Motivic Serre invariants and Weil restriction

We study the interactions between Weil restriction for formal schemes and rigid varieties, Greenberg schemes, and motivic Serre invariants, and their behavior with respect to finite extensions of the base ring R, which is a complete discrete valuation ring with perfect residue field.     

Motivic integral of K3 surfaces over a non-archimedean field

We prove a formula expressing the motivic integral (Loeser and Sebag, 2003) [34] of a K3 surface over C((t)) with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces.     

Motivic zeta functions of abelian varieties, and the monodromy conjecture

We prove for abelian varieties a global form of Denef and Loeser?s motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety A over a complete discretely valued field with algebraically closed residue field, its motivic zeta function has a unique pole at Chai?s base change conductor c(A) of A, and that the order of this pole equals one plus the potential toric rank of A. Moreover, we show that for every embedding of Q? in C, the value exp(2πic(A)) is an ?-adic tame monodromy eigenvalue of A. The main tool in the paper is Edixhoven?s filtration on the special fiber of the Néron model of A, which measures the behavior of the Néron model under tame base change.     

Multi-index Filtrations and Generalized Poincaré Series

A multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity C and for the so called divisorial filtration for a modification of by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ C or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious “motivic version” of an A’Campo type formula. First two authors were partially supported by the grant MEC, PN I + D + i MTM2004-00958. Partially supported by the grants RFBR-04-01-00762, NSh-4719.2006.1 The author is thankful to the University of Valladolid for hospitality.     

Galois ring extensions and localized modular rings of invariants of p-groups

We apply recent results on Galois-ring extensions and trace surjective algebras to analyze dehomogenized modular invariant rings of finite p-groups, as well as related localizations. We describe criteria for the dehomogenized invariant ring to be polynomial or at least regular and we show that for regular affine algebras with possibly non-linear action by a p-group, the singular locus of the invariant ring is contained in the variety of the transfer ideal. If V is the regular module of an arbitrary finite p-group, or V is any faithful representation of a cyclic p-group, we show that there is a suitable invariant linear form, inverting which renders the ring of invariants into a “localized polynomial ring” with dehomogenization being a polynomial ring. This is in surprising contrast to the fact that for a faithful representation of a cyclic group of order larger than p, the ring of invariants itself cannot be a polynomial ring by a result of Serre. Our results here generalize observations made by Richman [R] and by Campbell and Chuai [CCH].     

Multi-index Filtrations and Generalized Poincaré Series

A multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity C and for the so called divisorial filtration for a modification of (\Bbb C²,0)({\Bbb C}^2,0) by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ C or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious “motivic version” of an A’Campo type formula.     

Relative Regular Modules. Applications to von Neumann Regular Rings

We use the concept of a regular object with respect to another object in an arbitrary category, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories R −mod and S −mod, when S is an excellent extension of the ring R. Consequently, if S is an excellent extension of the ring R, then S is von Neumann regular ring if and only if R is also von Neumann regular ring. In the second part, using relative regular modules, we give a new proof of a classical result: the von Neumann regular property of a ring is Morita invariant.     

A trace formula for rigid varieties,and motivic Weil generating series for formal schemes

We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. Next, we show that the analytic Milnor fiber of a morphism f at a point x completely determines the formal germ of f at x. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal R-scheme of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. When is the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f. When is the formal completion of f at a closed point x of the special fiber , we obtain the local motivic zeta function of f at x. The research for this article was partially supported by ANR-06-BLAN-0183.     

Rigidity for orientable functors

In the following paper we introduce the notion of orientable functor (orientable cohomology theory) on the category of projective smooth schemes and define a family of transfer maps. Applying this technique, we prove that with finite coefficients orientable cohomology of a projective variety is invariant with respect to the base-change given by an extension of algebraically closed fields. This statement generalizes the classical result of Suslin, concerning algebraic K-theory of algebraically closed fields. Besides K-theory, we treat such examples of orientable functors as etale cohomology, motivic cohomology, algebraic cobordism. We also demonstrate a method to endow algebraic cobordism with multiplicative structure and Chern classes.     

Serre's Condition Rℓ for Affine Semigroup Rings

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_? of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of non-normal affine semigroup rings that satisfy R₂.     

Relèvement galoisien des revêtements¶de courbes nodales

Let R be a complete discrete valuation ring of mixed characteristics, with algebraically closed residue field k. We study the existence problem of equivariant liftings to R of Galois covers of nodal curves over k. Using formal geometry, we show that this problem is actually a local one. We apply this local-to-global principle to obtain new results concerning the existence of such liftings. Received: 10 February 2000 / Revised version: 13 September 2000     

The Néron component series of an abelian variety

We introduce the Néron component series of an abelian variety A over a complete discretely valued field. This is a power series in ${\mathbb{Z}}\left[\left[T\right]\right]$ , which measures the behaviour of the number of components of the Néron model of A under tame ramification of the base field. If A is tamely ramified, then we prove that the Néron component series is rational. It has a pole at T = 1, whose order equals one plus the potential toric rank of A. This result is a crucial ingredient of our proof of the motivic monodromy conjecture for abelian varieties. We expect that it extends to the wildly ramified case; we prove this if A is an elliptic curve, and if A has potential purely multiplicative reduction.     

