p-adic Weyl algebras [Elektronische Ressource] / vorgelegt von Alexis Pangalos

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Mathematikp-adic Weyl algebrasInaugural-Dissertationzur Erlangung des Doktorgradesder Naturwissenschaften im FachbereichMathematik und Informatikder Mathematisch-Naturwissenschaftlichen Fakultätder Westfälischen Wilhelms-Universität Münstervorgelegt vonAlexis Pangalosaus Herdecke/Ruhr- 2007 -Dekan: Prof. Dr. Dr. h.c. J. CuntzErster Gutachter: Prof. Dr. P. SchneiderZweiter Gutachter: Prof. Dr. Dr. h.c. J. CuntzTag der mündlichen Prüfung: 29. Januar 2008Tag der Promotion: 6. Februar 2008ContentsIntroduction 41 Weyl algebras 71.1 The classical Weyl algebra . . . . . . . . . . . . . . . . . . . . . 81.2 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Definition of completed Weyl algebras . . . . . . . . . . . . . . . 121.3 Division theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 13Division in the polynomial algebra . . . . . . . . . . . . . . . . 14Division in the classical Weyl algebra . . . . . . . . . . . . . . . 18Division in the completed Weyl algebraA . . . . . . . . . . . 20d,εDivision in the DMW-Weyl algebra . . . . . . . . . . . . . . . . 251.4 First properties of completed Weyl algebras . . . . . . . . . . . 29Tensor products of completed Weyl algebras . . . . . . . . . . . 322 Skew rings 332.1 Skew polynomial rings . . . . . . . . . . . . . . . . . . . . . . . 352.2 Rings of restricted skew power series . . . . . . . . . . . . . . . 38Alternative definition of complete Weyl algebras . . . . . . . . .

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• Mathematics

Informations

• Publié par: westfalische_wilhelms-universitat_munster
• Publié le: 01 janvier 2008
• Nombre de lectures: 10
• Langue: English

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Mathematik

p-adic Weyl algebras

Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Mathematik und Informatik der Mathematisch-Naturwissenschaftlichen Fakultät der Westfälischen Wilhelms-Universität Münster

vorgelegt von Alexis Pangalos aus Herdecke/Ruhr - 2007 -

Dekan:

Erster Gutachter:

Zweiter Gutachter:

Tag der mündlichen

Tag der Promotion:

Prüfung:

Prof.

Dr.

Dr.

h.c.

J.

Cuntz

Prof. Dr. P. Schneider

Prof. Dr. Dr. h.c. J. Cuntz

29. Januar 2008

6. Februar 2008

Contents

Introduction 4 1 Weyl algebras 7 1.1 The classical Weyl algebra . . . . . . . . . . . . . . . . . . . . . 8 1.2 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Definition of completed Weyl algebras . . . . . . . . . . . . . . . 12 1.3 Division theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Division in the polynomial algebra . . . . . . . . . . . . . . . . 14 Division in the classical Weyl algebra . . . . . . . . . . . . . . . 18 Division in the completed Weyl algebraA dε  20. . . . . . . . . . . Division in the DMW-Weyl algebra . . . . . . . . . . . . . . . . 25 1.4 First properties of completed Weyl algebras . . . . . . . . . . . 29 Tensor products of completed Weyl algebras . . . . . . . . . . . 32 2 Skew rings 33 2.1 Skew polynomial rings . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Rings of restricted skew power series . . . . . . . . . . . . . . . 38 Alternative definition of complete Weyl algebras . . . . . . . . . 39 Division in rings of skew power series . . . . . . . . . . . . . . . 40 Rings of restricted skew power series over a field . . . . . . . . . 46 3 Dimensions of Weyl algebras 48 3.1 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Upper bounds (d= 1 51) .. . . . . . . . . . . . . . . . . . . . . . 4 Filtration 55 4.1 Filtered rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Associated graded rings . . . . . . . . . . . . . . . . . . . . . . 57 4.3 Permanence properties . . . . . . . . . . . . . . . . . . . . . . . 61 5 A note on simple modules 66 References 69

