Locally nilpotent derivation

In mathematics, a derivation of a commutative ring is called a locally nilpotent derivation (LND) if every element of is annihilated by some power of .

One motivation for the study of locally nilpotent derivations comes from the fact that some of the counterexamples to Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring.[1]

Over a field of characteristic zero, to give a locally nilpotent derivation on the integral domain , finitely generated over the field, is equivalent to giving an action of the additive group to the affine variety . Roughly speaking, an affine variety admitting "plenty" of actions of the additive group is considered similar to an affine space.[2]

Definition

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Let   be a ring. Recall that a derivation of   is a map   satisfying the Leibniz rule   for any  . If   is an algebra over a field  , we additionally require   to be  -linear, so  .

A derivation   is called a locally nilpotent derivation (LND) if for every  , there exists a positive integer   such that  .

If   is graded, we say that a locally nilpotent derivation   is homogeneous (of degree  ) if   for every  .

The set of locally nilpotent derivations of a ring   is denoted by  . Note that this set has no obvious structure: it is neither closed under addition (e.g. if  ,   then   but  , so  ) nor under multiplication by elements of   (e.g.  , but  ). However, if   then   implies  [3] and if  ,   then  .

Relation to Ga-actions

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Let   be an algebra over a field   of characteristic zero (e.g.  ). Then there is a one-to-one correspondence between the locally nilpotent  -derivations on   and the actions of the additive group   of   on the affine variety  , as follows.[3] A  -action on   corresponds to a  -algebra homomorphism  . Any such   determines a locally nilpotent derivation   of   by taking its derivative at zero, namely   where   denotes the evaluation at  . Conversely, any locally nilpotent derivation   determines a homomorphism   by  

It is easy to see that the conjugate actions correspond to conjugate derivations, i.e. if   and   then   and  

The kernel algorithm

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The algebra   consists of the invariants of the corresponding  -action. It is algebraically and factorially closed in  .[3] A special case of Hilbert's 14th problem asks whether   is finitely generated, or, if  , whether the quotient   is affine. By Zariski's finiteness theorem,[4] it is true if  . On the other hand, this question is highly nontrivial even for  ,  . For   the answer, in general, is negative.[5] The case   is open.[3]

However, in practice it often happens that   is known to be finitely generated: notably, by the Maurer–Weitzenböck theorem,[6] it is the case for linear LND's of the polynomial algebra over a field of characteristic zero (by linear we mean homogeneous of degree zero with respect to the standard grading).

Assume   is finitely generated. If   is a finitely generated algebra over a field of characteristic zero, then   can be computed using van den Essen's algorithm,[7] as follows. Choose a local slice, i.e. an element   and put  . Let   be the Dixmier map given by  . Now for every  , chose a minimal integer   such that  , put  , and define inductively   to be the subring of   generated by  . By induction, one proves that   are finitely generated and if   then  , so   for some  . Finding the generators of each   and checking whether   is a standard computation using Gröbner bases.[7]

Slice theorem

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Assume that   admits a slice, i.e.   such that  . The slice theorem[3] asserts that   is a polynomial algebra   and  .

For any local slice   we can apply the slice theorem to the localization  , and thus obtain that   is locally a polynomial algebra with a standard derivation. In geometric terms, if a geometric quotient   is affine (e.g. when   by the Zariski theorem), then it has a Zariski-open subset   such that   is isomorphic over   to  , where   acts by translation on the second factor.

However, in general it is not true that   is locally trivial. For example,[8] let  . Then   is a coordinate ring of a singular variety, and the fibers of the quotient map over singular points are two-dimensional.

If   then   is a curve. To describe the  -action, it is important to understand the geometry  . Assume further that   and that   is smooth and contractible (in which case   is smooth and contractible as well[9]) and choose   to be minimal (with respect to inclusion). Then Kaliman proved[10] that each irreducible component of   is a polynomial curve, i.e. its normalization is isomorphic to  . The curve   for the action given by Freudenburg's (2,5)-derivation (see below) is a union of two lines in  , so   may not be irreducible. However, it is conjectured that   is always contractible.[11]

Examples

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Example 1

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The standard coordinate derivations   of a polynomial algebra   are locally nilpotent. The corresponding  -actions are translations:  ,   for  .

Example 2 (Freudenburg's (2,5)-homogeneous derivation[12])

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Let  ,  , and let   be the Jacobian derivation  . Then   and   (see below); that is,   annihilates no variable. The fixed point set of the corresponding  -action equals  .

