B = M ⊗A B, where all inclusions are proper as B is a faithfully ﬂat A-module by Proposi- tion 2.2.2. The successive quotients are given by

∼ ∼ ∼ Mi ⊗A B Mi−1 ⊗A B = (Mi/Mi−1) ⊗A B = A/mA ⊗A B = B/mAB,

where the ﬁrst isomorphism follows by the ﬂatness of B. By the usual argument using additivity (Proposition 2.4.2) we obtain

lenB (M ⊗A B) = ` · lenB B/mAB = lenA(M) lenB B/mAB .

The following lemma is another technical criterion for the ﬁniteness of the length of a module. It is used later in the machinery that shows that proper push-forward behaves well with rational equivalence.

Lemma 2.4.13. Let A be a Noetherian ring with some maximal ideal m and let n M be a ﬁnitely generated A-module such that m M = 0 for some n ∈ N. Then we have lenA(M) < ∞.

Proof: Consider the ﬁltration

0 = mnM ⊂ mn−1M ⊂ · · · ⊂ mM ⊂ M

i i+1 with successive quotients Ni := m M/m M. As A is Noetherian clearly the A-modules Ni are ﬁnitely generated and of course we have m · Ni = 0, hence we can apply Corollary 2.4.4 and get lenA(Ni) = dimA/m(Ni) < ∞. Now additivity (Proposition 2.4.2) applied to the exact sequences

i+1 i 0 −→ m M −→ m M −→ Ni −→ 0

provides i i+1 lenA(m M) = lenA(m M) + lenA(Ni).

– 42 – 2.4 Length

Put these equalities together to get

n−1 1 X lenA(M) = lenA(m M) + lenA(N1) = ··· = lenA(Ni) < ∞. i=0

As mentioned earlier we define cycles associated to coherent modules. To ensure that the coefficients – that will be given by certain lengths – are finite we need to show that if the support of a module is just one maximal ideal, then the module is of finite length. During the rest of this section we will elaborate this fact.

First we show a useful proposition due to [Bou, IV, §4 Thm. 1].

Proposition 2.4.14. Let A be a Noetherian ring and M be a ﬁnitely generated A-module. Then there exists a ﬁltration

0 = M0 ( ··· ( Mn = M ∼ of M with successive quotients Mi/Mi−1 = A/pi for some prime ideals pi ⊆ A.

Proof: Let F denote the set of all A-submodules of M that fulﬁll the claim. As for trivial reasons we have {0} ∈ F we get – as M is Noetherian – a maximal element N ∈ F. Assume for a contradiction that N ( M. Then M/N 6= 0, hence the set 0 F := AnnA(x) 0 6= x ∈ M/N

of proper ideals in A is non-empty. Again by Noetherian condition, this set 0 contains a maximal element a = AnnA(x) ∈ F . This ideal is prime, because assume ab ∈ a and b∈ / a. Clearly a ∈ AnnA(bx) and as b∈ / a we have bx 6= 0, 0 hence AnnA(bx) ∈ F . But obviously a ⊆ AnnA(bx), so a = AnnA(bx) by 0 maximality of a ∈ F , thus a ∈ AnnA(bx) = a and therefore a is prime.

Clearly A/a is isomorphic to the submodule N 0/N of M/N generated by x. So we can enlarge the ﬁltration of N by N 0 where the quotients are isomorphic to 0 A/pi and A/a, so N ∈ F contradicting the maximality of N.

S Lemma 2.4.15. In Proposition 2.4.14 we have suppA(M) = i V(pi). Proof: First we see that by Lemma 2.2.4

suppA(Mi/Mi−1) = suppA(A/pi) = V (AnnA(A/pi)) = V(pi).

S Now assume p ∈/ i V(pi), hence (Mi/Mi−1)p = 0 for all i and as localization is exact this induces Mi = Mi−1 for all i. Thus in particular

Mp = (Mn)p = (M0)p = 0,

hence p ∈/ suppA(M). – 43 – 2 Preparation

On the other hand suppose p ∈/ suppA(M), then Mp = 0. Further we have exact sequences

0 −→ (Mi−1)p −→ (Mi)p −→ (Mi/Mi−1)p −→ 0,

so we conclude from (Mn)p = 0, that (Mn−1)p = 0 and (Mn/Mn−1)p = 0. Apply this successively for all i and we obtain (Mi/Mi−1)p = 0 for all i, hence

[ [ p ∈/ suppA(Mi/Mi−1) = V(pi). i i

Corollary 2.4.16. Let A be a Noetherian ring and M be a ﬁnitely generated

A-module. Assume that suppA(M) = {m} for some maximal ideal m ⊆ A, then lenA(M) < ∞.

Proof: By Proposition 2.4.14 there is a ﬁltration

0 = M0 ( ··· ( Mn = M

and prime ideals pi ⊆ A such that

∼ Mi/Mi−1 = A/pi.

By Lemma 2.4.15 we obtain

[ {m} = suppA(M) = V(pi) i

hence pi = m for all i, so Corollary 2.4.6 ﬁnishes the proof.

2.5 Order function

It is easy to show that for a DVR A with valuation ν we have

lenA(A/(a)) = ν(a) for any a ∈ A \{0}. This motivates a generalization for the notion of valuation to all one-dimensional Noetherian domains. This will be elaborated in this very brief ∗ section. The so obtained group homomorphism Quot(A) → Z will be called the order function. This construction translates into algebraic geometry and we obtain the notion of an order function w.r.t. closed integral subschemes of codimension one. These order function will be used to deﬁne principal divisors, which are essential for the notion of rational equivalence on cycles.

– 44 – 2.6 Lattices

Construction 2.5.1. Let A be a Noetherian domain with dim(A) = 1 and quotient field K := Quot(A). Let further f ∈ K be a non-zero element. Then we can a write f = b for some non-zero elements a, b ∈ A. By Corollary 2.4.10 the lengths lenA(A/(a)) and lenA(A/(b)) are finite, so we can define the order of f as

ordA(f) := lenA A/(a) − lenA A/(b) .

c This is well-deﬁned because assume f = d for some other non-zero elements c, d ∈ A. Then – as A is integral – we get ad = bc. Using Proposition 2.4.11 twice we get

lenA(A/(a)) − lenA(A/(b)) = lenA(A/(ad)) − lenA(A/(d)) − lenA(A/(b))

= lenA(A/(bc)) − lenA(A/(d)) − lenA(A/(b))

= lenA(A/(c)) − lenA(A/(d)).

Construction 2.5.2. Let X be a locally Noetherian integral scheme, f ∈ K(X)∗ and Z ⊆ X be a closed integral subscheme with codim(Z,X) = 1.

Further let A := OX,ηZ . Then by Proposition 2.1.2 we have dim (A) = 1 and clearly Quot(A) = K(X). This means we can deﬁne the order of f w.r.t. Z as

ordZ (f) := ordA(f).

∗ This induces a group homomorphism ordZ : K(X) → Z:

Proposition 2.5.3. In the situation of Construction 2.5.2 assume furthermore g ∈ K(X)∗. Then we have

ordZ (fg) = ordZ (f) + ordZ (g).

a c Proof: Writing f = b and g = d for a, b, c, d ∈ A we obtain by using Proposi- tion 2.4.11 that

ordZ (fg) = lenA A/(ac) − lenA A/(bd)

= lenA(A/(a)) + lenA(A/(c)) − lenA(A/(b)) − lenA(A/(d))

= ordZ (f) + ordZ (g).

2.6 Lattices

The purpose of this section is to prove the “determinant lemma”. This is extremely important, because both main theorems – namely compatibility of proper push- forward and ﬂat pull-back with rational equivalence – will have a proof that bases heavily on this assertion.

– 45 – 2 Preparation

Throughout this section we fix a Noetherian local domain A with maximal ideal m := mA and dim(A) = 1. Further K := Quot(A) denotes the quotient field of A and we fix a K-vector space V of finite dimension.

We will deﬁne the notion of an A-lattice as well as the distance of two such A-lattices. The determinant lemma states that for an A-lattice M and a K-isomorphism φ: V → V the distance of M and φ(M) is given by the order of det(φ).

Deﬁnition 2.6.1. An A-lattice in V is a ﬁnitely generated A-submodule M ⊆ V such that V = M ⊗A K.

Remark 2.6.2. If in the deﬁnition above the ring A is an DVR, then any lattice is a free A-module. But we will not need this here.

Deﬁnition 2.6.3. Let M, M˜ be two A-lattices in V . Then we deﬁne their distance by d M, M˜ := lenA M M ∩ M˜ − lenA M˜ M ∩ M˜ .

If M˜ ⊆ M then we get d M, M˜ = lenA M M˜ .

The distance of two A-lattices is a well-deﬁned integer by the following lemma:

Lemma 2.6.4. (i) Let M, M˜ be two A-lattices in V , then M ∩ M˜ is also an A-lattice.

(ii) If M˜ ⊆ M are two A-lattices in V , then lenA M M˜ < ∞.

Proof: To show (i) ﬁrst notice that – as A is Noetherian – we get that M ∩ M˜ is ﬁnitely generated because M is. So we just have to verify that it contains a K-basis of V :

For this take a K-basis {b1, . . . , bn} ⊆ V . As

−1 V = M ⊗A K = S M (for S = A \{0})

we can write b = mi for some m ∈ M and s ∈ S. As {b } is a K-basis and i si i i i i ∗ si ∈ K we have that also

{sibi}i = {mi}i ⊆ M

forms a K-basis of V .

On the other hand V = S−1M˜ , so m = m˜ i for some m˜ ∈ M˜ and λ ∈ A \{0}, i λi i i

– 46 – 2.6 Lattices

hence

xi := λimi ∈ M ∩ M.˜

It is clear that {xi}i form a K-basis of V because {mi}i is one and λi 6= 0. So clearly (M ∩ M˜ ) ⊗A K = V and we are done.

To show (ii) let {m1, . . . , mn} ⊆ M generate M as A-module. As in the proof of (i) we get λ ∈ A \{0} and m˜ ∈ M˜ such that m = m˜ i . So the element i i i λi

n Y λ := λi ∈ A i=1

fulﬁlls λmi ∈ M˜ for all i and as the mi generate M we get λM ⊆ M˜ .

We can assume that λ ∈ m as otherwise we would have M˜ = M and the assertion is clear. We have just seen that M/M˜ is a well-deﬁned A/λA-module, but as 0 6= λ ∈ m we get dim(A/λA) = dim(A) − 1 = 0

by Krulls PIT (Theorem 2.1.5). So A/λR is Artinian and hence its maximal ideal m is nilpotent yielding an integer n such that mn = 0, which in turn means mn ⊆ (λ). As we have seen above we have λM ⊆ M˜ , hence mnM ⊆ M˜ and therefore

mn · M/M˜ = 0.

So Lemma 2.4.13 yields the claim.

Lemma 2.6.5. Let M1,M2,M3 be A-lattices in V .

(i) If M1 ⊆ M2 ⊆ M3, then the distance is “transitive”:

lenA(M3/M1) = lenA(M2/M1) + lenA(M3/M2).

(ii) In general, we have sort of a “triangle inequality”:

d(M1,M3) = d(M1,M2) + d(M2,M3).

Proof: For (i) use the isomorphism theorem to get an exact sequence

0 −→ M2/M1 −→ M3/M1 −→ M3/M2 −→ 0

to which we can apply additivity of lengths (Proposition 2.4.2) to get the result.

For the proof of (ii) we repeatedly use (i). First we apply it to the situation

M1 ∩ M2 ∩ M3 ⊆ M1 ∩ M2 ⊆ M1

– 47 – 2 Preparation

(which are A-lattices by Lemma 2.6.4) and we obtain

lenA M1 M1 ∩M2 ∩M3 = lenA M1 M1 ∩M2 +lenA M1 ∩M2 M1 ∩M2 ∩M3

and analogously

lenA M2 M1 ∩M2 ∩M3 = lenA M2 M1 ∩M2 +lenA M1 ∩M2 M1 ∩M2 ∩M3 .

Subtracting these two equations we obtain for N := M1 ∩ M2 ∩ M3

lenA M1 M1 ∩ M2 − lenA M2 M1 ∩ M2 = lenA(M1/N) − lenA(M2/N).

Now repeating the whole argument twice yields the equations

lenA M2 M2 ∩ M3 − lenA M3 M2 ∩ M3 = lenA(M2/N) − lenA(M3/N) lenA M1 M1 ∩ M3 − lenA M3 M1 ∩ M3 = lenA(M1/N) − lenA(M3/N).

Using these three equations we arrive at the claim:

d(M1,M3) = lenA(M1/M1 ∩ M3) − lenA(M3/M1 ∩ M3)

= lenA(M1/N) − lenA(M3/N)

= lenA(M1/N) − lenA(M2/N) + lenA(M2/N) − lenA(M3/N)

= lenA(M1/M1 ∩ M2) − lenA(M2/M1 ∩ M2) + lenA(M2/M2 ∩ M3)

− lenA(M3/M2 ∩ M3)

= d(M1,M2) + d(M2,M3).

Theorem 2.6.6 (Determinant lemma). Let φ: V →∼ V be a K-linear isomorphism and let M ⊆ V be an A-lattice. Then φ(M) ⊆ V is an A-lattice and

d M, φ(M) = ordA det(φ) .

Proof: As φ is an isomorphism, it is clear that φ(M) ⊆ V is an A-lattice, so we just have to show the formula for the distance.

(Step 1) The left-hand-side of the claim is independent of the choice of the A- lattice M. For that let M˜ ⊆ V be another A-lattice. Since φ is an isomorphism the sequence

0 −→ φM˜ ∩ M −→ φM˜ −→ φM˜ M˜ ∩ M −→ 0

is exact, hence

φM˜ (φ(M) ∩ φ(M˜ )) = φ(M˜ )/φ(M˜ ∩ M) =∼ φM˜ M˜ ∩ M

– 48 – 2.6 Lattices

and analogously

φM(φ(M) ∩ φ(M˜ )) =∼ φMM˜ ∩ M.

Therefore we obtain

d φ(M˜ ), φ(M) = lenA φ(M˜ ) φ(M˜ ) ∩ φ(M) − lenA φ(M) φ(M˜ ) ∩ φ(M) = lenA φ(M/˜ M˜ ∩ M) − lenA φ(M/M˜ ∩ M) = lenA M˜ M˜ ∩ M − lenA M M˜ ∩ M = d M,M˜ = − d M, M˜ ,

where we used that isomorphic modules clearly have the same length. Now using this and Lemma 2.6.5 we obtain

d(M, φ(M)) = d M, M˜ + d M,˜ φ(M˜ ) + d φ(M˜ ), φ(M) = d M, M˜ + d M,˜ φ(M˜ ) − d M, M˜ = d M,˜ φ(M˜ ).

Hence we see the independence of the A-lattice.

(Step 2) Both sides are additive: Let ψ : V →∼ V be another K-linear isomorphism then by Proposition 2.5.3

ordA det(φ ◦ ψ) = ordA det(φ) · det(ψ) = ordA(det(φ)) + ordA(det(ψ))

which shows the additivity of the right-hand-side.