Introduction

LetK consider the Webe a complete non-archimedean field.d-th classical Weyl algebraA d overK(see section 1.1 for a definition) and, for someε∈R 2>d 0 , endow it with the non-archimedeanK-vector space norm |f| ε = max|a αβ |ε (αβ) iff∈A d is written in the formf= P a αβ X α Y β  we require. Ifεto be an element ofR 2>d 0 such that|γ!|ε (−γ−γ) is bounded by some constant for allγ∈N d then the norm is in fact an algebra norm onA d  norm The(cf. lemma 1.2.1). is multiplicative if and only ifεsatisfiesε i ε d+i ≥1for all1≤i≤d(cf. lemma 1.2.4). We denote the completion ofA d with respect to this norm byA dε . The elements ofA dε can be written as formal power series in non-commuting variables such that the coefficients satisfy a certain convergence condition: A dε ={ X a αβ X α Y β :|a αβ |ε (αβ) →0for|α|+|β| → ∞} We call the elements ofA dε restricted power series. Different versions of com- pleted Weyl algebras appear in the literature. One can construct the algebra which is the union of allA dε withε 1 =  =ε d = 1andε i >1for all d+ 1≤i≤2dand the algebra which is the union of allA dε withε i >1for alli algebras are considered in [Ber] and [MN].. These latter version is The denoted byA † d and is called theDwork-Monsky-Washnitzer-Weylalgebra. The algebraA d(11) appears in [Nar2]. We call it theTate-Weylalgebra. The fact that we can define a whole family of algebra norms on the classical Weyl algebra defined over a non-archimedean field is in sharp contrast to the fact that the classical Weyl algebra defined over the field of complex numbers has no algebra norm at all (see for example [Cun]). The classical Weyl algebraA d defined over an arbitrary field has been exten- sively studied during the last 50 years. The classical Weyl algebraA d is a left and right Noetherian integral domain. The classical Weyl algebraA d is simple if defined over a field of characteristic zero. In this case the Krull and

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the global dimension ofA d ared; the Krull dimension ofA d was first deter- mined by Gabriel and Rentschler in [GR]. That the global dimension ofA d isdwas proved by Rinehart [Rin] ford= 1and in the general case by Roos [Roo]. The Krull and the global dimension ofA d are2difA d is defined over a field of characteristicp >0 classical Weyl algebra. TheA d is an Auslander regular ring. Stafford proved that any left ideal ofA d has a set of2generators ifA d  Theis defined over a field of characteristic zero [Sta]. simple modules over the classical Weyl algebraA 1  a longwere classified by Block [Blo]. For list of known and conjectured properties of the classical Weyl algebra see the introduction of [Bav]. In our thesis we are going to consider the question of which properties of the classical Weyl algebra over a complete non-archimedean field carry over to its various completions. For almost all results we will assume that the components ofεlie in the value group|K × |. We take this as a general assumption for this introduction and consider only the case where the norm onA dε is multiplicative. In [Nar2] Narváez Macarro proves division theorems for the Tate- and Dwork- Monsky-Washnitzer-Weyl algebra under the assumption that the fieldKis discretely valued. We prove a division theorem for all Weyl algebrasA dε defined over an arbitrary complete non-archimedean field (cf. theorem 1.3.14). It was suggested to me by L. Narváez Macarro how to prove the division theorem forA d † in the case of an arbitrary complete non-archimedean fieldK (cf. theorem 1.3.16). We use a technique similar to one used in [HM], [HN] and [NR]. In [Nar1] this technique is applied to the Dwork-Monsky-Washnitzer completion of the polynomial ring – a situation very similar toA † d . The division theorems enable us to prove some of the basic properties ofA dε andA † d  Weyl algebras. TheA dε andA † d are Noetherian (cf. proposition 1.4.1). An element ofA dε orA d † is a unit if and only if its exponent is zero (cf. proposition 1.4.3). We consider formal partial differentiation on elements of A dε andA † d  a consequence of Asshow that it respects two-sided ideals.and this result, together with the characterization of units, we get thatA dε and