Example 3

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Consider  . The locally nilpotent derivation   of its coordinate ring corresponds to a natural action of   on   via right multiplication of upper triangular matrices. This action gives a nontrivial  -bundle over  . However, if   then this bundle is trivial in the smooth category[13]

LND's of the polynomial algebra

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Let   be a field of characteristic zero (using Kambayashi's theorem one can reduce most results to the case  [14]) and let   be a polynomial algebra.

n = 2 (Ga-actions on an affine plane)

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Rentschler's theorem — Every LND of   can be conjugated to   for some  . This result is closely related to the fact that every automorphism of an affine plane is tame, and does not hold in higher dimensions.[15]

n = 3 (Ga-actions on an affine 3-space)

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Miyanishi's theorem — The kernel of every nontrivial LND of   is isomorphic to a polynomial ring in two variables; that is, a fixed point set of every nontrivial  -action on   is isomorphic to  .[16][17]

In other words, for every   there exist   such that   (but, in contrast to the case  ,   is not necessarily a polynomial ring over  ). In this case,   is a Jacobian derivation:  .[18]

Zurkowski's theorem — Assume that   and   is homogeneous relative to some positive grading of   such that   are homogeneous. Then   for some homogeneous  . Moreover,[18] if   are relatively prime, then   are relatively prime as well.[19][3]

Bonnet's theorem — A quotient morphism   of a  -action is surjective. In other words, for every  , the embedding   induces a surjective morphism  .[20][10]

This is no longer true for  , e.g. the image of a quotient map   by a  -action   (which corresponds to a LND given by   equals  .

Kaliman's theorem — Every fixed-point free action of   on   is conjugate to a translation. In other words, every   such that the image of   generates the unit ideal (or, equivalently,   defines a nowhere vanishing vector field), admits a slice. This results answers one of the conjectures from Kraft's list.[10]

Again, this result is not true for  :[21] e.g. consider the  . The points   and   are in the same orbit of the corresponding  -action if and only if  ; hence the (topological) quotient is not even Hausdorff, let alone homeomorphic to  .

Principal ideal theorem — Let  . Then   is faithfully flat over  . Moreover, the ideal   is principal in  .[14]

Triangular derivations

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Let   be any system of variables of  ; that is,  . A derivation of   is called triangular with respect to this system of variables, if   and   for  . A derivation is called triangulable if it is conjugate to a triangular one, or, equivalently, if it is triangular with respect to some system of variables. Every triangular derivation is locally nilpotent. The converse is true for   by Rentschler's theorem above, but it is not true for  .

Bass's example

The derivation of   given by   is not triangulable.[22] Indeed, the fixed-point set of the corresponding  -action is a quadric cone  , while by the result of Popov,[23] a fixed point set of a triangulable  -action is isomorphic to   for some affine variety  ; and thus cannot have an isolated singularity.

Freudenburg's theorem — The above necessary geometrical condition was later generalized by Freudenburg.[24] To state his result, we need the following definition:

A corank of   is a maximal number   such that there exists a system of variables   such that  . Define   as   minus the corank of  .

We have   and   if and only if in some coordinates,   for some  .[24]

Theorem: If   is triangulable, then any hypersurface contained in the fixed-point set of the corresponding  -action is isomorphic to  .[24]

In particular, LND's of maximal rank   cannot be triangulable. Such derivations do exist for  : the first example is the (2,5)-homogeneous derivation (see above), and it can be easily generalized to any  .[12]

Makar-Limanov invariant

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The intersection of the kernels of all locally nilpotent derivations of the coordinate ring, or, equivalently, the ring of invariants of all  -actions, is called "Makar-Limanov invariant" and is an important algebraic invariant of an affine variety. For example, it is trivial for an affine space; but for the Koras–Russell cubic threefold, which is diffeomorphic to  , it is not.[25]

