Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes

# Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes

Vincent Cossart and Uwe Jannsen and Shuji Saito
18th February 2013
11footnotetext: Université de Versailles, laboratoire de mathématiques UMR8100, 45 avenue des États-Unis, F78035 Versailles cedex, France, Vincent.Cossart@math.uvsq.fr22footnotetext: Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93053 Regensburg, Germany, uwe.jannsen@mathematik.uni-regensburg.de 33footnotetext: Interactive Research Center of Science, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan sshuji@msb.biglobe.ne.jp

## 0 Introduction

The principal aim of this paper is to show the following three theorems on the resolution of singularities of an arbitrary reduced excellent noetherian scheme of dimension at most two. In the following, all schemes will be assumed to be noetherian, but see the end of the introduction and section 16 for locally noetherian schemes.

###### Theorem 0.1

(Canonical controlled resolution) There exists a canonical finite sequence of morphisms

 π:X′=Xn−−−−→…−−−−→X1−−−−→X0=X

such that is regular and, for each , is the blow-up of in a permissible center which is contained in , the singular locus of . This sequence is functorial in the sense that it is compatible with automorphisms of and (Zariski or étale) localizations.

We note that this implies that is an isomorphism over , and we recall that a subscheme is called permissible, if is regular and is normally flat along (see 2.1). The compatibility with automorphisms means that every automorphism of extends to the sequence in a unique way. The compatibility with the localizations means that the pull-back via a localization is the canonical resolution sequence for after suppressing the morphisms which become isomorphisms over . It is well-known that Theorem 0.1 implies:

###### Theorem 0.2

(Canonical embedded resolution) Let be a closed immersion, where is a regular excellent scheme. Then there is a canonical commutative diagram

 X′i′−−−−→Z′π⏐⏐↓⏐⏐↓πZXi−−−−→Z

where and are regular, is a closed immersion, and and are proper and surjective morphisms inducing isomorphisms and . Moreover, the morphisms and are compatible with automorphisms of and (Zariski or étale) localizations in .

In fact, starting from Theorem 0.1 one gets a canonical sequence and closed immersions for all , such that is the blow-up in and is identified with the strict transform of in the blow-up . Then is proper (in fact, projective) and surjective, and is regular since and are.

For several applications the following refinement is useful:

###### Theorem 0.3

(Canonical embedded resolution with boundary) Let be a closed immersion into a regular scheme , and let be a simple normal crossings divisor such that no irreducible component of is contained in . Then there is a canonical commutative diagram

 X′i′−−−−→Z′⊃B′πX⏐⏐↓πZ⏐⏐↓Xi−−−−→Z⊃B

where is a closed immersion, and are projective, surjective, and isomorphisms outside , and , where is the exceptional locus of (which is a closed subscheme such that is an isomorphism over ). Moreover, and are regular, is a simple normal crossings divisor on , and intersects transversally on . Furthermore, and are compatible with automorphisms of and with (Zariski or étale) localizations in .

More precisely, we prove the existence of a commutative diagram

 πB:B′=BmBm−1…B1B0=B⏐⏐↓⏐⏐↓⏐⏐↓⏐⏐↓⏐⏐↓πZ:Z′=Zm−−−−→Zm−1−−−−→…−−−−→Z1−−−−→Z0=Z↑⏐⏐↑⏐⏐↑⏐⏐↑⏐⏐⏐⏐↓π:X′=Xm−−−−→Xm−1−−−−→…−−−−→X1−−−−→X0=X

where the vertical morphisms are closed immersions and, for each , is the blow-up of in a permissible center , is the blow-up of in (so that is regular and is identified with the strict transform of in ), and is the complete transform of , i.e., the union of the strict transform of in and the exceptional divisor of the blow-up . Furthermore, is -permissible, i.e., is permissible, and normal crossing with (see Definition 3.1), which implies that is a simple normal crossings divisor on if this holds for on .

In fact, the second main theorem of this paper, Theorem LABEL:thm.main.3, states a somewhat more general version, in which can contain irreducible components of . Then one can assume that is not contained in the strongly -regular locus (see Definition LABEL:def.B-reg), and one gets that is normal crossing with (Definition 3.1). This implies that is an isomorphism above , and, in particular, again over . In addition, this Theorem also treats non-reduced schemes , in which case is regular and normal crossing with and is normally flat along .