On the other hand we know that ψ(M) ⊆ V is another A-lattice, hence by the independence shown in (Step 1), we know

d(M, φ(M)) = d ψ(M), φ(ψ(M))

yielding (together with triangle inequality Lemma 2.6.5) that

d M, φ(ψ(M)) = d(M, ψ(M)) + d ψ(M), φ(ψ(M)) = d(M, ψ(M)) + d(M, φ(M)),

which shows the other additivity.

(Step 3) By (Step 2) it suﬃces to show the claim for generators of GlK (V ). But ∼ after ﬁxing a K-basis for V we have GlK (V ) = Gln(K), which is generated by elementary matrices of type I and III.

(Case 1) The elementary matrix is of type III: Then (after possibly changing to

– 49 – 2 Preparation

another basis) we can assume

1 1 0 ... 0   0 1 0 ... 0 E = . . . . . ......  . . . . . 0 0 0 ... 1

n We choose the A-lattice M := A (for n := dimK (V )) and we clearly have φ(M) = EM = M, hence

d(M, φ(M)) = 0 = ordA(1) = ordA(det(E))

and we have shown the claim in this case.

(Case 2) The elementary matrix is of type I: Then to ease notation we assume E = diag(λ, 1, 1,..., 1) for λ ∈ K∗. Again using the lattice M := An, we see that φ(M) = λA ⊕ An−1 and hence writing λ = λ1 for λ ∈ A \{0} we get λ2 i

n n n n n n d(M, φ(M)) = lenA A (A ∩ φ(A )) − lenA φ(A ) (A ∩ φ(A ))

= lenA(A/(λA ∩ A)) − lenA(λA/(λA ∩ A))

= lenA(A/λ1A) − lenA(A/λ2A) = ordA(λ)

= ordA(det(φ)),

∼ where we used λA/λ1A = A/λ2A because the surjective map

A −→ λA/λ1A, 1 7→ λ

has kernel λ2A.

As the claim is obviously true for the identity matrix, we are done.

– 50 – 3 S-schemes and cycles

In this chapter we begin with intersection theory. Fulton starts his book [Ful] deﬁn- ing cycles and rational equivalence. We will postpone the latter to the next chapter and in this chapter we will concentrate on cycles.

Fulton defines a cycle on a variety X over a field K as a finite formal sum of integral sub-varieties with integer coefficients. If all these sub-varieties are of the same (Krull-)dimension k, he calls the cycle a k-cycle. Furthermore, he constructs a push-forward of cycles along a proper morphism f : X → Y of varieties as well as a pull-back along a flat morphism.

Now in this chapter we will generalize these constructions. We reduce the assumptions on our schemes to a minimum and call them S-schemes. Those are only supposed to be schemes locally of finite type over some fixed locally Noetherian and universally catenary scheme S. We also have to introduce another notion of dimension (the S-dimension), as the usual Krull-dimension does not behave well enough in this relative situation. Moreover, in preparation for pulling back cycles we will introduce the notion of a morphism of relative dimension. It ensures that fibers and pre-images have a good dimensional behavior.

Taking into account that S-schemes need not be quasi-compact, we also have to allow infinite sums (which are required to be locally finite) as cycles. The subgroup of all k-cycles on an S-scheme X will be denoted by Zk(X). Furthermore, we introduce for any closed subscheme of S-dimension k an associated k-cycle using its irreducible components together with suitable coefficients, called multiplicities, which are given by certain lengths of modules.

In addition to the theory in [Ful] we also associate a cycle to any coherent OX -module which is carried on the support of that module. And we will see close relations of the cycle associated to subschemes and those associated to coherent modules. This indicates a link to K-theory, which we will not discuss in detail.

Having defined those objects, we proceed with the two basic constructions, namely if we are given a proper morphism f : X → Y of S-schemes, we will define the push- forward, which sends a k-cycle on X to a k-cycle on Y . By construction, this will become a group homomorphism f∗ : Zk(X) → Zk(Y ). The other construction will be the pull-back along a flat morphism f : X → Y of S-schemes whose fibers have a “good dimensional behavior”, which ensures that if

– 51 – 3 S-schemes and cycles we pull back a k-cycle on Y , we obtain a (k + `)-cycle on X (for some `). So again, ∗ we obtain a group homomorphism f : Zk(Y ) → Zk+`(X). Both constructions will be functorial. Moreover, taking the associated cycle to coherent modules will commute with proper push-forward and ﬂat pull-back.

We conclude this chapter with a few observations about combining push-forward and pull-back. Here we will elaborate the localization sequence, which can be seen as an analogue to excision in homology and we will evaluate the composition of push-forward and pull-back along a ﬁnite morphism of some degree.

3.1 S-schemes

This very brief section deﬁnes the notion of an S-scheme and morphisms between S-schemes.

Deﬁnition 3.1.1. (i) A ring A is called catenary, if for any two prime ideals 0 0 p ⊆ p in A, there exists a maximal chain p = p0 ( p1 ( ··· ( pn = p of prime ideals and all these maximal chains have the same length.

(ii) A Noetherian ring A is called universally catenary, if every ﬁnitely generated A-algebra is catenary.

(iii) Let X be a locally Noetherian scheme. Then X is called (universally) catenary,

if all local rings OX,x are (universally) catenary.

Notation. Throughout this thesis S will always denote a locally Noetherian and universally catenary scheme.

Deﬁnition 3.1.2. (i) An S-scheme is a scheme X together with a structure morphism ρX : X → S, which is locally of ﬁnite type.

(ii) By a morphism of S-schemes f : X → Y we mean a morphism locally of ﬁnite

type, compatible with the structure morphisms, i.e. ρY ◦ f = ρX .

Remark 3.1.3. As any S-scheme is assumed to be locally of ﬁnite type over S, it is locally Noetherian and catenary.

In some sense our assumptions on S-schemes are minimal: For all the theory we elaborate, we need some sensible notion of dimension (which we will introduce in section 3.2). For the existence of any sensible notion of dimension it is necessary that we are in the catenary setting, because one easily shows that a locally Noetherian

– 52 – 3.1 S-schemes scheme is catenary, if and only if its underlying topological space is catenary and thus the following lemma shows the necessity:

Lemma 3.1.4. Let X be a scheme and assume we have a sensible notion of dimension on X, i.e. a function dim0 that maps closed irreducible subschemes integers in such a way, that for any two irreducible closed subschemes W ( Z ⊆ X we have dim0(W ) < dim0(Z) and if there is no closed irreducible subscheme between W and Z, we want dim0(Z) = dim0(W ) + 1. Then X is a catenary topological space.

Proof: Let W ⊆ Z ⊆ X be two irreducible closed subsets and consider a chain

W = W0 ( ··· ( Wk = Z

of closed irreducible subsets. By assumption on dim0 we get

dim0(Z) ≥ dim0(W ) + k.

Or in other words k ≤ dim0(Z) − dim0(W ) which implies that

codim(W, Z) ≤ dim0(Z) − dim0(W ) < ∞.

So we may consider a maximal chain

W = W0 ( ··· ( Wr = Z

of closed irreducible subsets with r = codim(W, Z). By maximality, the second assumption on dim0 yields

codim(W, Z) = r = dim0(Z) − dim0(W ).

So the space X is catenary.

Remark 3.1.5. (i) For deeper study of intersection theory on schemes over S we will need that S is regular (which implies that S is universally catenary, [Mat, Thm 17.3]), as for example Hartshorne showed in [Ha, A.1.1.2] that there is no reasonable intersection theory on singular schemes in general. Anyhow within this thesis regularity is not necessary.

(ii) As mentioned in [Ful], the most important cases are S = Spec(K) for some ﬁeld K (in this case we get Fultons theory verbatim) or S = Spec(A) for some Dedekind domain A (e.g. A = Z).

– 53 – 3 S-schemes and cycles

3.2 Relative S-dimension

Considering S-schemes one cannot use use the notion of Krull-dimension in a suitable way: Take A to be a DVR with quotient ﬁeld K := Quot(A). Then for the base scheme S := Spec(A) we get an S-scheme Spec(K) with dim Spec(K) = 0, which lies dense in S, but dim(S) = 1. So we introduce a suitable notion of dimension due to [Ful, §20.1]: The so called (relative) S-dimension.

Further, we elaborate important properties of this dimension and show that in the case of varieties over ﬁelds it coincides with the usual notion of Krull-dimension. After that we have a look at morphisms of relative dimension. This will be important for the pull-back of cycles as these morphisms ensures that the pre-images are “dimensional well-behaved”.

3.2.1 S-dimension and its properties

The notion of S-dimension and their properties we will discuss are due to [Ful, §20.1]. For some properties we also followed [Web].

Deﬁnition 3.2.1. (i) Let f : X → Y be a morphism of S-schemes and V ⊆ X be a closed irreducible subscheme, then deﬁne the Y -dimension of V as

dimY (X) := TrDeg κ(ηV )/κ f(ηV ) − dim OY,f(ηV ) .

In particular if Y = S then

dimS(V ) = TrDeg κ(ηV )/κ f(ηV ) − dim OS,f(ηV ) = TrDeg K(V )/K(T ) − codim(T,S),

where T denotes the closure of f(V ) in S (for equality see Proposition 2.1.2 and Lemma 2.3.10).

(ii) An S-scheme X is called S-equidimensional of S-dimension d, if for all irre-

ducible components Xi of X we have dimS(Xi) = d.

(iii) For an arbitrary closed subscheme W ⊆ X we deﬁne its S-dimension by

dimS(W ) := sup{dimS(V )}, V

where the sum is over all irreducible components V ⊆ W .

This notion of dimension is well-behaved, as we will show during this section. We start with proving that – when talking about S-dimensions – we can always restrict

– 54 – 3.2 Relative S-dimension to the case of integral subschemes. And further we show that S-dimension behaves well with respect to dense subsets (cf. the “problem” discussed above when using Krull-dimension).

Lemma 3.2.2. Let X be an irreducible S-scheme.

(i) Then dimS(X) = dimS(Xred).

(ii) For any non-empty open subscheme U ⊆ X we have dimS(U) = dimS(X).

Proof: By construction, Xred has the same underlying topological space as X and also K(Xred) = K(X), hence the ﬁrst assertion is clear. The second follows from the fact that U and X have the same generic point.

An important property is that S-dimension also is compatible with codimension:

Proposition 3.2.3. Let X be an irreducible S-scheme and V ⊆ X be a closed irreducible subscheme. Then

dimS(X) = dimS(V ) + codim(V,X).

In particular dimS(V ) ≤ dimS(X).

Proof: Let T , U be the closures of the images of X, V in S respectively. Let

f := ρX denote the structure morphism, then f induces a dominant morphism f : X → T locally of ﬁnite type with f(ηV ) = ηU (cf. Lemma 2.3.10).

Applying the dimension formula (Corollary 2.1.9) to x = ηV , we get

dim (OX,ηV ) + TrDeg (κ(ηV )/κ(ηU )) = dim (OT,ηU ) + TrDeg (K(X)/K(T )) .

Using Proposition 2.1.2, this equation becomes

TrDeg K(X)/K(T ) + codim(U, T ) = TrDeg K(V )/K(U) + codim(V,X). (*)

As S is catenary we have

codim(U, T ) + codim(T,S) = codim(U, S).

Putting everything together we get

dimS(X) = TrDeg K(X)/K(T ) − codim(T,S) (∗) = TrDeg K(V )/K(U) + codim(V,X) − codim(U, T ) − codim(T,S)

= codim(V,X) + dimS(V ).

– 55 – 3 S-schemes and cycles

Intuitively we also would expect that the image of a morphism cannot have a bigger dimension than the domain. Indeed, the following lemma shows this property. It also is important for the push-forward later, as it ensures the ﬁniteness of the degree.

Proposition 3.2.4. Assume X,Y are irreducible S-schemes and f : X → Y is a dominant morphism of S-schemes. Then

dimS(X) = dimS(Y ) + TrDeg K(X)/K(Y ) .

In particular dimS(Y ) ≤ dimS(X) with equality iff the field extension K(Y ) ,→ K(X) is algebraic, i.e. iff the degree [K(X): K(Y )] < ∞ is finite.

Proof: As f is dominant we have f(ηX ) = ηY by Lemma 2.3.10, so

ρX (ηX ) = (ρY ◦ f)(ηX ) = ρY (ηY )

yielding dim OS,ρX (ηX ) = dim OS,ρY (ηY ) and κ (ρX (ηX )) = κ (ρY (ηY )) .

Now applying the formula for the degree of transcendence to the tower

κ (ρX (ηX )) = κ (ρY (ηY )) ,→ K(Y ) ,→ K(X)

gives (together with the above results)

dimS(X) = TrDeg K(X)/κ(ρX (ηX )) − dim OS,ρX (ηX ) = TrDeg K(X)/K(Y ) + TrDeg K(Y )/κ(ρY (ηY )) − dim OS,ρY (ηY ) = TrDeg K(X)/K(Y ) + dimS(Y ).

The last assertion follows as f is locally of finite type by definition, hence K(X) is finitely generated over K(Y ), hence it is finite iff it is algebraic.

Corollary 3.2.5. If f : X → Y is a morphism of irreducible S-schemes, then

dimS(X) = dimS(Y ) + dimY (X).

Proof: Denote by T the closure of f(X). Then by [GW, Rem. 10.32], the mor- g phism f factors as X −→h T −→ Y , where h is dominant and g is a closed immersion. Then by Proposition 3.2.4 applied to h we get

dimS(X) = dimS(T ) + TrDeg K(X)/K(T ) (*)

and applying Proposition 3.2.3 to g gives

dimS(Y ) = dimS(T ) + codim(T,Y ). (†)

– 56 – 3.2 Relative S-dimension

Putting this together we obtain

dimY (X) + dimS(Y ) = TrDeg K(X)/K(T ) − codim(T,Y ) + dimS(Y ) (†) = TrDeg K(X)/K(T ) + dimS(T ) (∗) = dimS(X).

So in our situation of S-schemes the notion of S-dimension is better behaved as the Krull-dimension. However in some cases both notions are connected. We just stated the following lemma for informative reasons, we will not need the results.

Lemma 3.2.6. (i) If S = Spec(K) for some ﬁeld K, then for S-schemes the S-dimension and the Krull dimension coincide.

(ii) Let X,Y be irreducible S-schemes and f : X → Y be a closed, dominant (hence set-theoretically surjective) morphism of ﬁnite type, then

dimY (X) = dim(X) − dim(Y ).

Proof: The ﬁrst assertion follows from Noether Normalization. For example this can be found in [GW, Thm 5.22]. For the second part observe that by surjectivity we have

dimY (X) = TrDeg K(X)/K(Y ) .

Now the claim follows from [EGA.IV, Cor 5.6.6].

3.2.2 Morphisms of relative dimension

Here we define when a morphism is said to be of (some) relative dimension. This will be the right condition for a flat morphism, if we want to define a pull-back of a cycle, as in that situation we will need or want that each pre-image of an integral subscheme of S-dimension k is S-equidimensional and therefore provides an `-cycle for some `.