References

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  1. ^ Daigle, Daniel. "Hilbert's Fourteenth Problem and Locally Nilpotent Derivations" (PDF). University of Ottawa. Retrieved 11 September 2018.
  2. ^ Arzhantsev, I.; Flenner, H.; Kaliman, S.; Kutzschebauch, F.; Zaidenberg, M. (2013). "Flexible varieties and automorphism groups". Duke Math. J. 162 (4): 767–823. arXiv:1011.5375. doi:10.1215/00127094-2080132. S2CID 53412676.
  3. ^ a b c d e f Freudenburg, G. (2006). Algebraic theory of locally nilpotent derivations. Berlin: Springer-Verlag. CiteSeerX 10.1.1.470.10. ISBN 978-3-540-29521-1.
  4. ^ Zariski, O. (1954). "Interprétations algébrico-géométriques du quatorzième problème de Hilbert". Bull. Sci. Math. (2). 78: 155–168.
  5. ^ Derksen, H. G. J. (1993). "The kernel of a derivation". J. Pure Appl. Algebra. 84 (1): 13–16. doi:10.1016/0022-4049(93)90159-Q.
  6. ^ Seshadri, C.S. (1962). "On a theorem of Weitzenböck in invariant theory". J. Math. Kyoto Univ. 1 (3): 403–409. doi:10.1215/kjm/1250525012.
  7. ^ a b van den Essen, A. (2000). Polynomial automorphisms and the Jacobian conjecture. Basel: Birkhäuser Verlag. doi:10.1007/978-3-0348-8440-2. ISBN 978-3-7643-6350-5. S2CID 252433637.
  8. ^ Deveney, J.; Finston, D. (1995). "A proper  -action on   which is not locally trivial". Proc. Amer. Math. Soc. 123 (3): 651–655. doi:10.1090/S0002-9939-1995-1273487-0. JSTOR 2160782.
  9. ^ Kaliman, S; Saveliev, N. (2004). " -Actions on contractible threefolds". Michigan Math. J. 52 (3): 619–625. arXiv:math/0209306. doi:10.1307/mmj/1100623416. S2CID 15020160.
  10. ^ a b c Kaliman, S. (2004). "Free  -actions on   are translations" (PDF). Invent. Math. 156 (1): 163–173. arXiv:math/0207156. doi:10.1007/s00222-003-0336-1. S2CID 15769378.
  11. ^ Kaliman, S. (2009). "Actions of   and   on affine algebraic varieties" (PDF). Algebraic geometry-Seattle 2005. Part 2. Proceedings of Symposia in Pure Mathematics. Vol. 80. pp. 629–654. doi:10.1090/pspum/080.2/2483949. ISBN 9780821847039.
  12. ^ a b Freudenburg, G. (1998). "Actions of   on   defined by homogeneous derivations". Journal of Pure and Applied Algebra. 126 (1): 169–181. doi:10.1016/S0022-4049(96)00143-0.
  13. ^ Dubouloz, A.; Finston, D. (2014). "On exotic affine 3-spheres". J. Algebraic Geom. 23 (3): 445–469. arXiv:1106.2900. doi:10.1090/S1056-3911-2014-00612-3. S2CID 119651964.
  14. ^ a b Daigle, D.; Kaliman, S. (2009). "A note on locally nilpotent derivations and variables of  " (PDF). Canad. Math. Bull. 52 (4): 535–543. doi:10.4153/CMB-2009-054-5.
  15. ^ Rentschler, R. (1968). "Opérations du groupe additif sur le plan affine". Comptes Rendus de l'Académie des Sciences, Série A-B. 267: A384–A387.
  16. ^ Miyanishi, M. (1986). "Normal affine subalgebras of a polynomial ring". Algebraic and Topological Theories (Kinosaki, 1984). pp. 37–51.
  17. ^ Sugie, T. (1989). "Algebraic Characterization of the Affine Plane and the Affine 3-Space". Topological Methods in Algebraic Transformation Groups (New Brunswick, NJ, 1988). Progress in Mathematics. Vol. 80. Birkhäuser Boston. pp. 177–190. doi:10.1007/978-1-4612-3702-0_12. ISBN 978-1-4612-8219-8.
  18. ^ a b D., Daigle (2000). "On kernels of homogeneous locally nilpotent derivations of  ". Osaka J. Math. 37 (3): 689–699.
  19. ^ Zurkowski, V.D. "Locally finite derivations" (PDF).
  20. ^ Bonnet, P. (2002). "Surjectivity of quotient maps for algebraic  -actions and polynomial maps with contractible fibers". Transform. Groups. 7 (1): 3–14. arXiv:math/0602227. doi:10.1007/s00031-002-0001-6.
  21. ^ Winkelmann, J. (1990). "On free holomorphic  -actions on   and homogeneous Stein manifolds" (PDF). Math. Ann. 286 (1–3): 593–612. doi:10.1007/BF01453590.
  22. ^ Bass, H. (1984). "A non-triangular action of   on  ". Journal of Pure and Applied Algebra. 33 (1): 1–5. doi:10.1016/0022-4049(84)90019-7.
  23. ^ Popov, V. L. (1987). "On actions of   on  ". Algebraic Groups Utrecht 1986. Lecture Notes in Mathematics. Vol. 1271. pp. 237–242. doi:10.1007/BFb0079241. ISBN 978-3-540-18234-4.
  24. ^ a b c Freudenburg, G. (1995). "Triangulability criteria for additive group actions on affine space". J. Pure Appl. Algebra. 105 (3): 267–275. doi:10.1016/0022-4049(96)87756-5.
  25. ^ Kaliman, S.; Makar-Limanov, L. (1997). "On the Russell-Koras contractible threefolds". J. Algebraic Geom. 6 (2): 247–268.

Further reading

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