p-adic Weyl algebras [Elektronische Ressource] / vorgelegt von Alexis Pangalos
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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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Publié par
Publié le 01 janvier 2008
Nombre de lectures 8
Langue English
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 algebraAd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 algebraAdoverK(see section 1.1 for a definition) and, for someεR2>d0, endow it with the non-archimedeanK-vector space norm |f|ε= max|aαβ|ε(αβ) iffAdis written in the formf=PaαβXαYβ we require. Ifεto be an element ofR2>d0such that|γ!|ε(γγ)is bounded by some constant for allγNd then the norm is in fact an algebra norm onAd norm The(cf. lemma 1.2.1). is multiplicative if and only ifεsatisfiesεiεd+i1for all1id(cf. lemma 1.2.4). We denote the completion ofAdwith respect to this norm byAdε. The elements ofAdεcan be written as formal power series in non-commuting variables such that the coefficients satisfy a certain convergence condition: Adε={XaαβXαYβ:|aαβ|ε(αβ)0for|α|+|β| → ∞}We call the elements ofAdεrestricted power series. Different versions of com-pleted Weyl algebras appear in the literature. One can construct the algebra which is the union of allAdεwithε1=  =εd= 1andεi>1for all d+ 1i2dand the algebra which is the union of allAdεwithεi>1for alli algebras are considered in [Ber] and [MN].. These latter version is The denoted byAdand is called theDwork-Monsky-Washnitzer-Weylalgebra. The algebraAd(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 algebraAddefined over an arbitrary field has been exten-sively studied during the last 50 years. The classical Weyl algebraAdis a left and right Noetherian integral domain. The classical Weyl algebraAdis simple if defined over a field of characteristic zero. In this case the Krull and
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the global dimension ofAdared; the Krull dimension ofAdwas first deter-mined by Gabriel and Rentschler in [GR]. That the global dimension ofAd isdwas proved by Rinehart [Rin] ford= 1and in the general case by Roos [Roo]. The Krull and the global dimension ofAdare2difAdis defined over a field of characteristicp >0 classical Weyl algebra. TheAdis an Auslander regular ring. Stafford proved that any left ideal ofAdhas a set of2generators ifAd Theis defined over a field of characteristic zero [Sta]. simple modules over the classical Weyl algebraA1 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 onAdεis 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 algebrasAd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 forAdin 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 toAd. The division theorems enable us to prove some of the basic properties ofAdε andAd Weyl algebras. TheAdεandAdare Noetherian (cf. proposition 1.4.1). An element ofAdεorAdis a unit if and only if its exponent is zero (cf. proposition 1.4.3). We consider formal partial differentiation on elements of AdεandAd a consequence of Asshow that it respects two-sided ideals.and this result, together with the characterization of units, we get thatAdεand
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Adare simple rings if we assume the characteristic ofKto be zero. We prove that the Krull dimension and the global dimension of the completed Weyl algebraAdεare 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 algebraAdit turns out to be very useful to consider the localizations ofAdwith respect to the Ore setsK[Xi]\{0}resp. K[Yi]\{0}. One might expect that the multiplicative subsetsKhXiiεi\{0}and KhYiiεi\{0}ofAdεthe sets of all non-zero restricted power series in, i.e. Xi resp.Yi, are Ore sets inAdεer, this is not the case (cf. lemma 2.0.1) . Howev which is equivalent to the fact that the localizations ofAdεwith 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 extensionsBεdXiandBYdεofAd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 ringsBεdXiresp.BYdiε are the microlocalizations ofAdεwith respect to the setsKhXiiεi\{0}resp. KhYiiεi\{0} We set(for the notion of microlocalizations see [LvO] or [Nag]). d d Bdε:=MBXdiεMBYdiεi=1i=1 With the assumption that the characteristic ofKis zero we prove the following lemma. For any maximal left idealI⊂ Ad εthe left idealBdεIgenerated byIis not the unit idealBdε(cf. lemma 3.2.1). proof involves both The the division theorem for the Weyl algebraAdε(cf. theorem 1.3.14) and the division theorems forBXdiεandBYdiε This(cf. theorem 2.2.4). lemma will be an important ingredient to obtain upper bounds for the Krull dimension and the global dimension ofAdεin section 4. In analogy to the fact that the localizations of the classical Weyl algebraA1 mentioned above are simple principal left and right ideal domains, the rings B1XεandB1εXare 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 onAdεcoming 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 algebraAdεis Auslander regular (cf. proposition 4.3.3). We show that the Krull dimension and the global dimension ofAdεare 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 ofAdεare2dif the characteristic ofKisp >0 prove. We that the Krull dimension and the global dimension ofAdεare bounded above by2d1if the characteristic ofK the Krull Henceis zero (cf corollary 4.3.8). dimension and the global dimension ofA1εare1. For some special cases we also prove our conjecture that the Krull dimension and the global dimension ofAdεaredtrue 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 forA1ε, i.e. any left ideal ofA1εhas a set of2generators if the characteristic ofKis zero (cf. corollary 4.3.9). In section 5 we show that the so called saturationSsatof the subsetKhXiε1\{0} ofA1εis an Ore set inA1d(cf. proposition 5.1). simple TheSsat-torsionfree A1d-modules are in bijection with the simple(Ssat)1A1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 byAd, is the algebra with2dgeneratorsX1     Xd Y1     Yd and relations YiXjXjYi=δij and XiXjXjXi=YiYjYjYi= 0whereδijdenotes 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 ofAdhave a unique expression as finite sums XaαβXαYβ αβNd with coefficientsaαβKand the notationXα=X1α1  XdαdandYα= d Y1α1  Ydα. Weform and get the following rules always write elements in this of multiplication. Lemma 1.1.1.We have YβXα=X γNdγ!βγαγXαγYβγγiiαβi Iff=Pa XαYβ,g=PbαβXαYβandf g=PcαβXαYβthen αβ αβ=Xaαβbα′′β′′γ! cβγα′′α  β α′′ β′′ γNdγ 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! :=γ1!  γd!andγα:=γα11  γαdd write. We |α|:=α1+  +αdfor elements inNd. For06=f=PaαβXαYβAdthe degreeis deg(f) := max{|α|+|β| ∈N:aαβ6= 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 algebraAdand the polynomial ring in2dvariables are isomorphic. distinguish the two To different algebra structures we writefor polynomial multiplication andfor Weyl algebra multiplication if we want to emphasize in which ring we multiply. Lemma 1.1.2.Letf gAd. Then deg(fg) = deg(f) + deg(g) and deg(fgfg)<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 algebraAdis a Noetherian integral do-main. If the fieldKhas characteristic zero thenAdis simple, i.e. has no two-sided ideals other than0andAd.