In contrast to Fultons definition of relative dimension, we decided to use the notion of fibers to introduce the theory. Afterwards we will show that our definition induces the one given by Fulton.

Definition 3.2.7. Let f : X → Y be a morphism locally of finite type. Then f is of relative dimension r, if every non-empty fiber

Xy = X ×Y Spec(κ(y)) is equidimensional with dim(Xy) = r.

– 57 – 3 S-schemes and cycles

Note that we could also deﬁne, when a morphism is of relative S-dimension r, namely if all non-empty ﬁbers are S-equidimensional of S-dimension r. But in this thesis we will not come to a point, where this notion is needed. Furthermore, as all non-empty

fibers Xy in this setting are schemes of finite type over the field κ(y), the notions of S-dimension and Krull-dimension does coincide anyway. A similar statement in the context of Fultons definition of relative dimension can be found in [Web, Remark A.1.3].

Fulton defines the notion of relative dimension by requiring that for any closed irreducible W ⊆ Y , each irreducible component Z of f −1(W ) is of dimension dim(W ) + r. This property also holds for our morphisms of relative dimension, if we assume in addition that they are flat. That restriction is not a big disadvan- tage because – as mentioned above – we only need this notion in context of the pull-back of cycles, which will only be defined along flat morphisms.

Proposition 3.2.8. Let f : X → Y be a ﬂat morphism of S-schemes. Assume f is of relative dimension r, then for any closed irreducible W ⊆ Y , any irreducible component Z of f −1(W ) dominates W and we have

dimS(Z) = dimS(W ) + r.

In particular for any closed subset W ⊆ Y we have

−1 dimS f (W ) = dimS(W ) + r.

Proof: The property “ﬂat of relative dimension r” is stable under base-change (cf. the following Proposition 3.2.9), so it remains valid if we replace Y by W and X −1 by f (W ) = W ×Y X. Then by ﬂatness, Corollary 2.3.12 implies f(ηZ ) = ηY for any irreducible component Z ⊆ X.

We have shown ηZ ∈ XηY , so we can apply Lemma 2.1.11 to obtain dimη (Xη ) = dim O ⊗ κ(η ) + TrDeg (κ(η )/κ(η )) . Z Y X,ηZ OY,ηY Y Z Y

But clearly OY,ηY = κ(ηY ) hence

O ⊗ κ(η ) =∼ O , X,ηZ OY,ηY Y X,ηZ

which is zero-dimensional as Z ⊆ X is an irreducible component. Together with

dimηZ (XηY ) = r (by assumption on f) we arrive at

r = dimηZ (XηY ) = TrDeg (κ(ηZ )/κ(ηY )) .

Now Proposition 3.2.4 yields

dimS(Z) = dimS(Y ) + TrDeg (K(Z)/K(Y )) = dimS(Y ) + r.

– 58 – 3.2 Relative S-dimension

In the previous proof we used stability of “ﬂat of relative dimension” under base- change. We now verify that fact:

Proposition 3.2.9. Consider the following cartesian diagram

f X / Y O O 0 g g W / Z f 0 where we assume f is ﬂat of relative dimension r. Then f 0 is ﬂat (see for example [Ha, Prop. 9.2]), so we have to verify that it is of relative dimension r.

For this let w ∈ W and denote x := g0(w), z := f 0(w) and y := f(x) = g(z). Thus we have a ﬁeld extension k := κ(y) ,→ κ(z) =: K.

We want to show that dimw(Wz) = dimx(Xy). Note that Xy is of ﬁnite type over k, hence we may assume Xy = Spec(A) for some A = k[X1,...,Xn]/I and Wz = Spec (A ⊗k K).

Let p ⊆ A belong to x and q belong to w (thus q lies over p). These primes correspond to prime ideals p ⊆ k[X1,...,Xn] and q ⊆ K[X1,...,Xn] containing 0 0 I. Let x ∈ Spec(k[X1,...,Xn]) correspond to p and w ∈ Spec(K[X1,...,Xn]) correspond to q, then we apply Corollary 2.1.12 twice to obtain

dimx0 (Spec(k[X1,...,Xn])) − dimx(Spec(A)) = ht(p) − ht(p)

dimw0 (Spec(K[X1,...,Xn])) − dimw (Spec (A ⊗k K)) = ht(q) − ht(q).

Note that clearly dimx0 (Spec(k[X1,...,Xn])) = dimw0 (Spec(K[X1,...,Xn])) = n, so if we subtract both equations, we obtain

dimw (Spec (A ⊗k K)) − dimx(Spec(A)) = ht(p) − ht(p) − ht(q) + ht(q) (†)

If we show that the right-hand-side of (†) vanishes, we are through.

For this observe the following diagram:

φ k[X1,...,Xn]p / K[X1,...,Xn]q

A / (A ⊗k K) . p ψ q

Both homomorphisms φ and ψ are ﬂat as they are a localization of a ﬂat homomor-

– 59 – 3 S-schemes and cycles phism. Hence we can apply Proposition 2.1.7 to both of them and obtain

ht(q) = dim (K[X1,...,Xn]q) = dim (k[X1,...,Xn]p) + dim (K[X1,...,Xn]q/p) = ht(p) + dim (K[X1,...,Xn]/p)q ht(q) = dim (A ⊗k K)q = dim Ap + dim (A ⊗k K)q /p = ht(p) + dim ((A ⊗k K) /p)q . ∼ Looking at the deﬁnitions, we see (K[X1,...,Xn]/p)q = ((A ⊗k K) /p)q, hence subtracting both equations above yields

ht(q) − ht(q) = ht(p) − ht(p) and we win.

3.3 Cycles

As the name suggests in this section we ﬁnally are going to deﬁne the notion of cycles on an S-scheme X. They are the basic objects of study in intersection theory – at least after we quotient out rational equivalence.

In simplicial homology for example the cycles are defined as formal sum of simplicial subcomplexes, likewise cycles in intersection theory will be defined as a formal sum of integral subschemes of X. As mentioned earlier – unlike Fulton – we will not require the sum to be finite, but only locally finite. One reason for this is that on non-quasi-compact schemes X a rational function can have infinitely many poles.

To any closed subscheme of X having S-dimension k we will associate a k-cycle

[W ]k and further we will do this for any coherent OX -module. Eventually we show that the cycles associated to a closed subscheme Z and associated to the coherent module OZ coincide, which will be very important for some proofs later on.

We mentioned that a cycle will be a locally finite sum, so first we have to introduce the notion of locally finiteness.

Definition 3.3.1. Let X be a topological space and let {Zi}i∈I be a family of subsets of X. This family is locally finite, if for any x ∈ X, there exists an open neighborhood x ∈ U ⊆ X such that U ∩ Zi 6= ∅ for only finitely many i ∈ I.

In many proofs it is more convenient to use an equivalent deﬁnition:

– 60 – 3.3 Cycles

Lemma 3.3.2. Let X be a topological space and {Zi}i∈I be a collection of subspaces of X. Then {Zi}i is locally ﬁnite, if and only if for any quasi-compact open subset U ⊆ X the intersection U ∩ Zi is non-empty for only ﬁnitely many i ∈ I.

Proof: Assume {Zi}i is locally ﬁnite and let ∅= 6 U ⊆ X be a quasi-compact open subset. For all x ∈ X there exists an open neighborhood x ∈ Ux ⊆ X such that the set

i ∈ I Zi ∩ Ux 6= ∅

S is ﬁnite by assumption. Now U can be covered U ⊆ x∈U Ux and by quasi-compactness of U there exists a ﬁnite subcover

U ⊆ Ux1 ∪ · · · ∪ Uxn .

If U ∩ Zi 6= ∅, then Uxj ∩ Zi 6= ∅ for some xj. But for all xj there are only ﬁnitely

many i ∈ I such that Uxj intersects Zi, hence U meets Zi for only ﬁnitely many i ∈ I.

For the other direction assume that every quasi-compact open meets only ﬁnitely

many Zi. To show that {Zi}i is locally finite let x ∈ X and U ⊆ X be an affine open neighborhood, so U is quasi-compact and by assumption we have U ∩Zi 6= ∅ for only finitely many i ∈ I.

Now let us deﬁne the notion of k-cycles on X:

Deﬁnition 3.3.3. Let X be an S-scheme and k ∈ Z.

(i)A k-cycle α on X is a formal sum

X α = nZ [Z](nZ ∈ Z), Z

where the sum is taken over all closed integral subschemes Z ⊆ X with

dimS(Z) = k. We further require that the collection

Z ⊆ X nZ 6= 0

is locally ﬁnite.

(ii) The set of all k-cycles on X becomes an Abelian group by summing the co-

eﬃcients. It is denoted by Zk(X).

(iii) Similarly a cycle of codimension k is deﬁned as a locally ﬁnite formal sum

X α = nZ [Z], Z

– 61 – 3 S-schemes and cycles

where the sum is taken over all closed integral subschemes Z of Y with codim(Z,Y ) = k. The group of cycles of codimension k is denoted by Zk(X).

In this thesis we will almost entirely work with the group of k-cycles Zk(X). However the grading via codimension will be important in the section about ﬂat pull-back, because pulling back a cycle will not preserve its S-dimension, but – at least in the S-equidimensional setting – it will preserve codimension.

We proceed with the definition of a k-cycle associated to a closed subscheme. It will be the sum over all irreducible components of S-dimension k where the coefficients are certain lengths. So first we have to ensure these coefficients are integers.

Lemma 3.3.4. Let X be an S-scheme and Z ⊆ X be a closed subscheme with irreducible component W ⊆ Z. Then

len (O ) < ∞. OX,ηW Z,ηW Proof: By Proposition 2.1.2 we have

dim (OZ,ηW ) = codim(W, Z) = 0,

so OZ,ηW is a Noetherian (local) ring of dimension zero, hence it is Artinian and therefore by Proposition 2.4.9 we have len (O ) < ∞. Now Proposi- OZ,ηW Z,ηW tion 2.4.8 applied to the local ring homomorphism

OX,ηW −→ OZ,ηW

gives the result.

Deﬁnition 3.3.5. (i) In the situation of Lemma 3.3.4 the integers

µZ := len (O ) W OX,ηW Z,ηW

are called the multiplicity of W in Z.

(ii) For dimS(Z) = k, we deﬁne the k-cycle associated to Z by

X Z [Z]k := µW [W ], W

where the sum is taken over all irreducible components of Z with dimS(W ) = k. This is a well-deﬁned element in Zk(X) by Lemma 3.3.6.

Lemma 3.3.6. Let X be a locally Noetherian scheme and Z ⊆ X be a closed subscheme. Then the collection of irreducible components of Z is locally ﬁnite.

– 62 – 3.3 Cycles

Proof: Let U be a quasi-compact open subscheme of X. Then U is Noetherian, hence has a Noetherian underlying topological space. So all subspaces of U are Noetherian as well.

Now let M denote the set of all closed subsets of U that have inﬁnitely many irreducible components and assume for a contradiction that M 6= ∅. Now M is partially ordered by inclusion. So – as U is Noetherian – we get by the descending chain condition that we ﬁnd a smallest element W in M. This W is not irreducible as W ∈ M, so write

W = W1 ∪ W2

with Wi ( W strictly smaller closed subsets. By minimality of W , we have that W1 and W2 have a ﬁnite number of irreducible components, so W has; a contradiction.

Remark 3.3.7. If in the situation of Deﬁnition 3.3.5 the closed subscheme Z is Z irreducible, then OZ,ηZ = κ(ηZ ) is a ﬁeld, hence µZ = 1 and we obtain

[Z]k = [Z].

Next, we similarly associate a k-cycle to a coherent OX -module using its support. Again, we first ensure that the definition we are going to give will be well-defined:

Lemma 3.3.8. Let X be an S-scheme and F be a coherent OX -module. Then

(i) The collection of irreducible components of the support supp(F) is locally ﬁnite.

(ii) If Z is an irreducible component of supp(F), then lenOX ,ηZ (FηZ ) < ∞. Proof: Let W := supp(F) be the support of F. It is closed by Proposition 2.2.5 and thus we may think of it as a closed reduced subscheme of X. Hence (i) follows from Lemma 3.3.6.

For (ii) ﬁrst notice that clearly Fξ = 0 for any proper generalization ηZ C ξ, but those ξ that generalize ηZ correspond one-to-one with non-maximal ideals in

OX,ηZ , hence

supp (FηZ ) = {mηZ } OX,ηZ

and by assumption the OX,ηZ -module FηZ is ﬁnitely generated, hence Corol- lary 2.4.16 yields the claim.

Deﬁnition 3.3.9. (i) In the situation of Lemma 3.3.8 the integers

F µ := len (Fη ) Z OX,ηZ Z

– 63 – 3 S-schemes and cycles

are called the multiplicity of Z in F.

(ii) If dimS(supp(F)) = k, then we deﬁne the k-cycle associated to F as

X F [F]k := µZ [Z], Z

where the sum is taken over all irreducible components Z of supp(F) that are of S-dimension k.

This is a well-deﬁned element in Zk(X) by Lemma 3.3.8.

The following result will be used in many proofs in the proceeding. It connects the associated cycles to a subscheme and those associated to their structure sheaf.

Proposition 3.3.10. Let X be an S-scheme and Z ⊆ X be a closed subscheme of X with dimS(Z) = k. Then supp(OZ ) = Z and [Z]k = [OZ ]k.

Proof: First note that the sloppy notation OZ means ι∗(OZ ) for the closed immersion ι: Z → X. This is a coherent OX -module by Theorem 2.2.6. If we assume x∈ / Z, then we ﬁnd an open neighborhood U ⊆ X \ Z of x and obtain

ι∗(OZ )(U) = OZ (U ∩ Z) = OZ (∅) = 0.

Hence OZ,x = 0 and it follows supp(OZ ) = Z.

The second assertion is clear as in this situation the deﬁnitions of the coeﬃcients coincide.

The next lemma is again stated for informative reasons only and will not be used in the thesis.

Lemma 3.3.11. Let 0 −→ G −→ F −→ H −→ 0 be a short exact sequence of coherent OX -modules. Assume that the S-dimension of the supports of F, G and H is at most k, then we have

[F]k = [G]k + [H]k.

Proof: Let Z ⊆ supp(F) be an irreducible component with generic point η. Then obtain an exact sequence

0 −→ Gη −→ Fη −→ Hη −→ 0.

– 64 – 3.4 Proper push-forward

Now we apply additivity of lengths (Proposition 2.4.2) to obtain

F G H µZ = lenOX,η (Fη) = lenOX,η (Gη) + lenOX,η (Hη) = µZ + µZ .

Thus this claim follows.

3.4 Proper push-forward

In this section we will deﬁne how we can push a k-cycle on X forward along a proper morphism f : X → Y of S-schemes to obtain a k-cycle on Y . This is one of the two basic constructions given in this thesis. By deﬁnition, this will become a group homomorphisms f∗ : Zk(X) → Zk(Y ). Further the push-forward will be functorial, hence Zk becomes a functor from the category of S-schemes together with proper morphisms into the category of Abelian groups.