Moreover, we obtain a variant, in which we only consider strict transforms for the normal crossings divisor, i.e., where is the strict transform of . Then we only get the normal crossing of (or in the non-reduced case) with the strict transform of in .

Theorem 0.1, i.e., the case where we do not assume any embedding for , will also be proved in a more general version: Our first main theorem, Theorem LABEL:thm.main.1, allows a non-reduced scheme as well as a so-called boundary on , a notion which is newly introduced in this paper (see section 4). Again this theorem comes in two versions, one with complete transforms and one with strict transforms.

Our approach implies that Theorem LABEL:thm.main.1 implies Theorem LABEL:thm.main.3. In particular, the canonical resolution sequence of Theorem LABEL:thm.main.3 for and strict transforms (or of Theorem 0.3 for this variant) coincides with the intrinsic sequence for from Theorem 0.1. Thus, the readers only interested in Theorems 0.1 and 0.2 can skip sections 3 and 4 and ignore any mentioning of boundaries/normal crossings divisors (by assuming them to be empty).

We note the following corollary.

###### Corollary 0.4

Let be a regular excellent scheme (of any dimension), and let be a reduced closed subscheme of dimension at most two. Then there exists a projective surjective morphism which is an isomorphism over , such that , with the reduced subscheme structure, is a simple normal crossings divisor on .

In fact, applying Theorem 0.3 with , we get a projective surjective morphism with regular , a regular closed subscheme and a simple normal crossings divisor on such that is an isomorphism over (in fact, over ), and . Moreover, and intersect transversally. In particular, is normal crossing with in the sense of Definition 3.1. Hence we obtain the wanted situation by composing with , the blow-up of in the -permissible (regular) subscheme , and letting which is a simple normal crossings divisor, see Lemma 3.2.

Moreover we mention that Theorem 0.3 is applied in a paper of the second and third author [JS], to prove a conjecture of Kato and finiteness of certain motivic cohomology groups for varieties over finite fields. This was a main motivation for these authors to work on this subject.

To our knowledge, none of the three theorems is known, at least not in the stated generality. Even for we do not know a reference for these results, although they may be well-known. For integral of dimension 1, Theorem 0.1 can be found in [Be] section 4, and a proof of Theorem 0.3 is written in [Ja].

In 1939 Zariski [Za1] proved Theorem 0.1 (without discussing canonicity or functoriality) for irreducible surfaces over algebraically closed fields of characteristic zero. Five years later, in [Za3], he proved Corollary 0.4 (again without canonicity or functoriality) for surfaces over fields of characteristic zero which are embedded in a non-singular threefold. In 1966, in his book [Ab4], Abhyankar extended this last result to all algebraically closed fields, making heavy use of his papers [Ab3] and [Ab5]. Around the same time, Hironaka [H6] sketched a shorter proof of the same result, over all algebraically closed ground fields, based on a different method. Recently a shorter account of Abhyankar’s results was given by Cutkosky [Cu2]. For all excellent schemes of characteristic zero, i.e., whose residue fields all have characteristic zero, and of arbitrary dimension, Theorems 0.1 and 0.3 were proved by Hironaka in his famous 1964 paper [H1] (Main Theorem , p. 138, and Corollary 3, p. 146), so Theorem 0.2 holds as well, except that the approach is not constructive, so it does not give canonicity or functoriality. These issues were addressed and solved in the later literature, especially in the papers by Villamayor, see in particular [Vi], and by Bierstone-Millman, see [BM1], by related, but different approaches. In these references, a scheme with a fixed embedding into a regular scheme is considered, and in [Vi], the process depends on the embedding. The last issue is remedied by a different approach in [EH]. In positive characteristics, canonicity was addressed by Abhyankar in [Ab6].