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A d † are simple rings if we assume the characteristic ofKto be zero. We prove that the Krull dimension and the global dimension of the completed Weyl algebraA dε are bounded below byd(cf. propositions 3.1.2 and 3.1.3). The lower bound is given by2dif we assume that the fieldKhas characteristic zero (cf. propositions 3.1.5 and 3.1.6). In the study of the classical Weyl algebraA d it turns out to be very useful to consider the localizations ofA d with respect to the Ore setsK[X i ]\{0}resp. K[Y i ]\{0}. One might expect that the multiplicative subsetsKhX i i ε i \{0}and KhY i i ε i \{0}ofA dε the sets of all non-zero restricted power series in, i.e. X i resp.Y i , are Ore sets inA dε er, this is not the case (cf. lemma 2.0.1) . Howev which is equivalent to the fact that the localizations ofA dε with respect to these sets do not exist. Section 2 provides us with a construction of restricted skew power series rings. i We use this construction to define ring extensionsB εd X i andB Y dε ofA dε (cf. section 3.2). These rings will to some extent play the role of the localizations in the case of the classical Weyl algebra. In fact, the ringsB εd X i resp.B Y d i ε are the microlocalizations ofA dε with respect to the setsKhX i i ε i \{0}resp. KhY i i ε i \{0} We set(for the notion of microlocalizations see [LvO] or [Nag]). d d B dε := M B X d i ε ⊕ M B Y d i ε  i=1i=1 With the assumption that the characteristic ofKis zero we prove the following lemma. For any maximal left idealI⊂ A d ε the left idealB dε Igenerated  byIis not the unit idealB dε (cf. lemma 3.2.1). proof involves both The the division theorem for the Weyl algebraA dε (cf. theorem 1.3.14) and the division theorems forB X d i ε andB Y d i ε  This(cf. theorem 2.2.4). lemma will be an important ingredient to obtain upper bounds for the Krull dimension and the global dimension ofA dε in section 4. In analogy to the fact that the localizations of the classical Weyl algebraA 1 mentioned above are simple principal left and right ideal domains, the rings B 1 X ε andB 1ε X are simple principal left and right ideal domains, too (cf. propo- sitions 3.2.4 and 2.2.8).

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Under the additional assumption that the fieldKis discretely valued it is possible to define a complete and separated filtration onA dε coming from the algebra norm. This allows us to apply the theory of filtered rings (for an introduction to the theory of filtered rings see [LvO]). We obtain the follow- ing. The completed Weyl algebraA dε is Auslander regular (cf. proposition 4.3.3). We show that the Krull dimension and the global dimension ofA dε are bounded above by2d(cf. proposition 4.3.6) which, when combined with the lower bounds computed in section 3, implies that the Krull dimension and the global dimension ofA dε are2dif the characteristic ofKisp >0 prove. We that the Krull dimension and the global dimension ofA dε are bounded above by2d−1if the characteristic ofK the Krull Henceis zero (cf corollary 4.3.8). dimension and the global dimension ofA 1ε are1. For some special cases we  also prove our conjecture that the Krull dimension and the global dimension ofA dε aredtrue for the Tate-Weyl algebra if the residue example, this is . For fieldkofK prove an analog of Wehas characteristic zero (cf. remark 4.3.10). Staffords theorem forA 1ε , i.e. any left ideal ofA 1ε has a set of2generators if the characteristic ofKis zero (cf. corollary 4.3.9). In section 5 we show that the so called saturationS sat of the subsetKhXi ε 1 \{0} ofA 1ε is an Ore set inA 1d (cf. proposition 5.1). simple TheS sat -torsionfree A 1d -modules are in bijection with the simple(S sat ) −1 A 1d -modules (cf. corol- lary 5.3). Acknowledgments.I would like to thank my thesis advisor Peter Schneider for his guidance. I am also grateful to Luis Narváez Macarro for discussions about division theorems, to Jan Kohlhaase for reading preliminary versions of this thesis and for his encouragement, to my friend Stefan Wiech for the idea to go to Münster, and to my friend Ralf Diepholz for his support.

1 Weyl algebras

LetKdenote a field.