Having deﬁned the push-forward, we can observe for any closed subscheme Z ⊆ X the push-forward of the associated cycle [Z]k. On the other hand we have seen in the previous section that [Z]k = [OZ ]k. Moreover, there is a push-forward of sheafs, so we can also observe the associated k-cycle to the pushed-forward sheaf [f∗(OZ )]k. In fact, we will verify that both cycles coincide.

The ﬁrst lemma is a technical observation that is necessary to ensure that, if we push a k-cycle forward summand by summand (as we will do), the resulting sum of integral subschemes of Y is again locally ﬁnite, hence a k-cycle.

Lemma 3.4.1. Let f : X → Y be a morphism of S-schemes, which is quasi- compact. Let further {Zi}i∈I be a locally finite collection of closed subsets of X. n o Then f(Zi) is a locally finite collection of closed subsets of Y . i∈I Proof: Let V ⊆ Y be an open quasi-compact subset. Then f −1(V ) ⊆ X is quasi- −1 compact by assumption on f and by local finiteness of {Zi}i, this set f (V ) meets only finitely many Zi (cf. Lemma 3.3.2).

−1 Further, f (V ) ∩ Zi 6= ∅ holds iﬀ V ∩ f(Zi) 6= ∅, so there are only ﬁnitely many i ∈ I such that f(Zi) ∩ V 6= ∅.

To arrive at the claim observe that, if f(Zi) ∩ V = ∅, then f(Zi) ⊆ V {. But V { is closed, so it follows that f(Zi) ⊆ V {, hence f(Zi) ∩ V = ∅. Therefore also the set

i ∈ I f(Zi) ∩ V 6= ∅

is ﬁnite.

– 65 – 3 S-schemes and cycles

Construction 3.4.2. Let f : X → Y be a proper morphism of S-schemes and let W ⊆ X be a closed integral subscheme with dimS(W ) = k. Then by properness of f, the image f(W ) ⊆ Y is closed and if we put on it the induced reduced structure, f(W ) becomes a closed integral subscheme of Y . We then deﬁne the degree of W over f(W ) as

  K(W ): K f(W ) , if dimS f(W ) = k, deg W/f(W ) :=  0 , else.

Note that Proposition 3.2.4 applied to the dominant morphism

f W : W −→ f(W ) yields the ﬁniteness of deg W/f(W ) in the case dimS(W ) = dimS(f(W )).

Further we deﬁne

f∗ [W ] := deg W/f(W ) · f(W ) ∈ Zk(Y ) and if we extend this deﬁnition additively, we obtain the push-forward

f∗ : Zk(X) −→ Zk(Y ) X X nW [W ] 7−→ nW deg W/f(W ) f(W ) . W W

This is well-deﬁned by Lemma 3.4.1 (which applies as f is quasi-compact by properness) and by deﬁnition it is a homomorphism of groups.

The proper push-forward is also functorial, as we will show in the following. Hence

Zk is a covariant functor from the category of S-schemes together with proper morphisms of S-schemes into the category of Abelian groups.

Proposition 3.4.3. Let f : X → Y and g : Y → Z be proper morphisms of S- schemes. Then h := g ◦ f is proper and we have

(g ◦ f)∗ = g∗ ◦ f∗ as maps Zk(X) → Zk(Z) for all k ∈ Z. Proof: By [GW, Prop. 12.58] the composition of proper morphisms is again proper, hence it suﬃces to show the second assertion.

As the push-forward is linear, it suﬃces to show that h∗ [W ] = g∗ f∗ [W ] for some integral subscheme W of X.

– 66 – 3.4 Proper push-forward

By construction we have

h∗ [W ] = deg W/h(W ) h(W )

and

g∗ f∗ [W ] = g∗ deg W/f(W ) f(W ) = deg W/f(W ) deg f(W )/h(W )h(W ).

Assume dimS(W ) > dimS h(W ) , then by Proposition 3.2.4 either

dimS(W ) > dimS f(W ) or dimS f(W ) > dimS h(W ) .

Hence deg W/h(W ) = 0 and either deg W/f(W ) or deg f(W )/h(W ) vanishes. In both cases, both sides equal zero and we are through.

Otherwise dimS(W ) = dimS f(W ) = dimS h(W ) and the claim follows as

deg(W/h(W )) = K(W ): Kh(W ) = K(W ): Kf(W ) · Kf(W ) : Kh(W ) = deg W/f(W ) · deg f(W )/h(W ) .

As mentioned in the prologue, given a coherent OX -module F we can either push- forward the associated cycle to F, or we could take the associated cycle of the push-forward of F and in both cases we will get the same k-cycle.

Proposition 3.4.4. Let f : X → Y be a proper morphism of S-schemes and let F be a coherent OX -module with dimS (supp(F)) = k, then we have

f∗ ([F]k) = [f∗(F)]k .

In particular for a closed subscheme Z ⊆ X with dimS(Z) = k we have

f∗ ([Z]k) = [f∗(OZ )]k . Proof: We ﬁrst show the latter assertion: By Proposition 3.3.10 we know that

[Z]k = [OZ ]k, hence if we show the ﬁrst assertion, then

f∗ ([Z]k) = f∗ ([OZ ]k) = [f∗(OZ )]k .

To show the ﬁrst assertion notice that the right-hand-side is well-deﬁned as f∗(F) is coherent by Theorem 2.2.6 and as f is proper supp(f∗(F)) = f(supp(F)), hence

dimS(supp(f∗(F))) = k.

– 67 – 3 S-schemes and cycles

We now do the proof in several cases.

(Case 1) Assume X = supp(F) and the morphism f is dominant. Let us write

X X f∗ ([F]k) = µZ [Z] and [f∗(F)]k = νZ [Z]. (*)

Let Z ⊆ Y be an closed irreducible subscheme that occurs in one of the sums

in (*) with non-zero coeﬃcient, hence dimS(Z) = k. But, as f is dominant, Proposition 3.2.4 provides

dimS(Y ) ≤ dimS(X) = k.

Thus it necessarily follows that Z ⊆ Y is an irreducible component. Let η := ηZ 0 denote its generic point. As dimS(X) = k, we have that each point η ∈ X with f(η0) = η is a generic point of some irreducible component of X.

Now let U ⊆ Y be a quasi-compact open neighborhood of η. Then f −1(U) ⊆ X is open and quasi-compact (as f is proper). So we can cover f −1(U) by ﬁnitely

many affine opens Ui, which are Noetherian, as X is locally Noetherian. So each of the Ui has only finitely many irreducible components (cf. Lemma 3.3.6), hence f −1(U) has only finitely many of them. In particular, f −1({η}) ⊆ X is finite. So we can apply Proposition 2.3.13 and obtain an affine open neighborhood V ⊆ Y of η such that −1 f f −1(V ) : f (V ) −→ V is a finite morphism. So if we replace X by f −1(V ) and Y by V , we may assume f is finite and Y = Spec(A) is affine. Further as finite morphisms are affine, we can write X = Spec(B) for some finite A-algebra B. In this situation we have

F = Mf for some ﬁnitely generated B-module M and f∗(F) corresponds to M viewed as A-module.

Let p ⊆ A be the minimal prime ideal belonging to η and let q1,..., qr ⊆ B be the primes lying over p. Then by checking the deﬁnitions we get

νZ = lenOY ,η (f∗(F)η) = lenAp (Mp) X X µ = deg(W/Z) len (Fη ) = [κ(qi): κ(p)] len (Mq ). Z OX,ηW W Bqi i W ⊆X qi over p f(W )=Z

Hence Proposition 2.4.8 (applied to Ap,Bp and Mp) shows νZ = µZ and we win this case.

(Case 2) Assume f : X → Y is a closed immersion. In this case if [Z] appears

in [f∗(F)]k, then necessarily we have ηZ ∈ X and f∗(F)ηZ = FηZ . Now again

– 68 – 3.5 Flat pull-back

applying Proposition 2.4.8 we obtain

len (f∗(F)η ) = len (Fη ) = len (Fη ) . OY,ηZ Z OY,ηZ Z OX,ηZ Z

Further, deg(Z/f(Z)) = 1 as f is a closed immersion and thus we see that this

equation is the equality of the coeﬃcients in [f∗(F)]k and f∗([F]k).

(General case) Let Z := supp(F) be the support of F and ι: Z → X be the 0 0 closed immersion and T := Im f be the scheme-theoretic image of f := f Z . 0 0 So we can write F = ι∗(F ) for a coherent OZ -module F and further we obtain a commutative diagram f 0 Z / T

ι ι0 X / Y. f We know the assertion for ι and ι0 by the (Case 2). Also by (Case 1), we know the assertion for f 0 as it is dominant (cf. [GW, Prop 10.30]) and supp(F 0) = Z by construction. So

0 0 0 0 0 f∗([F]k) = f∗([ι∗(F )]k) = f∗(ι∗([F ]k)) = (ι )∗((f )∗([F ]k)) 0 0 0 0 0 0 0 = (ι )∗([(f )∗(F )]k) = [(ι )∗((f )∗(F ))]k = [f∗(ι∗(F ))]k

= [f∗(F)]k.

This ﬁnishes the proof.

3.5 Flat pull-back

The structure of this section is very similar to those of the last section. We will construct the pull-back of k-cycles on an S-scheme Y along a ﬂat morphism f : X → Y of S-schemes. Moreover, we will require that f is of some relative dimension to ensure that the pull-back of a k-cycle is an k + `-cycle (for some `). By construction ∗ we obtain a group homomorphism f : Zk(Y ) → Zk+`(X).

Like for proper push-forward, the ﬂat pull-back will be functorial and compatible with associated cycles to coherent OY -modules in the sense that taking the cycle and pulling back yields the same result as taking the associated cycle to the sheaf- theoretic pull-back.

Again, ﬁrst we ensure that the pull-back summand by summand yields a locally ﬁnite sum.

– 69 – 3 S-schemes and cycles

Lemma 3.5.1. Let f : X → Y be a morphism of S-schemes and let {Zi}i∈I be a −1 locally ﬁnite collection of closed subsets of Y , then {f (Zi)}i∈I is a locally ﬁnite collection of closed subsets of X.

Proof: Let U ⊆ X be a quasi-compact open subset. Then f(U) ⊆ Y is quasi-

compact and by assumption f(U) meets only ﬁnitely many Zi. −1 But f (Zi) ∩ U 6= ∅ iﬀ Zi ∩ f(U) 6= ∅, hence the claim follows.

Construction 3.5.2. Let f : X → Y be a ﬂat morphism of S-schemes of relative dimension r. Let further Z ⊆ Y be a closed integral subscheme with dimS(Z) = k. Then all irreducible components of f −1(Z) have S-dimension k + r by Proposi- tion 3.2.8, hence h −1 i f (Z) ∈ Zk+r(X). k+r Thus we obtain a group homomorphism

∗ f : Zk(Y ) −→ Zk+r(X) X X h −1 i α = nZ [Z] 7−→ nZ f (Z) , k+r Z Z which is well-deﬁned by Lemma 3.5.1 and it is called the ﬂat pull-back of cycles.

By intuition the pull-back along an open immersion U,→ X should be the restriction to U and hence should be surjective. Indeed this (and even a little bit more) is true as we will see in Proposition 3.6.2.

We already saw in the last section that proper push-forward is functorial. Now we show that this is also true for ﬂat pull-back. Unlike in the case of proper push- forward this is more complex to verify. We need some preparatory work.

For the following technical lemma, I am indebted to Martin Brandenburg for ex- plaining me the proof.

Lemma 3.5.3. Let f : X → Y be a morphism of schemes. Further let Z ⊆ Y be ∗ ∼ a closed subscheme. Then f (OZ ) = Of −1(Z). Proof: Denote by i: Z → Y the closed immersion and consider the following diagram f X / Y O O j i

W := X ×Y Z g / Z. As a closed immersion is affine and “affine” is stable under base-change (cf. [GW, Prop. 12.3]) we have that both i and j are affine.

– 70 – 3.5 Flat pull-back

∗ ∗ By adjunction and using (co-)units id → g∗g , f f∗ → id we get a natural morphism

∗ ∗ ∗ ∼ ∗ ∗ ∗ ∼ f (i∗(OZ )) −→ f i∗g∗g OZ = f f∗j∗g OZ −→ j∗(g (OZ )) = j∗(OW ) (*)

where the latter isomorphism is due to [GW, Rem 7.10] or by Yoneda-lemma and adjunction as

∗ ∼ ∼ ∼ ∼ Hom(f (OZ ),M) = Hom(OZ , f∗(M)) = Γ(f∗(M)) = Γ(M) = Hom(OX ,M).

Now by aﬃne base-change (Proposition 2.2.7) the morphism (*) is an isomorphism.

Corollary 3.5.4. Let f : X → Y be a ﬂat morphism of S-schemes and let Z ⊆ Y be a closed irreducible subscheme of Y . Further, let W ⊆ f −1(Z) be an irreducible component. Then

∼ −1 OX,ηW /mY,ηZ OX,ηW = Of (Z),ηW .

Proof: By Corollary 2.3.12 we have f(ηZ ) = ηW . Now using Lemma 3.5.3 and basic deﬁnitions we easily calculate

∼ ∗ ∼ −1 O −1 = f (OZ )η = f (OZ )η ⊗ −1 OX,η f (Z),ηW W W f (OY )ηW W ∼ ∼ = O ⊗O OX,η = OZ,η ⊗O ,η OX,η Z,f(ηW ) Y,f(ηW ) W Z Y Z W ∼ = OX,ηW /mY,ηZ OX,ηW , ∼ where for the latter isomorphism we used the well-known fact M⊗AA/I = M/IM for any ring A, ideal I ⊆ A and A-module M. Further we used the fact that

O =∼ κ(η ) =∼ O /m . Z,ηZ Z Y,ηZ Y,ηZ

For the functoriality to make sense, we need that the pull-back along the composition is well-deﬁned in the ﬁrst place.

Proposition 3.5.5. Let X, Y and Z be S-schemes and f : X → Y , g : Y → Z be ﬂat morphisms of relative dimension n, m respectively. Then g ◦ f is ﬂat of relative dimension n + m.

Proof: Clearly g ◦ f is ﬂat, so it suﬃces to show that g ◦ f is of relative dimension n + m. For that let x ∈ X, y := f(x) ∈ Y and z := g(y) ∈ Z. We apply Lemma 2.1.11 twice to get

dimx(Xy) = dim OXy,x + TrDeg κ(x)/κ(y) dimy(Yz) = dim (OYz,y) + TrDeg κ(y)/κ(z) .

– 71 – 3 S-schemes and cycles

By assumption we have dimx(Xy) = n and dimx(Yz) = m, hence summing both equations above we get

n + m = dim OXy,x + dim (OYz,y) + TrDeg κ(x)/κ(z) . (*)

As f is ﬂat, the OYz,y-module OXz,x is ﬂat, hence we can apply Proposition 2.1.7 to obtain

dim (OXz,x) = dim (OYz,y) + dim (OXz,x/mYz,yOXz,x) = dim (OYz,y) + dim OXy,x . (†)

Subtracting (*) and (†) yields

n + m − dim (OXz,x) = TrDeg κ(x)/κ(z) .