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 algebraAdwe have for anyεR2>d0the non-archimedeanK-vector space norm| |εdefined by |f|ε:= max|aαβ|ε(αβ) forf=PaαβXαYβAd, whereε(αβ)=ε1α1  εαddεβd1+1  ε2dβd, i.e. we have
(i)|f|ε= 0ifff= 0,
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(ii)|af|ε=|a||f|ε, (iii)|f+g|εmax{|f|ε|g|ε}, for allf gAdandaK. Lemma 1.2.1.Letεbe an element ofR2>d0such that|γ!|ε(γγ)is bounded by some constantC >0for allγNd. Then |f g|εC|f|ε|g|εIfεiεd+i1for alli, we have|f g|ε≤ |f|ε|g|ε. Proof. |f g|ε= mαaβx|α β α′′Xβ′′ γNdaαβbα′′β′′γ!γβγα′′|ε(αβ) α+α′′γ=α β+β′′γ=β αβαmaβxγNd|aαβ||bα′′β′′||γ!βγγα′′|ε(α+αγβ+β′′γ) γsuNpd|γ!|ε(γγ)maxNd|aαβ|ε(αβ)α′′mβ′′axNd|bα′′β′′|ε(α′′β′′) αβ=C|f|ε|g|εIfεsatisfiesεiεd+i1for all1idwe have|γ!|ε(γγ)1for allγ, which gives the second part. Remark 1.2.2.If for exampleK=Qp, we know that|γ!|converges exponen-tially to zero as|γ| Hence we easily find angoes to infinity.εR2>d0with all εi<1such that|γ!|ε(γγ)is bounded by some constantC. Remark 1.2.3.Ifεiεd+i<1for somei, then| |εis not submultiplicative, for example 1 =|XiYi+ 1|ε=|YiXi|ε6≤ |Yi|ε|Xi|ε=εd+iεiHowever instead of| |εwe can take the equivalent norm| |εdefined by |f|ε:= sup{|f g|ε|g|ε1; 06=gAd}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| |εonAdis multiplicative if and only ifεiεd+i1 for all1id. Proof.The “only if” is remark 1.2.3. Letbe a total order onN2dcompatible with addition (see section 1.3). For06=f=PaαβXαYβAdwe define the ε-exponent to be ε-exp(f) := max{(α β)N2d;|aαβ|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=PaαβXαYβandg=PbαβXαYβinAdput (α1 β1) =ε-exp(f)and(α2 β2) =ε-exp(g) have. We |f g|ε≤ |f|ε|g|ε=|aα1β1bα2β2|ε(α1+α2β1+β2)hence the desired equality follows if we show that the(α1+α2 β1+β2)-th coefficient off ghas absolute value equal to|aα1β1bα2β2|. Recall by lemma 1.1.1 the(α1+α2 β1+β2)-th coefficient off gis given by the sum Xaαβbα′′β′′γ!γβγα′′β α′′ β′′ γNd α  α+α′′γ=α1+α2 β+β′′γ=β1+β2 We prove now the strict inequality |aαβbα′′β′′γ!βγαγ′′|<|aα1β1bα2β2| for allα β α′′ β′′ γNdwith(α β) + (α′′ β′′)(γ γ) = (α1 β1) + (α2 β2) and(α β α′′ β′′)6= (α1 β1 α2 β2) the following cases:. Consider (α β)(α1 β1)implies|aαβ|ε(αβ)<|aα1β1|ε(α1β1)((α1 β1) =ε-exp(f)). Further we have|bα′′β′′|ε(α′′γβ′′γ)≤ |bα′′β′′|ε(α′′β′′)≤ |bα2β2|ε(α2β2) gives. This |aαβ||bα′′β′′|ε(αβ)+(α′′γβ′′γ)<|aα1β1||bα2β2|ε(α1β1)+(α2β2) hence|aαβbα′′β′′γ!γβαγ′′| ≤ |aαβbα′′β′′|<|aα1β1bα2β2|. (α β)(α1 β1)implies(α′′ β′′)(α′′ β′′)(γ γ)(α2 β2)and we proceed as in the first case.
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