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1.1 The classical Weyl algebra

Thed-th Weyl algebra overK, here always calledclassical Weyl algebra and denoted byA d , is the algebra with2dgeneratorsX 1      X d  Y 1      Y d and relations Y i X j −X j Y i =δ ij and X i X j −X j X i =Y i Y j −Y j Y i = 0 whereδ ij denotes the Kronecker delta (see [McCR] section 1.3). Note that if objects or properties have left and right versions we restrict to the left version as [McCR] always use right versions. The elements ofA d have a unique expression as finite sums X a αβ X α Y β αβ∈N d with coefficientsa αβ ∈Kand the notationX α =X 1 α 1   X d α d andY α = d Y 1 α 1   Y d α . Weform and get the following rules always write elements in this of multiplication. Lemma 1.1.1.We have Y β X α = X γ∈N d γ!  β γ  α γ  X α−γ Y β−γ  γ i ≤ i αβ i Iff= P a X α Y β ,g= P b αβ X α Y β andf g= P c αβ X α Y β then αβ αβ = X a α ′ β ′ b α ′′ β ′′ γ! c  β ′  γ α ′′   α  β ′  α ′′  β ′′  γ∈N d γ ′ ′ α+α ′′ −γ=α β ′ +β ′′ −γ=β Proof. second statementThe first statement follows from [Dix] lemma 2.1. The follows from the first. Here we used the notationγ! :=γ 1 !  γ d !and  γ α  :=  γ α 1 1      γ α d d   write. We |α|:=α 1 +  +α d for elements inN d . For06=f= P a αβ X α Y β ∈A d the degreeis deg(f) := max{|α|+|β| ∈N:a αβ 6= 0}

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or−∞iff= 0in agreement with the usual definition of degree of polynomials in several variables. AsK-vector spaces the classical Weyl algebraA d and the polynomial ring in2dvariables are isomorphic. distinguish the two To different algebra structures we writefor polynomial multiplication and∗for Weyl algebra multiplication if we want to emphasize in which ring we multiply. Lemma 1.1.2.Letf g∈A d . Then deg(f∗g) = deg(f) + deg(g) and deg(fg−f∗g)<deg(f) + deg(g)

Proof.The first statement follows from [Dix] lemma 2.4.(ii). second state- The ment follows from [Dix] lemma 2.4.(i).

The classical Weyl algebra has the following basic algebraic properties. Theorem 1.1.3.The classical Weyl algebraA d is a Noetherian integral do- main. If the fieldKhas characteristic zero thenA d is simple, i.e. has no two-sided ideals other than0andA d .

Proof.[McCR] theorem 1.3.5 and theorem 1.3.8.(i).

1.2 Completions Let(K| |) On thedenote a complete non-archimedean field.d-th classical Weyl algebraA d we have for anyε∈R 2>d 0 the non-archimedeanK-vector space norm| | ε defined by |f| ε := max|a αβ |ε (αβ) forf= P a αβ X α Y β ∈A d , whereε (αβ) =ε 1 α 1   ε α d d ε β d 1 +1   ε 2d β d , i.e. we have

(i)|f| ε = 0ifff= 0,

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(ii)|af| ε =|a||f| ε , (iii)|f+g| ε ≤max{|f| ε |g| ε }, for allf g∈A d anda∈K. Lemma 1.2.1.Letεbe an element ofR 2>d 0 such that|γ!|ε (−γ−γ) is bounded by some constantC >0for allγ∈N d . Then |f g| ε ≤C|f| ε |g| ε  Ifε i ε d+i ≥1for alli, we have|f g| ε ≤ |f| ε |g| ε . Proof. |f g| ε = m α a β x| α ′  β ′  α ′′  X β ′′  γ∈N d a α ′ β ′ b α ′′ β ′′ γ!  γ β ′  γ α ′′  |ε (αβ) α ′ +α ′′ −γ=α β ′ +β ′′ −γ=β ′ ≤ α ′ β ′ α ′ m ′  a β ′ x ′ γ∈N d |a α ′ β ′ ||b α ′′ β ′′ ||γ!  β γ ′  γ α ′′  |ε (α ′ +α ′ −γβ ′ +β ′′ −γ) ≤ γ s ∈ u N p d |γ!|ε (−γ−γ) max N d |a α ′ β ′ |ε (α ′ β ′ ) α ′′  m β ′′ a ∈ x N d |b α ′′ β ′′ |ε (α ′′ β ′′ ) α ′ β ′ ∈ =C|f| ε |g| ε  Ifεsatisfiesε i ε d+i ≥1for all1≤i≤dwe have|γ!|ε (−γ−γ) ≤1for allγ, which gives the second part. Remark 1.2.2.If for exampleK=Q p , we know that|γ!|converges exponen- tially to zero as|γ| Hence we easily find angoes to infinity.ε∈R 2>d 0 with all ε i <1such that|γ!|ε (−γ−γ) is bounded by some constantC. Remark 1.2.3.Ifε i ε d+i <1for somei, then| | ε is not submultiplicative, for example 1 =|X i Y i + 1| ε =|Y i X i | ε 6≤ |Y i | ε |X i | ε =ε d+i ε i  However instead of| | ε we can take the equivalent norm| | ′ ε defined by |f| ′ ε := sup{|f g| ε |g| ε −1 ; 06=g∈A d } This norm is submultiplicative (see [BGR] §1.2.1., prop. 2, and note that the proof is the same in the non-commutative case).

