Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. I

In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers and the relative Jacobian of a family of curves . As one of the applications, we show that in the case of a single curve C this action induces a -form of a Lefschetz sl₂- action on the Chow groups of C ^[N]. Another application gives a new grading on the ring CH ₀(J) of 0-cycles on the Jacobian J of C (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes in CH and in CH and prove for them analogs of the properties established in the case of the Jacobian of a single curve by Beauville in [5]. We show that our algebras of operators preserve the subrings of tautological cycles and act on them as some explicit differential operators. This work was partially supported by the NSF grant DMS-0601034.     

A new proof of Faber?s intersection number conjecture

We give a new proof of Faber?s intersection number conjecture concerning the top intersections in the tautological ring of the moduli space of curves Mg. The proof is based on a very straightforward geometric and combinatorial computation with double ramification cycles.     

Proper actions on imprimitivity bimodules and decompositions of Morita equivalences

We consider a class of proper actions of locally compact groups on imprimitivity bimodules over C^*-algebras which behave like the proper actions on C^*-algebras introduced by Rieffel in 1988. We prove that every such action gives rise to a Morita equivalence between a crossed product and a generalized fixed-point algebra, and in doing so make several innovations which improve the applicability of Rieffel's theory. We then show how our construction can be used to obtain canonical tensor-product decompositions of important Morita equivalences. Our results show, for example, that the different proofs of the symmetric imprimitivity theorem for actions on graph algebras yield isomorphic equivalences, and this gives new information about the amenability of actions on graph algebras.     

Equivalences induced by graded bimodules

Let k be a commutative ringG a finite groupR and S fully G-graded k-algebras. In this paper we investigate Morita equivalences, derived equivalences and stable equivalences of Morita type between R and S, which are induced by G-graded R, 5-bimodules or complexes of G-graded bimodules. Such equivalences occur naturally in the case of group algebras in certain reduction steps for Broue's conjecture, and we show how they can be lifted from equivalences between R ₁ and S ₁.     

Nondurable K-Theory Equivalence and Bousfield Localization*

In this paper we consider what happens when Adams self maps are modified by adding certain unstable maps. The unstable maps which are added are trivial after a single suspension. We can choose the modification so that the maps are still K-theory equivalences but the loops on the map are no longer K-theory equivalences. As a corollary we note that the maps are K-theory equivalences but not v ₁-periodic equivalences. Another consequence is the behavior of the cobar spectral sequences for generalized homology theories. Tamaki shows that in certain cases a cobar-type spectral sequence for generalized homology theories is well behaved. The maps we construct give an example where despite the connectivity of the spaces the cobar spectral sequence is still poorly behaved. Finally we use our maps to construct spaces whose Bousfield class is distinct from the cofiber of the Adams map but which becomes the same after one suspension.     

Generating-Tree Isomorphisms for Pattern-Avoiding Involutions

We show that for k ≥ 5 and the permutations τ _k = (k − 1)k(k − 2). . .312 and J _k = k(k − 1). . .21, the generating tree for involutions avoiding the pattern τ _k is isomorphic to the generating tree for involutions avoiding the pattern J _k . This implies a family of Wilf equivalences for pattern avoidance by involutions.     

(n,t)-Quasi-Projective and Equivalences

We introduce the notion of (n, t)-quasi projective and show that they are intimately relative to equivalences of module categories. As applications, we give a classification of * ⁿ -modules and generalize Fuller's quasi-progenerators naturally.     

On the tautological ring of a Jacobian modulo rational equivalence

We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of whose ramification points are all of order d, and then for hyperelliptic curves.      

Persistence of freeness for Lie pseudogroup actions

The action of a Lie pseudogroup G\mathcal{G} on a smooth manifold M induces a prolonged pseudogroup action on the jet spaces J ⁿ of submanifolds of M. We prove in this paper that both the local and global freeness of the action of G\mathcal{G} on J ⁿ persist under prolongation in the jet order n. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the prolonged pseudogroup actions.     

Categorical and Simplicial Constructions and Dualization in Algebraic K-Theory

Given an exact category A, we introduce a bisimplicial set W A, which is self-dual and contains both the G-construction of A and its dualization. We prove that the embeddings of G.A and G ^op .A into W A are homotopy equivalences. If A is an exact category with duality, we calculate the induced action of duality on K ₁ (A). We also survey on the self-duality property of some known constructions.     

Elementary equivalence of some rings of definable functions

We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from R ⁿ to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition such as continuity.      

Nilpotent subgroups of self equivalences of torsion spaces

LetX be a finitep-torsion based connected nilpotent CW-complex. We give a criterion of a subgroup of ε(X), the group of self equivalences ofX, to be a nilpotent group, in terms of its action onE _*(X), whereE is a CW-spectrum, satisfying some technical conditions.     

Lattice generation of small equivalences of a countable set

Given a countable set A, let Equ(A) denote the lattice of equivalences of A. We prove the existence of a four-generated sublattice Q of Equ(A) such that Q contains all atoms of Equ(A). Moreover, Q can be generated by four equivalences such that two of them are comparable. Our result is a reasonable generalization of Strietz [5, 6] from the finite case to the countable one; and in spite of its essentially simpler proof it asserts more for the countable case than [2, 3].Dedicated to George Grätzer on his 60th birthdayThis research was supported by the NFSR of Hungary (OTKA), grant no. T7442.     

Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.     

Polygon Recognition and Symmetry Detection

We introduce an approach based on moving frames for polygon recognition and symmetry detection. We present detailed algorithms for the recognition of polygons in R² modulo the special Euclidean, Euclidean, equi-affine, skewed-affine, and similarity Lie groups. We also solve the case of polygons in the Poincar\'e half-plane under the action of SL(2) and explain a method applicable to Lie group actions in general. The time complexity of our algorithms is linear in the number of vertices and they are noise resistant. The signatures used allow the detection of partial, as well as approximate, equivalences.     

Some remarks on tautological sheaves on Hilbert schemes of points on a surface

We study tautological sheaves on the Hilbert scheme of points on a smooth quasi-projective algebraic surface by means of the Bridgeland–King–Reid transform. We obtain Brion–Danila’s Formulas for the derived direct image of tautological sheaves or their double tensor product for the Hilbert–Chow morphism; as an application we compute the cohomology of the Hilbert scheme with values in tautological sheaves or in their double tensor product, thus generalizing results previously obtained for tautological bundles.      

Quasi‐Random Oriented Graphs

We show that a number of conditions on oriented graphs, all of which are satisfied with high probability by randomly oriented graphs, are equivalent. These equivalences are similar to those given by Chung, Graham, and Wilson [5] in the case of unoriented graphs, and by Chung and Graham [3] in the case of tournaments. Indeed, our main theorem extends to the case of a general underlying graph G, the main result of [3] which corresponds to the case that G is complete. One interesting aspect of these results is that exactly two of the four orientations of a four cycle can be used for a quasi‐randomness condition, i.e., if the number of appearances they make in D is close to the expected number in a random orientation of the same underlying graph, then the same is true for every small oriented graph H.