Applying Lemma 2.1.11 once again yields

n + m = TrDeg κ(x)/κ(z) + dim (OXz,x) = dimx(Xz)

and that was the claim.

As mentioned in the beginning of this section, we again have some compatibility of pull-back and taking associated cycles to coherent modules. We show this now (and we will need it for example to show the functoriality of the pull-back).

Proposition 3.5.6. Let f : X → Y be a ﬂat morphism of S-schemes and assume f is of relative dimension r. Let F be a coherent OY -Module such that dimS supp(F) = k, then

∗ ∗ ∗ dimS supp(f (F)) = k + r and f [F]k = [f (F)]k+r.

∗ Proof: By Theorem 2.2.6 we see that f (F) is a coherent OX -module. Further we have supp (f ∗(F)) = f −1(supp(F)),

∗ ∼ because f (F)x = Ff(x). Hence the assertion about the dimension of the support follows from Proposition 3.2.8.

Now let W be an irreducible component of supp(F) with dimS(W ) = k and let Z be an irreducible component of f −1(W ). By deﬁnitions we have to check that

F f −1(W ) f ∗(F) µW · µZ = µZ .

For that set A := OY,ηW , B := OX,ηZ and M := FηW . Then M is an A-module with lenA(M) < ∞ by Lemma 3.3.8. Now as f(ηZ ) = ηW (cf. Corollary 2.3.12)

– 72 – 3.5 Flat pull-back

we have ∗ f (F)ηZ = FηW ⊗OY ,ηW OX,ηZ = M ⊗A B and therefore

f ∗(F) F f −1(W ) µZ = lenB (M ⊗A B) = lenA(M) · lenB (B/mA) = µW · µZ ,

where the second equation is Lemma 2.4.12 (which applies since A → B is a local ring homomorphism) and the last equation is because of

∼ −1 B/mAB = Of (W ),ηZ

by Corollary 3.5.4.

Corollary 3.5.7. Let f : X → Y be a ﬂat morphism of S-schemes and let f be of relative dimension r. Let further Z ⊆ Y be a closed subscheme with dimS(Z) = k. Then we have

−1 ∗ h −1 i dimS f (Z) = k + r and f ([Z]k) = f (Z) . k+r Proof: The assertion on the dimension follows from Proposition 3.2.8. For the second claim note that by Proposition 3.3.10 we have

[Z]k = [OZ ]k ,

so by Proposition 3.5.6 we get

∗ ∗ ∗ h i h −1 i f ([Z]k) = f ([OZ ] ) = [f (OZ )] = Of −1(Z) = f (Z) , k k+r k+r k+r

∗ where we used that Of −1(Z) = f (OZ ) by Lemma 3.5.3.

Now our desired functoriality follows as a corollary:

Proposition 3.5.8. Let f : X → Y and g : Y → Z be ﬂat morphisms of S- schemes. Assume f and g are of relative dimension n, m respectively. Then

(g ◦ f)∗ = f ∗ ◦ g∗ as maps Zk(Z) → Zk+n+m(X) for all k ∈ Z. Proof: The pull-back along the composition g ◦ f is well-deﬁned by Proposi- tion 3.5.5. Further, by construction it is linear, hence it suﬃces to prove that

(g ◦ f)∗([W ]) = f ∗ (g∗([W ]))

for some closed integral subscheme W ⊆ Z with dimS(W ) = k.

– 73 – 3 S-schemes and cycles

To show this, we notice that g−1(W ) is a closed subscheme of Y with

−1 dimS g (W ) = k + m

by Proposition 3.2.8. Hence by Corollary 3.5.7 we obtain

h i h i f ∗ g−1(W ) = f −1 g−1(W ) . k+m k+m+n

So the claim follows as

h i h i (g ◦ f)∗([W ]) = (g ◦ f)−1(V ) = f ∗ g−1(W ) = f ∗ (g∗ ([W ])) . k+n+m k+m

We see that ﬂat pull-back does not preserve S-dimension. However – at least in nice situations, where the schemes are S-equidimensional – it does preserve codimension. This is the reason, why sometimes the cycle group (or later the Chow group) is graded via codimension.

Lemma 3.5.9. Let f : X → Y be a ﬂat morphism of S-schemes which is of relative dimension r. Assume X and Y are S-equidimensional. Then for a cycle α ∈ Zk(Y ) of codimension k the pull-back f ∗(α) ∈ Zk(X) is also of codimension k, hence Zk becomes a contra-variant functor from the category of S-equidimensional S- schemes together with ﬂat morphisms of S-schemes of some relative dimension into the category of Abelian groups.

Proof: It suﬃces to show the assertion for α = [Z] for some closed integral subscheme Z ⊆ Y with codim(Z,Y ) = k. By Proposition 3.2.8 we have

−1 dimS f (Z) = dimS(Z) + r = dimS(Y ) − codim(Z,Y ) + r,

where for the latter equation we used Proposition 3.2.3. Another application of Proposition 3.2.8 yields

r = dimS(X) − dimS(Y ),

so if we plug this into the equation above, we get

−1 dimS f (Z) = dimS(X) − codim(Z,Y ) = dimS(X) − k

and the claim follows.

– 74 – 3.6 Pull-back and push-forward

3.6 Pull-back and push-forward

In this section we examine the composition of proper push-forward and flat pull-back. We start with a proposition about base-change. Then we elaborate the localization sequence, which can be seen as an analogue to excision in homology and finally we verify that the composition of push-forward and pull-back along a finite morphism of degree d is just multiplication by d.

Proposition 3.6.1. Consider the following cartesian diagram

g0 X0 / X

0 f f 0 Y g / Y of S-schemes, where f is proper and g is ﬂat of relative dimension d. Then for any

α ∈ Zk(X) we have

∗ 0 0 ∗ 0 g (f∗(α)) = (f )∗ (g ) (α) in Zk+d(Y ).

Proof: Properness is stable under base-change (cf. [GW, Prop. 12.58]), so f 0 is proper. By Proposition 3.2.9 g0 is ﬂat of relative dimension d so the assertion makes sense.

By linearity it suﬃces to show the claim for α = [W ] for some closed integral sub-

scheme W ⊆ X with dimS(W ) = k. Now the key to this proposition is using ﬂat base-change of quasi-coherent modules (Theorem 2.2.8), the elaborated connection between the cycles associated to closed subschemes and the cycle associated to their structure sheaf (Proposition 3.3.10) and the compatibility of push-forward and pull-back with associated cycles to coherent modules (Proposition 3.4.4 and Proposition 3.5.6). All together yield:

0 0 ∗ 0 0 ∗ 0 0 ∗ (f )∗ (g ) (α) = (f )∗ (g ) ([OW ]k) = (f )∗ (g ) (OW ) k+d ∗ ∗ ∗ = [g (f∗(OW ))]k+d = g (f∗ ([OW ]k)) = g (f∗(α)) .

Next we remember that we stated that by intuition the pull-back along an open immersion is just the restriction, so should be surjective. More precisely:

Proposition 3.6.2 (Localization sequence). Let X be an S-scheme and U ⊆ X be an open subscheme with open immersion i: U → X. Let Z := X \U be the closed complement equipped with the reduced structure, then we have a closed immersion j : Z → X and for all k ∈ Z, we obtain an exact sequence

∗ j∗ i Zk(Z) −→ Zk(X) −→ Zk(U) −→ 0.

– 75 – 3 S-schemes and cycles

Proof: For U = ∅ the assertion is clear, so assume U 6= ∅. For the surjectivity of i∗ let X α = nW [W ] ∈ Zk(U). W

For any closed integral subscheme W ⊆ U with dimS(W ) = k deﬁne

0 X W := W red

as the closure of W in X equipped with the reduced structure. Now the element

X 0 β := nW [W ] ∈ Zk(X) W

is a pre-image, as i−1(W 0) = W 0 ∩ U = W and thus

∗ X ∗ 0 X h −1 0 i X i (β) = nW · i [W ] = nW i (W ) = nW [W ]k k W W W X = nW [W ] = α, W

where the first equality in the second line is by Remark 3.3.7. Hence we just 0 have to verify that β ∈ Zk(X) is well-defined. First W are closed and integral subschemes of X by definition. By Lemma 3.2.2 we have

0 0 dimS(W ) = dimS(W ∩ U) = dimS(W ) = k,

so the only thing left to show is that the collection 0 0 W n 6= 0 W

is locally ﬁnite. For this let V ⊆ X be an open quasi-compact subset and let V 0 := V ∩ U. By assumption V 0 ∩ W 6= ∅ for only ﬁnitely many W occurring in α and further, if ∅ = W ∩ V 0 = W ∩ V,

then W ⊆ V {, where the latter set is closed in X yielding W 0 ⊆ V {, hence W 0 ∩ V = ∅. Thus there are only ﬁnitely many W 0 with W 0 ∩ V 6= ∅ and we win.

Now for the exactness at Zk(X) observe the ﬁber diagram

i0 ∅ = U ∩ Z U ×X Z / Z

0 j j U / X. i

Applying Proposition 3.6.1 we get for all α ∈ Zk(Z) that

∗ 0 0 ∗ 0 i j∗ α = (j )∗ (i ) (α) = (j )∗(0) = 0. – 76 – 3.6 Pull-back and push-forward

∗ Hence im(j∗) ⊆ ker(i ).

On the other hand let X α = nW [W ] ∈ Zk(X) W be an k-cycle on X with i∗(α) = 0. As

∗ X h −1 i X i (α) = nW i (W ) = nW [W ∩ U] k k W W

we have that – whenever nW 6= 0 – then W ∩ U = ∅, otherwise W ∩ U would be irreducible and dimS(W ∩ U) = dimS(W ) = k by Lemma 3.2.2, hence

[W ∩ U]k = [W ∩ U] ∈ Zk(U),

∗ 0 a contradiction to i (α) = 0 in Zk(U), because there is no other W with

W ∩ U = W 0 ∩ U

as this would induce W = W 0.

So for all W with nW 6= 0 we have W ⊆ Z, hence we view α as an element in ∗ Zk(Z) and clearly j∗(α) = α, thus ker(i ) ⊆ im(j∗).

The last observation is that composition of push-forward and pull-back along a flat finite morphism of degree d is just the multiplication by d. We first introduce the notions.

Definition 3.6.3. Let f : X → Y be a morphism of schemes. Then f is a called finite if for any affine open V = Spec(A) ⊆ Y the pre-image f −1(V ) ⊆ X is an affine open, say f −1(V ) = Spec(B) such that the induced ring homomorphism A → B is finite.

If the rank rkA(B) of B over A is d for any choice of V = Spec(A), then f is said to be of degree d.

Lemma 3.6.4. Let f : X → Y be a ﬂat morphism of S-schemes. Further assume that f is ﬁnite of degree d. Then f is of relative dimension 0.

Proof: Let y ∈ Y be a point with Xy 6= ∅ and choose some aﬃne open neighborhood U = Spec(A) ⊆ Y of y. By assumption on f we have f −1(U) = Spec(B) and the induced ring homomorphism φ: A → B is ﬁnite. In particular, it is inte-

gral. Let p ⊆ A be the prime ideal belonging to y, then Xy consist of those prime ideals in B that lie over p. It is a well-known fact about commutative algebra that – whenever A → B is integral – there are no two primes q ⊆ q0 ⊆ B both lying over p (e.g. [AM, Cor 5.9]). Hence dim(Xy) = 0 and we win.

– 77 – 3 S-schemes and cycles

Proposition 3.6.5. Let f : X → Y be a ﬂat morphism of S-schemes and assume f is ﬁnite of degree d. Then for all α ∈ Zk(Y ) we have

∗ f∗ (f (α)) = d · α in Zk(Y ).

Proof: First a finite morphism is proper (e.g. [GW, Cor 13.82]) and by assumption f is flat. Further, by Lemma 3.6.4, it is of relative dimension 0, hence push- forward and pull-back are well-defined and the formula makes sense.

To show the claim it suﬃces (due to additivity) to prove the assertion for α = [Z]

for some closed integral subscheme Z ⊆ Y with dimS(Z) = k.

By Corollary 2.3.18, the morphism f is ﬁnite locally free of degree d and by Lemma 2.3.20 this is also true for the base-change

0 −1 f : f (Z) = Z ×Y X −→ Z.

Hence there is some open neighborhood U ⊆ Z of ηZ such that

d 0 ∼ M (f )∗ Of −1(Z) = OZ . U U i=1

Therefore we obtain

d ∗ ∼ ∼ M f∗ f (OZ ) = f∗ Of −1(Z) = OZ,η , ηZ η Z Z i=1

where we used Lemma 3.5.3. Thus additivity of length (Proposition 2.4.2) and Remark 3.3.7 yield

∗ d f∗(f (OZ )) ∗ M µ = lenO f∗ f (OZ ) = lenO OZ,η Z Y,ηZ ηZ Y,ηZ i=1 Z = d · len (O ) = d · µZ = d. OY,ηZ Z,ηZ Z

Furthermore, by Remark 3.3.7 and Proposition 3.3.10 we have

[OZ ]k = [Z]k = [Z].

Finally, ∗ −1 supp f∗ f (OZ ) = f f (Z) ⊆ Z,

∗ hence the only possible irreducible component of supp (f∗(f (OZ ))) having S- dimension k is Z, so

∗ ∗ f∗(f (OZ )) f∗ f (OZ ) k = µZ · [Z] = d · [Z].

Now we conclude the proof using compatibility of push-forward and pull-back

– 78 – 3.6 Pull-back and push-forward with coherent sheafs (Proposition 3.4.4 and Proposition 3.5.6) again:

∗ ∗ ∗ f∗ f Z = f∗ f Z k = f∗ f OZ k ∗ = f∗ f (OZ ) k = d · [Z].

– 79 –

4 Rational equivalence and functoriality of Chow groups

In this final chapter we introduce the notion rational equivalence. We will use the order function defined in the second chapter, to define principal divisors. Then we say, a cycle is rationally equivalent to zero, if it can be written as a sum of principal divisors. Here again, we have to carefully allow locally finite sums. The set of all k-cycles rationally equivalent to zero forms a subgroup Ratk(X) of the cycle group Zk(X), hence they establish an equivalence relation on Zk(X): Two cycles will be rationally equivalent, if their difference is given by a locally finite sum of principal divisors. The quotient group by this relation is the k-th Chow group CHk(X) on X, which is the core of intersection theory. For a smooth S-schemes, this group actually becomes a ring with multiplication given by the intersection product defined for cycles modulo rational equivalence. Therefore the Chow group is often called the Chow ring instead.

