Wilberd van der Kallen

A module structure on certain orbit sets of unimodular rows

Abstract An algebraic version of cohomotopy groups is developed. Further the stabilization problem for the K 1 of Bass is studied for matrices that are much smaller than those treated classically.

A group structure on certain orbit sets of unimodular rows

Let R be a commutative noetherian d-dimensional ring. Recall that for n > d + 2 the group E,(R) (the subgroup of GL,(R) generated by elementary matrices) acts transitively on Urn,,(R), the set of unimodular rows of length .YI over R. If d > 2, we will describe an abelian group structure on Urxd+ ,(R)/ Ed+,(R). This group structure will be closely related with the higher Mennicke symbols of Suslin. In fact this article is mainly an elaboration of a theme in Suslin ( 141 (in particular [ 14, Sect. 11 j, Recall that for d = I, b] the Bass-Kubota theorem there is a bijection MS,(R) ++ Umz(R)/SLz(R) fl E(R), where MSz(R) denotes the target group of the universal Mennicke symbol, as in Suslin [ 13, Sect. 51. We will see that more generally hfS,+ ,(R) * um,, , (R)/SL,+ ,(R) f’E(R) if d is odd. Now let d = 2. Then, by a theorem of Vaserstein, Um,(R)/E,(R) . IS in bijective correspondence with a certain Witt group [ 17, Cor. 7.41. In particular Um,(R)/E,(R) gets the structure of an abelian group. We will derive from this (inductively) the structure of an abelian group on Urn,, ,(R)/E,+ ,(R) for d > 3. (It would be desirable to have an interpretation of these abelian groups in terms of Witt groups or of similar Grothendieck groups of categories.) As an intriguing by-product we get an abelian group structure on the set of isomorphism classes of oriented stably free rank d projective modules. (If d is odd one does not need the orientations. See 4.8.) We will borrow heavily from the work of Suslin and Vaserstein. For the convenience of the reader we have included a proof of Vaserstein’s prestabilization theorem for K, , making no restriction on the presence of zero-divisors. (In Vaserstein’s original proof such restrictions were made, but he has long since been able to remove them. Because of the crucial role his theorem plays in connecting higher Mennicke symbols with ordinary K363 OOZl-3693183

Constant terms in powers of a Laurent polynomial

3.00

Descent for theK-theory of polynomial rings

Abstract We classify the complex Laurent polynomials with the property that their powers have no constant term. The result confirms a conjecture of Mathieu for the case of tori. (A different case would imply Kellers Jacobian Conjecture.)

Heeke Eigenforms in the Cohomology of Congruence Subgroups of SL (3, Z)

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On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic p.

We list here Hecke eigenvalues of several automorphic forms for congruence subgroups of Sl(3; Z). To compute such tables, we describe an algorithm that combines techniques developed by Ash, Grayson and Green with the Lenstra–Lenstra–Lovasz algorithm. With our implementation of this new algorithm we were able to handle much larger levels than those treated by Ash, Grayson and Green and by Top and van Geemen in previous work. Comparing our tables with results from computations of Galois representations, we find some new numerical evidence for the conjectured relation between modular forms and Galois representations.

Complexity of the Havas, Majewski, Matthews LLL Hermite Normal Form Algorithm

This paper is about sheaf cohomology for varieties (schemes) in characteristic

On the Schur multipliers of Steinberg and Chevalley groups over commutative rings

p>0

Adjoint and coadjoint orbits of the Poincare group

. We assume the presence of a Frobenius splitting. (See V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Annals of Math. 122 (1985), 27--40). The main result is that a non-zero higher direct image under a proper map of the ideal sheaf of a compatibly Frobenius split subvariety can not have a support whose inverse image is contained in that subvariety. Earlier vanishing theorems for Frobenius split varieties were based on direct limits and Serres vanishing theorem, but our theorem is based on inverse limits and Grothendiecks theorem on formal functions. The result implies a Grauert--Riemenschneider type theorem.

Infinitesimal fixed points in modules with good filtration

We consider the complexity of the LLL HNF algorithm Havas et al.(1998, Algorithm 4). This algorithm takes as input an m by n matrix G of integers and produces as output a matrixb?GLm(Z) so thatA=bG is in Hermite normal form (upside down). The analysis is similar to that of an extended LLL algorithm as given in van der Kallen (1998).