There are further results on a weaker type of resolution for surfaces, replacing the blowups in regular centers by different techniques. In [Za2] Zariski showed how to resolve a surface over a not necessarily algebraically closed field of characteristic zero by so-called local uniformization which is based on valuation-theoretic methods. Abhyankar [Ab1] extended this to all algebraically closed fields of positive characteristics, and later [Ab2] extended several of the results to more general schemes whose closed points have perfect residue fields. In 1978 Lipman [Li] gave a very simple procedure to obtain resolution of singularities for arbitrary excellent two-dimensional schemes in the following way: There is a finite sequence of proper surjective morphisms such that is regular. This sequence is obtained by alternating normalization and blowing up in finitely many isolated singular points. But the processes of uniformization or normalization are not controlled in the sense of Theorem 0.1, i.e., not obtained by permissible blow-ups, and it is not known how to extend them to an ambient regular scheme like in Theorem 0.2. Neither is it clear how to get Theorem 0.3 by such a procedure. In particular, these weaker results were not sufficient for the mentioned applications in [JS]. This is even more the case for the weak resolution of singularities proved by de Jong [dJ].

It remains to mention that there are some results on weak resolution of singularities for threefolds over a field by Zariski [Za3] (char = 0), Abhyankar [Ab4] ( algebraically closed of characteristic – see also [Cu2]), and by Cossart and Piltant [CP1], [CP2] ( arbitrary), but this is not the topic of the present paper.

Our approach is roughly based on the strategy of Levi-Zariski used in [Za1], but more precisely follows the approach (still for surfaces) given by Hironaka in the paper [H6] cited above. The general strategy is very common by now: One develops certain invariants which measure the singularities and aims at constructing a sequence of blow-ups for which the invariants are non-increasing, and finally decreasing, so that in the end one concludes one has reached the regular situation. The choices for the centers of the blow-ups are made by considering the strata where the invariants are the same. In fact, one blows up ‘the worst locus’, i.e., the strata where the invariants are maximal, after possibly desingularising these strata. The main point is to show that the invariants do finally decrease. In characteristic zero this is done by a technique introduced by Hironaka in [H1], which is now called the method of maximal contact (see [AHV] and Giraud’s papers [Gi2] and [Gi3] for some theoretic background), and an induction on dimension.

But it is known that the theory of maximal contact does not work in positive characteristic. There are some theoretic counterexamples in [Gi3], and some explicit counterexamples for threefolds in characteristic two by Narasimhan [Na1], see also [Co2] for an interpretation in our sense. It is not clear if the counterexamples in [Na2], for threefolds in any positive characteristic, can be used in the same way. But in section 15 of this paper, we show that maximal contact does not even exists for surfaces, in any characteristic, even if maximal contact is considered in the weakest sense. Therefore the strategy of proof has to be different, and we follow the one outlined in [H6], based on certain polyhedra (see below). That paper only considers the case of a hypersurface, but in another paper [H3] Hironaka develops the theory of these polyhedra for ideals with several generators, in terms of certain ‘standard bases’ for them (which also appear in [H1]). The introduction of [H3] expresses the hope that this theory of polyhedra will be useful for the resolution of singularities, at least for surfaces. Our paper can be seen as a fulfilment of this program.

In his fundamental paper [H1], Hironaka uses two important invariants for measuring the singularity at a point of an arbitrary scheme . The primary is the -invariant , and the secondary one is the dimension (with ) of the so-called directrix of at . Both only depend on the cone of at . Hironaka proves that for a permissible blow-up and a point with image the -invariant is non-increasing: . If equality holds here (one says is near to ), then the (suitably normalized) -invariant is non-decreasing. So the main problem is to show that there cannot be an infinite sequence of blow-ups with ‘very near’ points (which means that they have the same - and -invariants).

To control this, Hironaka in [H3] and [H6] introduces a tertiary, more complex invariant, the polyhedron associated to the singularity, which lies in . It depends not just on , but on the local ring of at itself, and also on various choices: a regular local ring having as a quotient, a system of regular parameters for such that are ‘parameters’ for the directrix , and equations for as a quotient of (more precisely, a -standard base of  ). In the situation of Theorem 0.2, is naturally given as , but in any case, such an always exists after completion, and the question of ruling out an infinite sequence of very near points only depends on the completion of as well. In the case considered in section 13, it is not a single strictly decreasing invariant which comes out of these polyhedra, but rather the behavior of their shape which tells in the end that an infinite sequence of very near points cannot exits. This is sufficient for our purpose, but it might be interesting to find a strictly decreasing invariant also in this case. In the particular situation considered in [H6] (a hypersurface over an algebraically closed field), this was done by Hironaka; see also [Ha] for a variant.