The hard work will be sections two and three: We have defined proper push-forward and flat pull-back for cycles and in fact these constructions behave well with rational equivalence, hence they induce a proper push-forward and a flat pull-back on the Chow group of X. In both sections we first give a preparatory calculation of lengths on which the main proof will heavily depend on. Both preparatory lemmas boil down (in some sense) to the determinant lemma given in the second chapter. We want to note that the proof in the case of proper push-forward follows – in a way – the same idea, as the one given by Fulton. However, for the proof in the case of flat pull-back we decided to go a completely different way.

We conclude this chapter with a collection of properties of the Chow group. Beyond the functorialities w.r.t. proper morphisms and ﬂat morphisms of relative dimension, we see that many properties shown for the cycle groups carry on to the Chow group. For example the compatibility with the construction of cycles associated to coherent modules, the assertion about base-change and the composition of push-forward and pull-back along ﬁnite morphisms of some degree. Furthermore, we elaborate the localization sequence again, as a part of the assertion is stronger in the context of Chow groups. On top of that, we elaborate a new property, namely an analogue to the Mayer-Vietoris sequence in homology.

– 81 – 4 Rational equivalence and functoriality of Chow groups

4.1 Principal divisors and rational equivalence

Deﬁnition 4.1.1. Let X be an integral S-scheme. Then a closed integral subscheme Z of X is called a prime divisor, if codim(Z,X) = 1, which is by Proposi- tion 3.2.3 equivalent to dimS(Z) = dimS(X) − 1.

A principal divisor to a rational function f on an integral scheme X will be the formal sum over all prime divisors of X, where the coeﬃcients will be the orders of f w.r.t. the prime divisors. As always we have to be careful with local ﬁniteness, so as a preparatory work:

Lemma 4.1.2. Let X be an integral S-scheme and f ∈ K(X)∗. Then the collection

F := Z ⊆ X Z prime divisor and ordZ (f) 6= 0 is locally ﬁnite.

Proof: There is a non-empty open subscheme U ⊆ X such that f corresponds ∗ to some element in Γ(U, OX ). Thus every prime divisor in F is a subscheme of X \ U. By dimension reasons, it would be an irreducible component there. But the set of irreducible components of the closed X \ U is locally ﬁnite as we saw in Lemma 3.3.6.

Deﬁnition 4.1.3. Let X be an integral S-scheme with dimS(X) = n. For a rational function f ∈ K(X)∗ the principal divisor associated to f is deﬁned as

X divX (f) := ordZ (f)[Z] ∈ Zn−1(X), Z⊆X where the sum is taken over all prime divisors Z of X implying dimS(Z) = n − 1. For the deﬁnition of ordZ (f), see Construction 2.5.2. That this is a well-deﬁned element in Zn−1(X) follows from Lemma 4.1.2.

Now the set of all principal divisors form a subgroup by the following:

Lemma 4.1.4. Let X be an integral S-scheme and f, g ∈ K(X)∗. Then

−1 divX (fg) = divX (f) + divX (g) and divX f = −divX (f).

Proof: Fix a prime divisor Z ⊆ X. Then by Proposition 2.5.3 we have

ordZ (fg) = ordZ (f) + ordZ (g) −1 −1 ordZ (f) + ordZ (f ) = ordZ (ff ) = ordZ (1) = 0.

– 82 – 4.1 Principal divisors and rational equivalence

Using the notion of principal divisors, we can define rational equivalence on an S- scheme X. For this, we first introduce the group of k-cycles rationally equivalent to zero. Then two k-cycles will be said rationally equivalent, if their difference is rationally equivalent to zero.

Definition 4.1.5. Let X be an S-scheme and let {Wj}j∈J be a locally finite collection of closed integral subschemes of X with dimS(Wj) = k + 1 for some k. For ∗ any j ∈ J let fj ∈ K(Wj) and denote by ιj the closed immersion Wj ,→ X. In this situation we obtain a well-defined (cf. Lemma 4.1.6) k-cycle

X (ιj)∗ divWj (fj) ∈ Zk(X). j∈J

Any k-cycle on X, which is of this form, is called rationally equivalent to zero.

The set of all k-cycles on X that are rationally equivalent to zero is denoted by

Ratk(X).

Lemma 4.1.6. In the situation of Deﬁnition 4.1.5 above, the cycle is locally ﬁnite.

Proof: Let U ⊆ X be an open quasi-compact subset. Then, by Lemma 3.3.2

the local finiteness of {Wi} implies that U meets only finitely many Wi. The summands to each of those finitely many Wi are locally finite by Lemma 4.1.2 and Lemma 3.4.1. The claim follows.

Like the set of principal divisors forms a subgroup, also the set of cycles rationally equivalent to zero is a subgroup:

Lemma 4.1.7. The set Ratk(X) in Deﬁnition 4.1.5 forms a subgroup of Zk(X).

Proof: Take two cycles rationally equivalent to zero, say

X X 0 α = (ιj)∗ divWj (fj) and β = (ιj)∗ divWj (fj) . j∈J j∈J

We can assume both sums are taken over the same closed integral subschemes

otherwise formally add the missing Wj with fj = 1 (note that this preserves the local ﬁniteness of {Wj}). Now for the sum α + β observe that by Lemma 4.1.4

X 0 X 0 (ιj)∗ divWj fj · fj = (ιj)∗ divWj (fj) + divWj (fj) j∈J j∈J X X 0 = (ιj)∗ divWj (fj) + (ιj)∗ divWj (fj) j∈J j∈J = α + β.

– 83 – 4 Rational equivalence and functoriality of Chow groups

Hence α + β ∈ Ratk(X). Further, applying Lemma 4.1.4 again yields

X X −1 −α = (ιj)∗ −divWj (fj) = (ιj)∗ divWj fj . j∈J j∈J

Deﬁnition 4.1.8. Let X be an S-scheme and k ∈ Z. The k-th Chow group on X is deﬁned as the quotient

CHk(X) := Zk(X)/ Ratk(X).

Two k-cycles α, β ∈ Zk(X) are called rationally equivalent, if they represent the same class in CHk(X). In this case we write α ∼ β. The (total) Chow group is deﬁned as

M CH(X) := CHk(X). k∈Z By deﬁnition, it is graded by S-dimension, but – as in the situation of cycles in the previous chapter – we can also grade CH(X) via codimension. In this case we use the notation CHk(X) := Zk(X)/ Ratk(X).

4.2 Proper push-forward

In this section we show that proper push-forward is compatible with rational equivalence. That means, we have to show that, if we take a k-cycle which is rationally equivalent to zero, then its push-forward is also rationally equivalent to zero.

To show that each principal divisor divX (h) is pushed forward to a cycle rationally equivalent to zero, we show that either the push-forward vanishes (by dimension reasons) or that it equals the principal divisor to the norm of h. In this case the determinant lemma (Theorem 2.6.6) comes into play.

We start formulating the determinant lemma in a (for this situation) suitable way:

Lemma 4.2.1. Let A be a local Noetherian domain with dim(A) = 1 and assume φ: A,→ B is a finite extension of domains such that we obtain a finite field extension ∗ L K := Quot(A) ,→ L := Quot(B) of the quotient fields. For y ∈ L we let x := NK (y) denote its norm. In this situation B is semi-local and

X [κ(m): κ(mA)]ordBm (y) = ordA(x) m where the sum is taken over all maximal ideals m ⊆ B.

– 84 – 4.2 Proper push-forward

Proof: That B is semi-local follows as a special case of Corollary 2.1.10.

The order function is a homomorphism (Proposition 2.5.3) and the norm is mul- tiplicative, so it suﬃces to show the assertion for y ∈ B. Exactness of localization ∼ implies Bm/yBm = (B/yB)m and thus the left-hand-side of the claim becomes

X X [κ(m): κ(mA)]ordBm (y) = [κ(m): κ(mA)] lenBm ((B/yB)m) = lenA(B/yB), m m

where the latter equality is due to Proposition 2.4.8, which applies, since all

maximal ideals in B lie over mA, because A → B is ﬁnite (cf. [AM, Cor. 5.8]).

Now L is by assumption a ﬁnite-dimensional vector space over K containing the

finitely generated A-module B. Further, as A → B is finite, also K → B ⊗A K is finite yielding that the latter is a field itself. Hence we obtain B ⊗A K = L meaning that B is an A-lattice in L (cf. Definition 2.6.1). Moreover as y ∈ L∗ we obtain a K-linear isomorphism

φ: L −→ L, ` 7→ y`.

Now we apply the determinant lemma (Theorem 2.6.6) to φ. This yields that the right-hand-side of the claim equals

ordA(x) = ordA det(·y) = d B, φ(B) = lenA(B/yB)

and we are through.

Next we have to provide a preparatory lemma, which will be needed to reduce one case of the main proof to the algebraic lemma we just elaborated.

Lemma 4.2.2. Let f : X → Y be a proper dominant morphism of integral S- schemes and assume that the induced field extension K(Y ) ,→ K(X) is finite. Then −1 for every point y ∈ Y with dim (OY,y) = 1 the fiber f ({y}) is finite and there is an open neighborhood U ⊆ Y of y such that

−1 f f −1(U) : f (U) −→ U is a ﬁnite morphism.

−1 Proof: Let y ∈ Y with dim (OY,y) = 1. We show that f ({y}) ⊆ X is finite. For this we may replace Y by an affine open neighborhood V = Spec(A) of y. By quasi-compactness of f we see that X can be covered by finitely many affine opens

X = Spec(B1) ∪ · · · ∪ Spec(Bn).

By assumptions, the induced ring homomorphisms φi : A → Bi of Noetherian

– 85 – 4 Rational equivalence and functoriality of Chow groups

domains are injective and of finite type and induce finite field extensions of their quotient fields. In this situation y corresponds to some prime ideal p ⊆ A with

1 = dim (OY,y) = dim (Ap) = ht(p).

So for all (ﬁnitely many) i we apply Corollary 2.1.10 to get that there are only

ﬁnitely many primes qi ⊆ Bi lying over p, hence there are only ﬁnitely many points in X that map onto y.

A corollary to Zariski’s main theorem is that for a proper morphism f : X → Y the set −1 U := y ∈ Y f {y} is ﬁnite ⊆ Y

is open and the restriction

−1 f f −1(U) : f (U) −→ U

is a ﬁnite morphism. Hence by what we have shown above this yields the claim. For a proof of that corollary we refer to [GW, Cor. 12.90].

As mentioned above the strategy is to show that the push-forward of a principal divisor either vanishes or equals another principal divisor. The following proposition covers the latter case. It is a geometric version of the determinant lemma.

Proposition 4.2.3. Let X, Y be integral S-schemes of the same S-dimension n. Further let f : X → Y be a proper dominant morphism and h ∈ K(X)∗. Then K(X) is a ﬁnite extension of K(Y ) and we get

K(X) f∗ divX (h) = divY (g) for g := NK(Y ) (h).

Proof: The ﬁrst claim follows from Proposition 3.2.4, as dimS(X) = dimS(Y )

For the second, let Z ⊆ Y be a closed integral subscheme with dimS(Z) = n − 1. We have to compare the coeﬃcients of [Z] in both divY (g) and f∗ divX (h) . Combining Proposition 2.1.2 and Proposition 3.2.3 we get

dim(OY,ηZ ) = codim(Z,Y ) = 1

so we can apply Lemma 4.2.2 to f : X → Y and ηZ ∈ Y which yields an aﬃne open neighborhood U ⊆ Y of ηZ such that

−1 f f −1(U) : f (U) −→ U

−1 is a ﬁnite morphism. Therefore, by replacing X, Y , f by f (U) ,U, f f −1(U) we

– 86 – 4.2 Proper push-forward

can assume f is a ﬁnite morphism (note that clearly the assumptions “proper” and “dominant” remain valid!). Hence we write

f : X = Spec(B) → Spec(A) = Y

φ and φ for the ﬁnite ring homomorphism A → B, which induces f.

In this situation Z corresponds to some prime ideal p ⊆ A with

dim (Ap) = dim (OY,ηZ ) = 1.

By deﬁnition the coeﬃcient of [Z] in divY (g) is given by

ord (g) = ord (g) = ord (g). Z OY,ηZ Ap P On the other hand write divX (h) = W ordW (h)[W ] and observe

X f∗ divX (h) = ordW (h) deg W/f(W ) f(W ) . W

All subschemes W ⊆ X with f(W ) = Z correspond to prime ideals q ⊆ B such −1 that φ (q) = p, hence the coeﬃcient of [Z] in f∗ divX (h) is given by

X X ordW (h) deg(W/Z) = ordBq (h)[κ(q): κ(p)]. W ⊆X q⊆B f(W )=Z φ−1(q)=p

Now f is dominant, hence φ is injective and induces an injective ring homomor- 0 0 0 0 0 phism φ : A → B for A := Ap and B := Bp. Clearly φ0 : A0 → B0 is a ﬁnite extension of domains, A0 is Noetherian and local and we have dim(A0) = 1. The maximal ideals in B0 come exactly from those ﬁnitely many (cf. Lemma 4.2.2) primes in B that lie over p. As Quot(A0) = K(Y ) and Quot(B0) = K(X) we have already seen [Quot(B0) : Quot(A0)] < ∞. All together we can apply Lemma 4.2.1 to get

X [κ(q): κ(p)]ord 0 (h) = ord 0 (g) = ord (g). Bq A Ap q

0 Note that Bq = Bq and we get the desired equality of the coeﬃcients and arrive at the claim.

Theorem 4.2.4. Let f : X → Y be a proper morphism of S-schemes and let α ∈ Zk(X) with α ∼ 0. Then f∗(α) ∼ 0 in Zk(Y ). In particular f∗ induces a well-deﬁned group-homomorphism

f∗ : CHk(X) −→ CHk(Y ).

– 87 – 4 Rational equivalence and functoriality of Chow groups

Proof: By deﬁnition we have a locally ﬁnite collection {Wj}j of closed integral subschemes of X with dimS(W ) = k + 1 together with closed immersions

ιj : Wj ,→ X

∗ and rational functions fj ∈ K(Wj) such that

X α = (ιj)∗ divWj (fj) . j

We claim that X 0 ∼ f∗(α) = (f ◦ ιj)∗ divWj (fj) . j 0 Now f is proper, hence Wj := f (ιj(Wj)) (equipped with the reduced structure) 0 are closed integral subschemes of Y and the collection {Wj}j is locally ﬁnite in Y 0 0 by Lemma 3.4.1. Let ιj : Wj → Y denote the corresponding closed immersions. We will show that for any j either (f ◦ ιj)∗ divWj (fj) = 0

0 ∗ 0 or there is some gj ∈ K(Wj) (and dimS(Wj) = k + 1) such that

0 0 (f ◦ ιj) divWj (fj) = ιj divW (gj) . ∗ ∗ j So, in order to ease notation, we will omit the index j in the following and write 0 0 h for fj. Let f : W → W be the induced morphism (which is again proper) and we get a commutative diagram

f X / Y O O ι ι0 W / W 0. f 0

0 0 By Proposition 3.4.3 we have f∗ ◦ ι∗ = ι∗ ◦ f∗, hence what we want to show is that either 0 (f )∗ (divW (h)) = 0

0 0 or there is some g ∈ K(W ) (and we have dimS(W ) = k + 1) such that

0 (f )∗ (divW (h)) = divW 0 (g).