Automorphism Group of q-Quantum Torus Lie Algebra with q a Root of Unity*

Let α be an automorphism of a ring R. We study the skew Armendariz of Laurent series type rings (α-LA rings), as a generalization of the standard Armendariz condition from polynomials to skew Laurent series. We study on the relationship between the Baerness and p.p. property of a ring R and these of the skew Laurent series ring R[[x, x ^?1; α]], in case R is an α-LA ring. Moreover, we prove that for an α-weakly rigid ring R, R[[x, x ^?1; α]] is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of S _?(R) has a generalized countable join in R. Various types of examples of α-LA rings are provided.     

Multiplicity-free actions and the geometry of nilpotent orbits

Let G be a reductive Lie group. Take a maximal compact subgroup K of G and denote their Lie algebras by and respectively. We get a Cartan decomposition . Let be the complexification of , and the complexified decomposition. The adjoint action restricted to K preserves the space , hence acts on , where denotes the complexification of K. In this paper, we consider a series of small nilpotent -orbits in which are obtained from the dual pair ([R. Howe, Transcending classical invariant theory. J. Amer. Math. Soc. 2 (1989), no. 3, 535–552]). We explain astonishing simple structures of these nilpotent orbits using generalized null cones. For example, these orbits have a linear ordering with respect to the closure relation, and acts on them in multiplicity-free manner. We clarify the -module structure of the regular function ring of the closure of these nilpotent orbits in detail, and prove the normality. All these results naturally comes from the analysis on the null cone in a matrix spaceW , and the double fibration of nilpotent orbits in and . The classical invariant theory assures that the regular functions on our nilpotent orbits are coming from harmonic polynomials on W with repspect to or . We also provide many interesting examples of multiplicity-free actions on conic algebraic varieties. Received November 1, 1999 / Published online October 30, 2000     

Enriched simplicial presheaves and the motivic homotopy category

We construct models for the motivic homotopy category based on simplicial functors from smooth schemes over a field to simplicial sets. These spaces are homotopy invariant and therefore one does not have to invert the affine line in order to get a model for the motivic homotopy category.     

Motivic zeta functions of infinite-dimensional Lie algebras

The paper shows how to associate a motivic zeta function with a large class of infinite dimensional Lie algebras. These include loop algebras, affine Kac-Moody algebras, the Virasoro algebra and Lie algebras of Cartan type. The concept of a motivic zeta functions provides a good language to talk about the uniformity in p of local p-adic zeta functions of finite dimensional Lie algebras. The theory of motivic integration is employed to prove the rationality of motivic zeta functions associated to certain classes of infinite dimensional Lie algebras.     

Free Pairs of Bicyclic Units in the Integral Group Ring of the Dihedral Group

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings.     

On inverse skew Laurent series extensions of weakly rigid rings

Let R be a ring equipped with an automorphism α and an α-derivation δ. We studied on the relationship between the quasi Baerness and (α, δ)-quasi Baerness of a ring R and these of the inverse skew Laurent series ring R((x^?1; α, δ)), in case R is an (α, δ)-weakly rigid ring. Also we proved that for a semicommutative (α, δ)-weakly rigid ring R, R is Baer if and only if so is R((x^?1; α, δ)). Moreover for an (α, δ)-weakly rigid ring R, it is shown that the inverse skew Laurent series ring R((x^?1; α, δ)) is left p.q.-Baer if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.     

Armendariz Properties Relative to a Ring Endomorphism

For an endomorphism α of a ring R, we introduce the notion of an α-Armendariz ring to investigate the relative Armendariz properties. This concept extends the class of Armendariz rings and gives us an opportunity to study Armendariz rings in a general setting. It is obvious that every Armendariz ring is an α-Armendariz ring, but we shall give an example to show that there exists a right α-Armendariz ring which is not Armendariz. A number of properties of this version are established. It is shown that if I is a reduced ideal of a ring R such that R/I is a right α-Armendariz ring, then R is right α-Armendariz. For an endomorphism α of a ring R, we show that R is right α-Armendariz if and only if R[x] is right α-Armendariz. Moreover, a weak form of α-Armendariz rings is considered in the last section. We show that in general weak α-Armendariz rings need not be α-Armendariz.     

On Von Neumann Regular Rings of Skew Generalized Power Series

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S?{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]].