As a counterpart to this local question, one has to consider a global strategy and the global behavior of the invariants, to understand the choice of permissible centers and the global improvement of regularity. Since the -invariants are nice for local computations, but their geometric behavior is not so nice, we use the Hilbert-Samuel invariant as an alternative primary invariant here. They were extensively studied by Bennett [Be], who proved similar non-increasing results for permissible blow-ups, which was then somewhat improved by Singh [Si1]. Bennett also defined global Hilbert-Samuel functions , which, however, only work well and give nice strata in the case of so-called weakly biequidimensional excellent schemes. We introduce a variant (Definition 1.28) which works for arbitrary (finite dimensional) excellent schemes. This solves a question raised by Bennett. The associated Hilbert-Samuel strata

are then locally closed, with closures contained in . In particular, is closed for maximal (here if for all ).

Although our main results are for two-dimensional schemes, the major part of this paper is written for schemes of arbitrary dimension, in the hope that this might be useful for further investigations. Only in part of section 5 and in sections 10 through 14 we have to exploit some specific features of the low-dimensional situation. According to our understanding, there are mainly two obstructions against the extension to higher-dimensional schemes: The fact that in Theorem 2.14 (which gives crucial information on the locus of near points) one has to assume or , and the lack of good invariants of the polyhedra for , or of other suitable tertiary invariants in this case.

We have tried to write the paper in such a way that it is well readable for those who are not experts in resolution of singularities (like two of us) but want to understand some results and techniques and apply them in arithmetic or algebraic geometry. This is also a reason why we did not use the notion of idealistic exponents [H7]. This would have given the extra burden to recall this theory, define characteristic polyhedra of idealistic exponents, and rephrase the statements in [H5]. Equipped with this theory, the treatment of the functions defining the scheme and the functions defining the boundary would have looked more symmetric; on the other hand, the global algorithm clearly distinguishes these two.

We now briefly discuss the contents of the sections. In section 1 we discuss the primary and secondary invariants (local and global) of singularities mentioned above. In section 2 we discuss permissible blow-ups and the behavior of the introduced invariants for these, based on the fundamental results of Hironaka and Bennett (and Singh).

In section 3 we study similar questions in the setting of Theorem 0.3, i.e., in a ‘log-situation’ where one has a ‘boundary’: a normal crossings divisor on . We define a class of log-Hilbert-Samuel functions , depending on the choice of a ‘history function’ characterizing the ‘old components’ of at . Then , where is the number of old components at . This gives associated log-Hilbert-Samuel strata

 X(˜ν)={x∈X∣HOX(x)=˜ν}for˜ν∈NN×N.

For a -permissible blow-up , we relate the two Hilbert-Samuel functions and strata, and study some transversality properties.

In section 4 we extend this theory to the situation where we have just an excellent scheme and no embedding into a regular scheme . It turns out that one can also define the notion of a boundary on : it is just a tuple (or rather a multiset, by forgetting the ordering) of locally principal closed subschemes of . In the embedded situation , with a normal crossings divisor on , the associated boundary on is just given by the traces of the components of on and we show that they carry all the information which is needed. Moreover, this approach makes evident that the constructions and strategies defined later are intrinsic and do not depend on the embedding. All results in section 3 can be carried over to section 4, and there is a perfect matching (see Lemma LABEL:lem.comp.emb-nemb). We could have started with section 4 and derived the embedded situation in section 3 as a special case, but we felt it more illuminating to start with the familiar classical setting; moreover, some of the results in section 4 (and later in the paper) are reduced to the embedded situation, by passing to the local ring and completing (see Remark LABEL:rem.emb-nemb, Lemma LABEL:lem.comp.emb-nemb and the applications thereafter).