0 By Proposition 3.2.3 we have dimS(W ) ≤ dimS(W ) = k + 1, hence there are three cases to consider:

– 88 – 4.2 Proper push-forward

0 (Case 1) We have dimS(W ) = k + 1. In this case we can apply Proposition 4.2.3 to f 0 : W → W 0 and derive

0 (f )∗ (divW (h)) = divW 0 (g)

for some g ∈ K(Y )∗ and we are through.

0 (Case 2) We have dimS(W ) < k. Then for any closed integral subscheme Z ⊆ W with dimS(Z) = k appearing in divW (h) we have

0 0 dimS(f (Z)) ≤ dimS(W ) < k

by Proposition 3.2.3, hence deg Z/f 0(Z) = 0 and we get

0 (f )∗ (divW (h)) = 0.

0 0 (Case 3) In this case we have dimS(W ) = k. Then for η := ηW 0 observe the ﬁber diagram f 0 W / W 0 O O

0 Wη0 γ / Spec (κ(η )) .

0 In this situation Wη0 is a scheme of ﬁnite type over the ﬁeld κ(η ) with dimension

0 0 dim Wη0 = TrDeg κ(ηW )/κ(η ) = dimS(W ) − dimS(W ) = k + 1 − k = 1,

where the ﬁrst equality is due to [GW, Thm 5.22] and the second is by Proposi- tion 3.2.4.

0 Further, as f is dominant, the generic ﬁber Wη0 is integral and – as a base-change of a proper morphism – γ is proper as well (cf. [GW, Prop. 12.58]). All together 0 Wη0 is an integral proper curve over κ(η ).

Observe

0 X 0 0 (f )∗ (divW (h)) = ordZ (h) deg Z/(f )(Z) (f )(Z) . (†) Z

0 0 As the degree of Z over (f )(Z) vanishes for dimS((f )(Z)) < dimS(Z) = k, we may restrict to those subschemes Z whose image have dimension k. But 0 0 0 0 dimS(W ) = k, hence (f )(Z) = W for all of these Z. So the coeﬃcient of [W ] in (†) is given by

X ord (h)[K(Z): K(W 0)] = X ord (h)[κ(η ): κ(η0)]. (♣) Z OW,ηZ Z Z Z

– 89 – 4 Rational equivalence and functoriality of Chow groups

Now any prime divisor Z of W that maps onto W 0 correspond one-to-one to ∼ closed points ζ in Wη0 , and with the canonical identiﬁcations K(W ) = K(Wη0 ) and O ∼ O we obtain that W,ηZ = Wη0 ,ζ

X 0 0 (♣) = ordO (h )[κ(ζ): κ(η )], Wη0 ,ζ ζ∈Wη0 closed

where h0 corresponds to h under the identification of the function fields. But this integer is exactly the coefficient of [Spec(κ(η0))] in γ div (h0) by definitions. ∗ Wη0 We arrived at the situation where we have a proper push-forward over a field, hence by Fultons original theory [Ful, Thm 1.4] this integer vanishes and that completes the proof.

4.3 Flat pull-back

This section shows that flat pullback behaves well with rational equivalence. The proof differs completely from those given by Fulton, who first gives another definition for rational equivalence and then does the proof with that definition. In lecture notes (that were not too detailed though) by Prof. Dr. Yichao Tian at University of Bonn, I saw that it should be possible to prove the theorem by on-foot calculations of lengths and that will be the method we use to do the proof.

Like in the previous section, we prove a preparatory lemma, whose proof bases on the determinant lemma (Theorem 2.6.6).

Lemma 4.3.1. Let A be a Noetherian local ring and M be a ﬁnitely generated

A-module. Suppose that the support suppA(M) = {mA, p1,..., pr} is at most one- dimensional. Then for any a ∈ A with dim(suppA(M/aM)) ≤ 0 we have

r len (M/aM) − len ker(M −→·a M) = X ord (a) len (M ). A A A/pi Api pi i=1 Proof: We do the proof in several steps. First we show the claim for modules of ﬁnite length. Then we show that it is a “two out of three”-property. The whole proof then bases on the third step, where we assume r = 1: In this case the claim is true by the determinant lemma (Theorem 2.6.6). The last step reduces to the third one.

(Step one) Assume lenA(M) < ∞. In this case we have suppA(M) = {mA} by Corollary 2.4.7, so the right-hand-side of the claim vanishes and so does the left-hand-side by additivity of length (Proposition 2.4.2) applied to

0 −→ ker(·a) −→ M −→·a M −→ M/aM −→ 0.

– 90 – 4.3 Flat pull-back

(Step two) Let us assume we have a short exact sequence of ﬁnitely generated A-modules 0 −→ M 0 −→ M −→ M 00 −→ 0

all of which fulﬁll the assumption on the support. If the assertion is true for two out of these three modules, then it holds for the third, as we will show now: Consider the following diagram

0 / M 0 / M / M 00 / 0

·a ·a ·a 0 / M 0 / M / M 00 / 0.

The snake lemma yields an exact sequence

0 −→ ker(M 0 −→·a M 0) −→ ker(M −→·a M) −→ ker(M 00 −→·a M 00) −→ −→ M 0/aM 0 −→ M/aM −→ M 00/aM 00 −→ 0.

We can apply additivity of lengths (Proposition 2.4.2) to this sequence and obtain

·a 0 0 0 ·a 0 lenA(M/aM) − lenA ker(M −→ M) = lenA(M /aM ) − lenA ker(M −→ M )

00 00 00 ·a 00 + lenA(M /aM ) − lenA ker(M −→ M ) .

Say the assertion holds for M 0 and M 00 (the other cases follow analogously), then the right-hand-side of this equation above equals

X 0 X 00 ordA/p(a) lenAp (Mp) + ordA/p(a) lenAp (Mp ) = 0 00 p∈suppA(M ) p∈suppA(M ) p6=mA p6=mA X 0 00 = ordA/p(a) lenAp (Mp) + lenAp (Mp ) = p∈F X = ordA/p(a) lenAp (Mp).

p∈suppA(M) p6=mA

For the ﬁrst equality we set

0 00 F := suppA(M) ∪ suppA(M ) ∪ suppA(M ) \{mA}

and the second equality is additivity of lengths (Proposition 2.4.2) applied to the sequence 0 00 0 −→ Mp −→ Mp −→ Mp −→ 0. So we see that the assertion holds for M as well. That’s what we wanted to show.

(Step three) Assume r = 1, i.e. suppA(M) = {mA, p}. By Proposition 2.4.14 we

– 91 – 4 Rational equivalence and functoriality of Chow groups

have a ﬁltration 0 = M0 ( M1 ( ··· ( Mn = M ∼ with successive quotients Mi/Mi−1 = A/pi for some prime ideals pi ⊆ A. But suppA(M) = {mA, p}, so by Lemma 2.4.15 we have either ∼ ∼ Mi/Mi−1 = A/mA or Mi/Mi−1 = A/p.

If we show the assertion for each quotient Mi/Mi−1 then (Step two) successively applied to the short exact sequences

0 −→ Mi−1 −→ Mi −→ Mi/Mi−1 −→ 0

yield the claim for M. Thus we are reduced to verifying the assertion for A/mA and A/p. By (Step one) we already have the claim for A/mA, so it remains the case A/p. Therefore, we may replace M by A/p and hence assume pM = 0, so M is also an A/p-module. Thus by replacing A by A/p, we may assume A is a Noetherian local domain and dim(A) = 1 and p = (0).

Let V := M ⊗A Quot(A) and T := ker(M → V ). Again by (Step one) and (Step two) we may replace M by M/T and therefore assume that M ⊆ V is an A-lattice.

In this situation let φ: V → V be the multiplication by a. Then the left-hand-side of the assertion becomes (note that in this case M →·a M is injective)

dim (V ) lenA(M/aM) = d(M, φ(M)) = ordA(det(φ)) = ordA a Quot(A)

= dimQuot(A)(V )ordA(a) = lenAp (Mp)ordA/p(a).

Here the essential ingredient is the determinant lemma (Theorem 2.6.6) used for the second equality. The third one is because a ∈ A and the forth is the homomorphism-property of the order (Proposition 2.5.3). For the last equality remember that we are in the case p = (0).

(Step four) Based on (Step three) we show the assertion by induction on r. Say

we know the claim up to r and assume suppA(M) = {mA, p1,..., pr+1}.

Let M1 denote the image of M in Mp1 and M2 denote the image of M in

Mp2∪···∪pr+1 . Then we have

suppA(M1) = {mA, p1} and suppA(M2) = {mA, p2,..., pr+1}. (*)

Thus, by induction hypothesis, we know the assertion for M1 and M2, so by (Step two) we also know it for M1 ⊕ M2.

– 92 – 4.3 Flat pull-back

Let ψ : M → M1 ⊕M2, then ker(ψ) and coker(ψ) both are supported on {mA} by construction and (*). Therefore they have ﬁnite length over A by Corollary 2.4.16, hence (Step one) yields the assertion for both ker(ψ) and coker(ψ). So we apply (Step two) to the exact sequence

0 −→ im(ψ) −→ M1 ⊕ M2 −→ coker(ψ) −→ 0

and obtain the assertion for im(ψ), hence another application of (Step two) to the exact sequence

0 −→ ker(ψ) −→ M −→ im(ψ) −→ 0

yields the claim for M and we win.

Theorem 4.3.2. Let f : X → Y be a ﬂat morphism of S-schemes and assume f is of relative S-dimension r. Let α ∈ Zk(Y ) be rationally equivalent to zero, then ∗ f (α) ∼ 0 in Zk+r(X).

Proof: By assumption we have a locally ﬁnite collection {Wj}j of closed integral subschemes of Y with dimS(Wj) = k + 1, together with closed immersions

ιj : Wj → Y

∗ and rational functions fj ∈ K(Wj) such that

X α = (ιj)∗ divWj (fj) . j

Consider the ﬁber squares f X / Y O O ι0 ι j j 0 W / Wj. j f 0

0 0 and let {Wj`}` be the collection of irreducible components of Wj. As f is of relative dimension r, Proposition 3.2.8 implies that

0 dimS(Wj`) = dimS(W ) + r = k + r + 1.

0 0 So, if we write ηj` for the generic point of Wj`, we get the (k + r + 1)-cycle 0 associated to Wj by

0 X 0 [W ]k+r+1 = nj`[W ] for nj` = lenO 0 O 0 0 . j j` X,η Wj ,ηj` ` j`

– 93 – 4 Rational equivalence and functoriality of Chow groups

0 Now the collection {Wj`}j,` is locally finite in X as we will show in the following: 0 As {Wj}j is locally finite in Y , Lemma 3.5.1 implies that {Wj}j is locally finite in X. Now take an open quasi-compact subset U ⊆ X. Then by Lemma 3.3.2 we 0 have that U ∩Wj 6= ∅ for only finitely many j. Fix such a j, then by Lemma 3.3.6 0 0 the collection {Wj`}` is locally finite in Wj. We again apply Lemma 3.3.2 to 0 0 0 {Wj`}` and U ∩ Wj to get that U ∩ Wj` 6= ∅ for only finitely many `. As we had only finitely many j we win.

0 Now, by Proposition 3.2.8 the induced morphism Wj` → Wj is dominant, hence 0 0 we get a field extension κ(Wj) ,→ K(Wj`) and we denote by fj` ∈ K(Wj`) the 0 image of fj under this extension. Together with the local finiteness of {Wj`} shown above we obtain a well-defined (k + r)-cycle

X 0 nj` β := ιj` divW 0 f ∈ Zk+r(X) ∗ j` j` j,`

and we claim that f ∗(α) = β.

0 0 For that we choose some closed integral subscheme Z ⊆ X with dimS(Z ) = k+r and we denote by µ the coeﬃcient of [Z0] in f ∗(α) and by ν the coeﬃcient of [Z0] in β. If we show µ = ν we are through.

0 We are only interested in those Z whose generic point ηZ0 is contained in some 0 Wj`, since otherwise both coeﬃcients µ and ν vanish. Hence in the following we restrict to those Z0.

Let Z := f(Z0) be endowed with the reduced structure, then Z ⊆ Y is a closed

integral subscheme. As ηZ0 ∈ Wj` for some j, `, we obtain f(ηZ0 ) ∈ Wj, thus

0 Z = f(Z ) = f {ηZ0 } = {f(ηZ0 )} ⊆ Wj

and therefore dimS(Z) ≤ dimS(Wj) = k + 1 by Proposition 3.2.3. Moreover dimS(Z) = k + 1 would imply Z = Wj and hence contradiction to the fact 0 0 dimS(Z ) < dimS(Wj). Thus we have dimS(Z) ≤ k. On the other hand we have Z0 ⊆ f −1(Z), so by Proposition 3.2.3 and Proposi- tion 3.2.8 0 −1 k + r = dimS(Z ) ≤ dimS f (Z) = dimS(Z) + r

0 hence we deduce dimS(Z ) ≥ k. Together we obtain

dimS(Z) = k.

0 In order to ease notation, we abbreviate η := ηZ and η := ηZ0 . Further deﬁne A := OY,η and B := OX,η0 . Then A and B are Noetherian local rings and we get

– 94 – 4.3 Flat pull-back

an induced ring homomorphism

φ: A = OY,η = OY,f(η0) −→ OX,η0 = B

which is ﬂat and local (note f(η0) = η by Lemma 2.3.10).

Now by local ﬁniteness of {Wj}j there are only ﬁnitely many j1, . . . , js such that

η ∈ Wji . By abuse of notation we abbreviate those by Wi := Wji and we note that all these Wi correspond to prime ideals pi ⊆ A (because η ∈ Wi). For now, we fix some index i in the following. With the same argument as above 0 0 0 0 we have η ∈ Wi` for only finitely many `. Denote them by Wi1,...,Wit. By 0 construction Wi` correspond to prime ideals qi` ⊆ B that are minimal w.r.t. containing piB. We further fix an k ∈ {1, . . . , t} and we have a closer look at nik bearing in mind the identifications and correspondences above:

. 0 nik = lenOX,η OW ,η 0 = lenOX,η OX,η 0 mY,ηW OX,η 0 W 0 i W W 0 W i W ik ik ik ik ik = lenBq Bq piBq = lenBq B/piB . (*) ik ik ik ik qik

Here the second equality follows from Corollary 3.5.4 and the last equality by exactness of localization. Moreover, under the above correspondences, we can ∗ ∗ view fi ∈ K(Wi) as an element in κ(pi) .

Next let us have a closer look at the coeﬃcients µ and ν.

(The coeﬃcient µ) By deﬁnitions we have

∗ X X h −1 i f (α) = ordV (fj) f (V ) , k+r j V ⊆Wj prime div.

where we sloppily omitted the push-forward along ιj. Let V ⊆ Wj be a prime 0 −1 0 divisor with Z ⊆ f (V ), then Z = f(Z ) ⊆ V , but as dimS(Z) = dimS(V ) we get Z = V . That means, if we are looking for the coeﬃcient of [Z0] in f ∗(α), the only interesting summands are those with V = Z.