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Lemma 1.2.4.The norm| | ε onA d is multiplicative if and only ifε i ε d+i ≥1 for all1≤i≤d. Proof.The “only if” is remark 1.2.3. Let≺be a total order onN 2d compatible with addition (see section 1.3). For06=f= P a αβ X α Y β ∈A d we define the ε-exponent to be ε-exp(f) := m ≺ ax{(α β)∈N 2d ;|a αβ |ε (αβ) =|f| ε } We will define this exponent again in a slightly more general situation in section 1.3. For non-zero elementsf= P a αβ X α Y β andg= P b αβ X α Y β inA d put (α 1  β 1 ) =ε-exp(f)and(α 2  β 2 ) =ε-exp(g) have. We |f g| ε ≤ |f| ε |g| ε =|a α 1 β 1 b α 2 β 2 |ε (α 1 +α 2 β 1 +β 2 )  hence the desired equality follows if we show that the(α 1 +α 2  β 1 +β 2 )-th coefficient off ghas absolute value equal to|a α 1 β 1 b α 2 β 2 |. Recall by lemma 1.1.1 the(α 1 +α 2  β 1 +β 2 )-th coefficient off gis given by the sum X a α ′ β ′ b α ′′ β ′′ γ!  γ β ′  γ α ′′   ′ β ′  α ′′  β ′′  γ∈N d α  α ′ +α ′′ −γ=α 1 +α 2 β ′ +β ′′ −γ=β 1 +β 2 We prove now the strict inequality |a α ′ β ′ b α ′′ β ′′ γ!  β γ ′  α γ ′′  |<|a α 1 β 1 b α 2 β 2 | for allα ′  β ′  α ′′  β ′′  γ∈N d with(α ′  β ′ ) + (α ′′  β ′′ )−(γ γ) = (α 1  β 1 ) + (α 2  β 2 ) and(α ′  β ′  α ′′  β ′′ )6= (α 1  β 1  α 2  β 2 ) the following cases:. Consider (α ′  β ′ )≻(α 1  β 1 )implies|a α ′ β ′ |ε (α ′ β ′ ) <|a α 1 β 1 |ε (α 1 β 1 ) ((α 1  β 1 ) =ε-exp(f)). Further we have|b α ′′ β ′′ |ε (α ′′ −γβ ′′ −γ) ≤ |b α ′′ β ′′ |ε (α ′′ β ′′ ) ≤ |b α 2 β 2 |ε (α 2 β 2 )  gives. This |a α ′ β ′ ||b α ′′ β ′′ |ε (α ′ β ′ )+(α ′′ −γβ ′′ −γ) <|a α 1 β 1 ||b α 2 β 2 |ε (α 1 β 1 )+(α 2 β 2 ) hence|a α ′ β ′ b α ′′ β ′′ γ!  γ β ′  α γ ′′  | ≤ |a α ′ β ′ b α ′′ β ′′ |<|a α 1 β 1 b α 2 β 2 |. (α ′  β ′ )≺(α 1  β 1 )implies(α ′′  β ′′ )(α ′′  β ′′ )−(γ γ)≻(α 2  β 2 )and we proceed as in the first case.

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