In section 5 we state the Main Theorems LABEL:thm.main.1 and LABEL:thm.main.3, corresponding to somewhat more general versions of Theorem 0.1 and 0.3, respectively, and we explain the strategy to prove them. Based on an important theorem by Hironaka (see [H2] Theorem (1,B) and the following remark), it suffices to find a succession of permissible blowups for which the Hilbert-Samuel invariants decrease. Although this principle seems to be well-known, and might be obvious for surfaces, we could not find a suitable reference and have provided a precise statement and (short) proof of this fact in any dimension (see Corollaries LABEL:cor.max-elim and LABEL:cor.max-elim2 for the case without boundary, and Corollaries LABEL:cor.Omax-elim and LABEL:cor.Omax-elim1 for the case with boundary). The problem arising is that the set of Hilbert functions is ordered by the total (or product) order ( iff for all ), and that with this order there are infinite decreasing sequences in the set of all functions . This is overcome by the fact that the subset of Hilbert functions of quotients of a fixed polynomial ring is a noetherian ordered set, see Theorem LABEL:thm.ordHP. After these preparations we define a canonical resolution sequence (see Remark LABEL:rem.strategy for the definition of so-called -eliminations, and Corollaries LABEL:cor.max-elim and LABEL:cor.max-elim2 for the definition of the whole resolution sequence out of this).

We point out that in Remark LABEL:rem.strategy, we define these canonical resolution sequences, i.e., an explicit strategy for resolution of singularities for any dimension. It would be interesting to see if this strategy always works.

The proof of the finiteness of these resolution sequences for dimension two is reduced to two key theorems, Theorem LABEL:fu.thm0 and LABEL:Thm2, which exclude the possibility of certain infinite chains of blow-ups with near (or -near) points. The key theorems concern only isolated singularities and hence only the local ring of at a closed point , and they hold for of arbitrary dimension, but with the condition that the ‘geometric’ dimension of the directrix is (which holds for ). As mentioned above, for this local situation we may assume that we are in an embedded situation.

As a basic tool for various considerations, we study a situation as mentioned above, where a local ring (of arbitrary dimension) is a quotient of a regular local ring . In section 6 we discuss suitable systems of regular parameters for and suitable families of generators for . A good choice for is obtained if is admissible for (Definition LABEL:def2.11) which means that are affine parameters of the directrix of (so that is the -invariant recalled above). We study valuations associated to and initial forms (with respect to these valuations) of elements in and their behavior under change of the system of parameters. As for , in the special case that is generated by one element (case of hypersurface singularities), any choice of is good. In general, some choices of are better than the other. A favorable choice is a standard basis of (Definition 1.17) as introduced in [H1]. In [H3] Hironaka introduced the more general notion of a -standard base of which is more flexible to work with and plays an important role in our paper.

In section 7 we recall, in a slightly different way, Hironaka’s definition [H3] of the polyhedron associated to a system of parameters and a -standard basis , and the polyhedron which is the intersection of all for all choices of and as above (with fixed ). We recall Hironaka’s crucial result from [H3] that if is admissible and is what Hironaka calls well-prepared, namely normalized (Definition LABEL:def.normalized) and not solvable (Definition LABEL:def.solvable) at all vertices. Also, there is a certain process of making a given normalized (by changing ) and not solvable (by changing ) at finitely many vertices, and at all vertices, if is complete. One significance of this result is that it provides a natural way of transforming a -standard base into a standard base under the assumption that is admissible.

As explained above, it is important to study permissible blow-ups and near points and . In this situation, to a system at we associate certain new systems at . A key result proved in section 8 is that if is a standard base, then is a -standard base. The next key result is that the chosen is admissible. Hence, by Hironaka’s crucial result mentioned above, we can transform into a system , where is a standard base.