0 −1 In that situation the multiplicity of Z in [f (V )]k+r is given by

f −1(Z) µ = len O −1 0 = len O 0 /m O 0 = len (B/m B), Z0 OX,η0 f (Z),η OX,η0 X,η Y,η X,η B A

where the second equality is Corollary 3.5.4. Together we obtain

s −1 X Wi f (Z) X µ = ordZ (fi) · µZ0 = ordA/pi (fi) lenB(B/mAB), i=1 i

Wi where by ordZ (fi) we mean the order of fi w.r.t. Z ⊆ Wi.

– 95 – 4 Rational equivalence and functoriality of Chow groups

(The coeﬃcient ν) By deﬁnitions, together with the homomorphism-property of the order (Proposition 2.5.3), we obtain

s t X X nik X X ν = ordB/qik fik = ordB/qik (fik) · nik i=1 k=1 i k (*) = X X ord (f ) · len ((B/p B) ) . B/qik ik Bqik i qik i k

(Equality of coeﬃcients) In order to show µ = ν, it clearly suﬃces to verify that every ith-summand in µ and ν coincides.

a Writing fi ∈ κ(pi) as fi = b for a, b ∈ A\p and using the homomorphism-property of the order (Proposition 2.5.3), we see that it is enough to show the following equation

X ordA/p (a) lenB(B/mAB) = ordB/q (φ(a)) lenBq B/piB . (†) i ik ik qik k

Now ﬁrst observe

ordA/pi (a) = lenA/pi (A/pi)/(a) = lenA/pi A/(pi + (a)) = lenA A/(pi + (a)) ,

where the last equation is Lemma 2.4.3. So the left hand side of (†) equals

ordA/pi (a) lenB(B/mAB) = lenA A/(pi + (a)) lenB(B/mAB) = lenB B ⊗A (A/(pi + (a))) = lenB B/(pi + (a))B = lenB B/(piB + (φ(a))) ,

where the second equation is valid by Lemma 2.4.12. Plugging this into (†) we see that, what we have to show has become

X lenB B/(piB + (φ(a))) = ordB/q (φ(a)) lenBq B/piB . (♦) ik ik qik k

Finally, by Lemma 4.3.1 applied to the ring B with module B/piB, whose support is given by V(pi) = {qi1,..., qit, mB} and to the element φ(a) ∈ B, which is a non-zero-divisor as A → B is ﬂat, we see that (♦) holds, hence we are done.

4.4 Properties of the Chow group

This conclusive section should collect properties of the Chow group. We already have seen many of them as properties of the cycle group Zk(X). Almost all of those properties on the level of cycles stay valid after passing to the Chow group. For

– 96 – 4.4 Properties of the Chow group a better overview we nevertheless stated all of them below and gave a reference to their proof in the thesis. But there are also new or stronger results, such as the localization sequence and an analogue to the Mayer-Vietoris sequence (both again highlighting the connection of Chow groups and homology).

Taking the Chow group associates to every S-scheme X and every integer k an Abelian group

CHk(X) = Zk(X)/ Ratk(X). Therefore, we obtain a graded Abelian group

M CH(X) := CHk(X). k∈Z We can also grade everything by codimension. In this case we obtain

CH(X) = M CHk(X). k∈Z

Every S-scheme X has the same integral subschemes as its reduced subscheme Xred. ∼ Also the function ﬁelds coincide, hence we obtain CHk(X) = CHk(Xred) for all k and therefore ∼ CH(X) = CH(Xred).

Furthermore, if X is a disjoint union of S-schemes X1,...,Xn, then clearly

∼ M CHk(X) = CHk(Xi). i

This reminds one to the additivity axiom for homology.

Next, the Chow group is functorial in the following sense:

Proposition 4.4.1. Let f : X → Y and g : Y → Z be morphisms of S-schemes and let k ∈ Z. (i) Assume f is proper, then we get an induced group homomorphism

f∗ : CHk(X) −→ CHk(Y ).

If in addition g is proper, then also g ◦ f is proper and we get and equality of group homomorphisms

(g ◦ f)∗ = g∗ ◦ f∗ : CHk(X) −→ CHk(Z).

(ii) Assume f is ﬂat of relative dimension r, then we get an induced group homomorphism ∗ f : CHk(Y ) −→ CHk+r(X).

– 97 – 4 Rational equivalence and functoriality of Chow groups

If in addition g is ﬂat of relative dimension r0, then g ◦ f is ﬂat of relative dimension r + r0 and we get an equality of group homomorphisms

∗ ∗ ∗ (g ◦ f) = f ◦ g : CHk(Z) −→ CHk+r+r0 (X).

(iii) If in the situation of (ii) the schemes are S-equidimensional (e.g. if they are irreducible), we get

(g ◦ f)∗ = g∗ ◦ f ∗ : CHk(Z) −→ CHk(X).

Proof: See Proposition 3.4.3, Proposition 3.5.8, Lemma 3.5.9 together with The- orem 4.2.4 and Theorem 4.3.2.

Now taking Chow group associated to a coherent OX -module commutes with proper push-forward and ﬂat pull-back, so there is a link to K-theory:

Proposition 4.4.2. Let f : X → Y be a morphism of S-schemes.

(i) Assume f is proper and F is a coherent OX -module with dimS(supp(F)) = k, then

f∗ ([F]k) = [f∗(F)]k ∈ CHk(Y ).

For a closed subscheme Z ⊆ X with dimS(Z) = k we have

f∗ ([Z]k) = [f∗(OZ )]k .

(ii) Assume f is ﬂat of relative dimension r and F is a coherent OY -module whose support has S-dimension k. In this case we have

∗ ∗ f ([F]k) = [f (F)]k+r ∈ CHk+r(X).

For a closed subscheme Z ⊆ Y with dimS(Z) = k we obtain

∗ h −1 i f ([Z]k) = f (Z) . k+r

(iii) Assume we have a short exact sequence

0 −→ G −→ F −→ H −→ 0

of coherent OX -modules, whose support is of S-dimension at most k, then

[F]k = [G]k + [H]k in CHk(X).

Proof: See Proposition 3.4.4, 3.5.6, Lemma 3.3.11 and Corollary 3.5.7.

– 98 – 4.4 Properties of the Chow group

Next we obtain an assertion related to base-change:

Proposition 4.4.3. Consider a cartesian diagram of S-schemes:

f X / Y O O 0 g g W / Z. f 0

Assume f is proper and g is ﬂat of relative dimension r. Then we have

∗ 0 0 ∗ g ◦ f∗ = (f )∗ ◦ (g ) : CHk(X) −→ CHk+r(Z).

Proof: See Proposition 3.6.1.

Until now every result immediately followed from the results on the cycle group

Zk(X). The next result needs a proof, as it is partially stronger than the assertion for cycles. We will elaborate the so-called localization sequence. This sequence again shows a slight connection to homology. However, in homology it would be part of a long exact sequence. An attempt to extend this localization sequence to the left, leads – at least for schemes over a ﬁeld – to higher Chow groups and motivic cohomology.

Proposition 4.4.4 (localization sequence). Let X be an S-scheme and U ⊆ X be an open subscheme with open immersion i: U → X. Denote by Z := X \ U the closed complement equipped with the reduced structure, then we have a closed immersion j : Z → X and the sequence

∗ j∗ i CHk(Z) −→ CHk(X) −→ CHk(U) −→ 0 is exact for all k ∈ Z. Proof: By Proposition 3.6.2 we already have the surjectivity of i∗ and the complex ∗ property im(j∗) ⊆ ker(i ).

∗ Let α ∈ Zk(X) such that i (α) ∼ 0 in CHk(U). Then by deﬁnition we have a locally ﬁnite collection {Wj}j of subschemes of U of S-dimension k + 1 and ∗ rational functions fj ∈ K(Wj) such that

∗ X i (α) = (ιj)∗ divWj (fj) ∈ Zk(U). j

With the same arguments as in the proof of Proposition 3.6.2, we see that taking 0 the closure of Wj in X yields a locally ﬁnite collection {Wj} of subschemes of 0 X of S-dimension k + 1 and clearly K(Wj) = K(Wj) so we obtain a well-deﬁned

– 99 – 4 Rational equivalence and functoriality of Chow groups

element X 0 γ := (ι )∗ div 0 (fj) ∈ Z (X) j Wj k j 0 0 (where ιj : Wj → X is the induced closed immersion), which is by deﬁnition rationally equivalent to zero and by construction (note for example that taking

the closure induces a bijection between the prime divisors of Wj and the prime 0 divisors of Wj) we have

∗ X ∗ i (γ) = (ιj)∗ divWj (fj) = i (α). j

∗ Therefore we have α − γ ∈ ker(i ) = im(j∗) (exactness on the level of cycles!). Thus there exists a k-cycle β ∈ Zk(Z) such that j∗(β) = α − γ, hence we obtain

α − j∗(β) = γ ∼ 0 ∈ Zk(X).

So α = j∗(β) in CHk(X).

Sticking at similarities between homology and Chow groups we also can state an analogue to the Mayer-Vietoris sequence:

Proposition 4.4.5 (Mayer-Vietoris). Let X be an S-scheme and let X1, X2 be two closed subschemes. Then for any k ∈ Z we obtain an exact sequence

φ ψ CHk(X1 ∩ X2) −→ CHk(X1) ⊕ CHk(X2) −→ CHk(X1 ∪ X2) −→ 0, where the maps φ and ψ are given as follows:

Let i` : X1 ∩ X2 → X` and j` : X` → X1 ∪ X2 be the closed immersions. Then φ(α) := (i1)∗(α) , (i2)∗(α) , ψ(α, β) := (j1)∗(α) − (j2)∗(β).

Proof: Since all the push-forwards in the assertion are along closed immersions,

the degree is always 1, so by abuse of notation we write i∗([Z]) = [Z] for any closed immersion i.

Now observe that clearly ψ(φ(α)) = α − α = 0

for all α ∈ Zk(X1 ∩ X2). This implies that the sequence is a complex.

To show surjectivity, let

X α = nZ [Z] ∈ Zk(X1 ∪ X2), Z

where the sum is over all subschemes Z ⊆ X1 ∪ X2 with dimS(Z) = k. Now

– 100 – 4.4 Properties of the Chow group

deﬁne X α1 := nZ [Z] Z⊆X1 dimS (Z)=k

which we can view as an element in both Zk(X1) and Zk(X1 ∪ X2). Now by construction the cycle

α2 := α1 − α

has non-zero coeﬃcients only for subschemes Z with Z 6⊆ X1. But clearly we can write Z as a union

Z = (Z ∩ X1) ∪ (Z ∩ X2)

of two closed subsets of Z, hence by irreducibility of Z (and Z 6⊆ X1) we obtain Z ⊆ X2. Therefore we can view α2 as an element in Zk(X2) and clearly

ψ(α1, α2) = α1 − α2 = α.

This provides surjectivity on the level of cycles and therefore also modulo rational equivalence.

For the last step let (α1, α2) ∈ ker(ψ). That means there is a locally ﬁnite collection {Wj}j∈J of subschemes of X1 ∪ X2 having S-dimension k + 1, together ∗ with rational functions fj ∈ K(Wj) and closed immersions ιj : Wj → X1 ∪ X2 such that X α1 − α2 = (ιj)∗ divWj (fj) ∈ Zk(X1 ∪ X2). j∈J 0 Now set J := j ∈ J Wj ⊆ X1 . Then by the above we have

X X α1 − (ιj)∗ divWj (fj) = α2 + (ιj)∗ divWj (fj) =: α. (*) j∈J0 j∈J\J0

With the same arguments as for the surjectivity we see that the left-hand-side of

(*) can be viewed as an element in Zk(X1) and the right-hand-side as an element in Zk(X2), hence α can be viewed as an element in Zk(X1 ∩ X2). Passing to the Chow group we obtain that

φ(α) = (α, α) = (α1, α2),

as by construction and (*) we have α ∼ αi in Zk(Xi).

Finally, we evaluate the composition of ﬂat pull-back and proper push-forward along a ﬁnite morphism of some degree:

– 101 – 4 Rational equivalence and functoriality of Chow groups

Proposition 4.4.6. Let f : X → Y be a ﬂat morphism of S-schemes and assume it is ﬁnite of degree d. Then it is of relative dimension 0 and for all α ∈ CHk(Y ) we have ∗ f∗ (f (α)) = d · α in CHk(Y ).

Proof: See Proposition 3.6.5.

– 102 – Bibliography

[AM] Michael Francis Atiyah and Ian Grand Macdonald. Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.- London-Don Mills, Ont., 1969, pp. ix+128. [Bou] Nicolas Bourbaki. Commutative algebra. Chapters 1–7. Elements of Mathematics (Berlin). Translated from the French, Reprint of the 1989 English translation. Springer-Verlag, Berlin, 1998, pp. xxiv+625. isbn: 3-540-64239-0. [Ei] David Eisenbud. Commutative algebra. Vol. 150. Graduate Texts in Mathematics. With a view toward algebraic geometry. Springer-Verlag, New York, 1995, pp. xvi+785. isbn: 978-0-387-94269-8. [Ful] William Fulton. Intersection theory. Second. Vol. 2. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Sur- veys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1998, pp. xiv+470. isbn: 0-387-98549-2. [GH] Phillip Griﬃths and Joseph Harris. Principles of algebraic geometry. Wiley Classics Library. Reprint of the 1978 original. John Wiley & Sons, Inc., New York, 1994, pp. xiv+813. isbn: 0-471-05059-8. [EGA.III] Alexander Grothendieck. “Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I”. In: Inst. Hautes Études Sci. Publ. Math. 11 (1961), p. 167. issn: 0073-8301. [EGA.IV] Alexander Grothendieck. “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II”. In: Inst. Hautes Études Sci. Publ. Math. 24 (1965), p. 231. issn: 0073-8301. [GW] Ulrich Görtz and Torsten Wedhorn. Algebraic geometry I. Advanced Lec- tures in Mathematics. Schemes with examples and exercises. Vieweg + Teubner, Wiesbaden, 2010, pp. viii+615. isbn: 978-3-8348-0676-5. [Ha] Robin Hartshorne. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977, pp. xvi+496. isbn: 0-387-90244-9. [Hei] Daniel Heiß. “Dimension theory for commutative rings”. 235/SK 230 H473. Bachelors Thesis. Universität Regensburg, 2013.

– 103 – [Mat] Hideyuki Matsumura. Commutative ring theory. Vol. 8. Cambridge Studies in Advanced Mathematics. Translated from the Japanese by M. Reid. Cambridge University Press, Cambridge, 1986, pp. xiv+320. isbn: 0-521-25916-9. [Sta] The Stacks Project Authors. Stacks Project. http://stacks.math. columbia.edu. 2017. [Web] Andreas Weber. “Intersection Theory on Regular Schemes via Alter- ations and Deformation to the Normal Cone”. Dissertation. Universität Regensburg, 2015.

– 104 –