The Key Theorems LABEL:fu.thm0 and LABEL:Thm2 concern certain sequences of permissible blowups, which arise naturally from the canonical resolution sequence. We call them fundamental sequences of -permissible blowups (Definition LABEL:Def.fupb) and fundamental units of permissible blowups (Definition LABEL:Def.fupb2) and use them as a principal tool. These are sequences of -permissible blowups

 Xm→…→X1→X0=X,

where the first blowup is in a closed point (the initial point), and where the later blowups are in certain maximal -permissible centers , which map isomorphically onto each other, lie above , and consist of points near to . For a fundamental sequence there is still a -permissible center with the same properties; for a fundamental unit there is none, but only a chosen closed point (the terminal point) which is near to . In section 9 we study some first properties of these fundamental sequences. In particular we show a certain bound for the -invariant of the associated polyhedra. This suffices to show the first Key Theorem LABEL:fu.thm0 (dealing with the case ), but is also used in section 14.

For the second Key Theorem LABEL:Thm2 (dealing with the case ), one needs some more information on the (-dimensional) polyhedra, in particular, some additional invariants. These are introduced in section 10. Then Theorem LABEL:Thm2 is proved in the next three sections. It states that there is no infinite sequence

 …→X2→X1→X0

of fundamental units of blow-ups such that the closed initial points and terminal points match and are isolated in their Hilbert-Samuel strata. After some preparations in section 11, section 12 treats the case where the residue field extension is trivial (or separable). This is very much inspired by [H6], which however only treats the special situation of a hypersurface in a regular threefold over an algebraically closed field and does not contain proofs for all claims. Then section 13 treats the case where there occur inseparable residue field extensions . This case was basically treated in [Co1] but we give a more detailed account and fill gaps in the original proof, with the aid of the results of section 8, and Giraud’s notion of the ridge [Gi1], [Gi3] (faîte in French) a notion which generalizes the directrix.

In section 14, we show that maximal contact does not exist for surfaces in positive characteristic . For each a counterexample is given which then works for any field of that characteristic.

In section 15 we give a more algebro-geometric proof of the fact that it suffices to show how to eliminate the maximal Hilbert-Samuel stratum.

Finally, in section 16 we give a re-interpretation of the functoriality we obtain for our resolution, for arbitrary flat morphisms with regular fibers, and we apply this to show resolution for excellent schemes which are only locally noetherian, and for excellent algebraic stacks with atlas of dimension at most two.

It will be clear from the above how much we owe to all the earlier work on resolution of singularities, in particular to the work of Hironaka which gave the general strategy but also the important tools used in this paper.

Conventions and concluding remarks: All schemes are assumed to be finite dimensional. Regular schemes are always assumed to be locally noetherian. Recall also that excellent schemes are by definition locally noetherian.

In this introduction and in sections 1 to 15, the readers should best assume that all schemes are noetherian. At some places we write locally noetherian to indicate that certain definitions make sense and certain results still hold for schemes which are only locally noetherian. Resolution for such schemes is treated in section 16.

## 1 Basic invariants for singularities

In this section we introduce some basic invariants for singularities.

1.1 Invariants of graded rings and homogeneous ideals in polynomial rings

Let be a field and be a polynomial ring with variables. Let be the -subspace of the homogeneous polynomials of degree (including ). Fix a homogeneous ideal .

###### Definition 1.1

For integers we define as the supremum of the satisfying the condition that there exist homogeneous such that

 Sμ∩I=Sμ∩<φ1,…,φi−1>for all μ<ν.

By definition we have . We write

 ν∗(I)=(ν1(I),ν2(I),…,νm(I),∞,∞,…)

and call it the -invariant of . We have the following result (cf. [H1] Ch. III §1, Lemma 1).

###### Lemma 1.2

Let with homogeneous elements of degree such that:

• for all ,

• .

Then we have

 νi(I)={νi,i≤m,∞,i>m.
###### Definition 1.3

Let be a system of homogeneous elements in and be the ideal that it generates.

• is weakly normalized if it satisfies the condition of Lemma 1.2.

• is a standard base of if it satisfies the conditions and of Lemma 1.2.

We have the following easy consequence of 1.2.

###### Corollary 1.4

Let be a homogeneous ideal and let be a system of homogeneous elements in which is weakly normalized.

• The following conditions are equivalent:

• for .

• For all , has minimal degree in such that .

• If the conditions of (1) are satisfied, then can be extended to a standard base of .

By the lemma a standard base of and are obtained as follows:
Put and pick .
Put and pick .
Proceed until we get . Then is a standard base of and .

###### Remark 1.5

Let be homogeneous generators of such that , where . Then the above considerations show that

 (ν1,ν2,…,νℓ,∞,…)≤ν∗(I),

because a standard base of is obtained by possibly omitting some of the for .

In what follows, for a -vector space (or a -algebra) and for a field extension we write . From Lemma 1.2 the following is clear.

###### Lemma 1.6

For the ideal we have

 ν∗(I)=ν∗(IK).

A second invariant is the directrix. By [H1] Ch. II §4, Lemma 10, we have:

###### Lemma 1.7

Let be a field extension. There is a smallest -subvector space such that

 (IK∩K[T(I,K)])⋅SK=IK,

where . In other words is the minimal -subspace such that is generated by elements in . For we simply write .

Recall that is called the cone of the graded ring .

###### Definition 1.8

For any field extension , the closed subscheme

 DirK(S/I)=Dir(SK/IK)⊆C(SK/IK)=C(S/I)×kK

defined by the surjection is called the directrix of in over . By definition

 DirK(S/I)≅Spec(SymK((S1)K/T(I,K))).

We define

 e(S/I)K=dim(DirK(S/I))=dim(S)−dimK(T(I,K))=n−dimK(T(I,K)),

so that , and simply write for with .

###### Remark 1.9

(a)  By definition is determined by the pair , but indeed it has an intrinsic definition depending on for only: Let , which is a polynomial ring over . Then the surjection factors through the canonical surjection , and the directrix as above is identified with the directrix defined via and . In this way is defined for any graded -algebra which is generated by elements in degree 1.

(b)  Similarly, for any standard graded -algebra , i.e., any finitely generated graded -algebra which is generated by , we may define its intrinsic -invariant by , where is the canonical epimorphism. In the situation of Definition 1.1 we have

 ν∗(I)=(1,…,1,ν1(S/I),ν2(S/I),…),

with entries of 1 before , where , and .

(c)  If is a variable, then obviously and in the situation of 1.1. On the other hand, , i.e., .

###### Lemma 1.10

Let the assumptions be as in 1.8.

• .

• For field extensions , we have .

• The equality holds if one of the following conditions holds:

• is separable (not necessarily algebraic).

• .

In particular it holds if is a perfect field.

Proof The inequality in (1) is trivial, and (2) follows since , This in turn implies claim (3) for condition (ii). Claim (3) for condition (i) is proved in [H1] II, Lemma 12, p. 223, for arbitrary degree of transcendence. The case of a finite separable extension is easy: It is obviously sufficient to consider the case where is finite Galois with Galois group . Then, by Hilbert’s theorem 90, for any -vector space on which acts in a semi-linear way the canonical map is an isomorphism. This implies that and the claim follows.

Finally we recall the Hilbert function (not to be confused with the Hilbert polynomial) of a graded algebra. Let be the set of the natural numbers (including ) and let be the set of the functions . We endow with the product order defined by:

 (1.1) ν≥μ⇔ν(n)≥μ(n) for any n∈N.
###### Definition 1.11

For a finitely generated graded -algebra its Hilbert function is the element of defined by

 H(0)(A)(n)=H(A)(n)=dimk(An)(n∈N).

For an integer we define inductively by:

 H(t)(A)(n)=n∑i=0H(t−1)(A)(i).

We note

 H(t−1)(A)(n)=H(t)(A)(n)−H(t)(A)(n−1)≤H(t)(n).
###### Remark 1.12

(a)  Obviously, for any field extension we have

 H(0)(AK)=H(0)(A).

(b)  For a variable we have

 H(t)(A[X])=H(t+1)(A).

(c)  For any and any define inductively by .

In a certain sense, the Hilbert function measures how far is away from being a polynomial ring:

###### Definition 1.13

Define the function by and for . Define for inductively as above, i.e., by .

Then one has

 Φ(t)(n)=H(0)(k[X1,…,Xt])(n)=(n+t−1n)for% all n≥0.

and:

###### Lemma 1.14

Let be a finitely generated graded algebra of dimension over a field , which is generated by elements in degree one (i.e., is a standard graded algebra).

• Then , and equality holds if and only if .

• For a suitable integer one has .

• If for , then .

Proof (a): We may take a base change with an extension field , and therefore may assume that is infinite. In this case there is a Noether normalization such that the elements are mapped to , the degree one part of , see [Ku], Chap. II, Theorem 3.1 d). This means that is a monomorphism of graded -algebras. But then , and equality holds if and only if is an isomorphism.

(b): Since is a finitely generated -module, it also has finitely many homogenous generators of degrees . This gives a surjective map of graded -modules

 ⊕mi=1S[−di]↠A,

where is with grading , and hence

 H(A)(n)≤m∑i=1Φ(d)(n−di)≤mΦ(d)(n).

(c) follows from (a) and (b), because and have different asymptotics for .

We shall need the following property.

###### Theorem 1.15

Let be a field and for , let be the set of all Hilbert functions of all standard graded -algebras with .

(a) is independent of .

(b) is a noetherian ordered set, i.e.,

• is well-founded, i.e., every strictly descending sequence in is finite.

• For every infinite subset there are elements with .

Proof If is a standard graded -algebra, then holds if and only if is a quotient of , i.e., for a homogeneous ideal . On the other hand, it is known that for some monomial ideal , i.e., an ideal generated by monomials in the variables (see [CLO] 6 §3, where this is attributed to Macaulay; to wit, one may take for the ideal of leading terms for , with respect to the lexicographic order on monomials, loc. cit. Proposition 9). This proves (a). Moreover, for (b) we may assume that all considered Hilbert functions are of the form for some monomial ideal . Thus in (b) (ii) we are led to the consideration of an infinite set of monomial ideals in . But then the main theorem in [Macl] (Theorem 1.1) says that there are with , so that . For (b) (i) we may again assume that all are of the form for some monomial ideals , and by [Macl] 1.1 that one finds an infinite chain among these ideals, necessarily strictly increasing since the sequence of the is, which is a contradiction because is noetherian.

###### Remark 1.16

(a) For the functions instead of the functions property (b) (i) was shown in [BM2] Theorem 5.2.1.
(b) Our proof was modeled on [AP] Corollary 3.6, which is just formulated for monomial ideals. See also [AP] Corollary 1.8 for another argument that the set of monomial ideals in , with respect to the reverse inclusion, is a noetherian ordered set. Finally, in [AH] it is shown, by more sophisticated methods, that even the set of all Hilbert functions is noetherian.

1.2 Invariants for local rings

For any ring and any prime ideal we set

 grp(R)=⨁n≥0pn/pn+1,

which is a graded algebra over .

In what follows we assume that is a noetherian regular local ring with maximal ideal and residue field . Moreover, assume that is regular. Then we have

 grp(R)=SymR/p(gr1p(R)),gr1p(R)=p/p2.

where denotes the symmetric algebra of a free -module . Concretely, let be a system of regular parameters for such that . Then is identified with a polynomial ring over :

 grp(R)=(R/p)[Y1,…,Yr]where Yi=yimodp2(1≤i≤r).

Fix an ideal . In case we set

and define an ideal by the exact sequence

 0→Inp(J)→grp(R)→grp(R/J)→0.

Note

 Inp(J)=⨁n≥0(J∩pn+pn+1)/pn+1.

For and a prime ideal put

 (1.2) vp(f)={max{ν|f∈pν},f≠0,∞,f=0

called the order of at . For prime ideals , we have the following semi-continuity result (cf. [H1] Ch.III §3):

 (1.3) vp(f)≤vq(f) for ∀f∈R.

Define the initial form of at as

 inp(f):=fmodpvp(f)+1∈pvp(f)/pvp(f)+1∈grp(R).

In case we have

 Inp(J)={inp(f)|f∈J}.
###### Definition 1.17
• A system of elements in is a standard base of , if

is a standard base of in the polynomial ring .

• We define as the -invariant (cf. Definition 1.1) for .

• The absolute -invariant of a noetherian local ring with maximal ideal is defined as the absolute -invariant (cf. Remark 1.9 (b))