Fundamental Classes in Algebraic Cobordism

We recall our construction of fundamental classes in the algebraic cobordism of smooth quasi-projective varieties, and show by example that it is not possible to extent this to fundamental classes, functorial for local complete intersection morphisms, for Cohen–Macaulay varieties.     

Algebraic Equivalence and Homology Classes of Real Algebraic Cycles

In this note we study the spectral properties of a multiplication operator in the space L_p(X)^m which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called “floating singularities”.     

Function Spaces and Continuous Algebraic Pairings for Varieties

Given a quasi-projective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of 'smooth curves') for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including (1) the existence of cap and join product pairings in topological cycle theory; (2) the agreement of cup product and intersection product for topological cycle theory; (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product; and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles.     

Galois Cohomology of Unipotent Algebraic Groups and Field Extensions

In this note we discuss, in the case of unipotent groups over nonperfect fields k of characteristic p, an analog of a theorem of Steinberg (formely a Serre's conjecture) for unipotent algebraic group schemes, which relates properties of Galois (or flat) cohomology of unipotent group schemes to finite extensions of k of degree divisible by p.     

Cohomology Theories of Hopf Bimodules and Cup-Product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by M. Gerstenhaber and S. D. Schack, and by C. Ospel. We prove, when A is finite-dimensional, that they are all equal to the Ext functor on the module category of an associative algebra associated to A, described by C. Cibils and M. Rosso. We also give an expression for a cup-product in the cohomology defined by C. Ospel, and prove that it corresponds to the Yoneda product of extensions.     

Cohomology of Real Algebraic Varieties

A class of effective spaces with involution introduced by the author is studied. The cohomology ring of the fixed-point set of an effective space is completely determined by the spectral sequence of involution. Real algebraic varieties admitting a “cellular decomposition” are effective M-spaces. Under certain restrictions, one calculates the spectral sequence of involution and the total ℤ₂ Betti number of the real part for real subvarieties of real algebraic varieties that are effective GM-spaces. Bibliography: 14 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2002, pp. 112–151.     

Hopf Algebra Equivariant Cyclic Cohomology, K-theory and Index Formulas

For an algebra with an action of a Hopf algebra we establish the pairing between equivariant cyclic cohomology and equivariant K-theory for . We then extend this formalism to compact quantum group actions and show that equivariant cyclic cohomology is a target space for the equivariant Chern character of equivariant summable Fredholm modules. We prove an analogue of Julg's theorem relating equivariant K-theory to ordinary K-theory of the C^*-algebra crossed product, and characterize equivariant vector bundles on quantum homogeneous spaces.     

Affine Algebraic Varieties Relative to an Algebraic Theory

Affine algebraic varieties relative to an algebraic theory are introduced and described as irreducible components of affine algebraic sets. Their category is shown to be dually equivalent to the category of irreducible functional algebras.     

Differential Characters and Multiplicative Cohomology

We compare the groups of differential characters of Cheeger and Simons to Karoobi's multiplicative cohomology.     

Algebraic quantum kk-theory

Let 𝒢 be an algebraic quantum group. We introduce an equivariant algebraic kk-theory for 𝒢-module algebras. We study an adjointness theorem related with smash product and trivial action. We also discuss a duality property.     

The Quantum Cohomology Ring of Flag Varieties

We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work Quantum cohomology of flag varieties (Internat. Math. Res. Notices, no. 6 (1995), 263-277). We also give a geometric proof of the quantum Monk formula.

     

非自伴算子代数间的等距代数同构

设Ｇｉ是满足第二可数性公理的、Ｈａｕｓｄｏｒｆｆ的、顺从的、ｒ－离散的、主的局部紧群胚，并且有一个紧开Ｇ－集覆盖；设Ｐｉ是Ｇｉ中含Ｇ＿ｉ￣０的开闭集，且满足及相应的模是具有性质ＤＣ的Ｃ（Ｇｉ）的子代数（ｉ＝１，２）．本文证明从Ａ（Ｐ１）到Ａ（Ｐ２）上的每一个等距代数同构可以扩张成从Ｃ（Ｇ１）到Ｃ（Ｇ２）上的Ｃ－同构，进一步，可以对Ｃ（Ｇ２）重新坐标化，使得这个Ｃ－同构可由一个群胚同构生成．     

Simplicial Cohomology with Coefficients in Symmetric Categorical Groups

In this paper we introduce and study a cohomology theory {H ⁿ (–,A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A,n)} _n0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n3, the functor K(–,n) is right adjoint to the functor _n , where _n (X ) is defined as the fundamental groupoid of the n-loop complex ⁿ (X ). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with _i (Y)=0 for all in,n+1 and n3; and also we obtain a classification theorem for those spaces: [–,Y]H ⁿ (–, _n (Y)).     

Some Remarks on Real and Algebraic Cobordism

We investigate connections between Real cobordism, algebraic cobordism, quadratic forms, the Rost Motive, Morava K(n)-theories and analogues of homotopy classes of Hopf invariant 1.     

The de Rham Bundle on a Compactification of Moduli Space of Abelian Varieties

In this paper, we show that the Chern classes c _k of the de Rham bundle defined on any good toroidal compactification of the moduli space of Abelian varieties of dimension g are zero in the rational Chow ring of , for g=4, 5 and k>0.     

On the First Galois Cohomology Group of the Algebraic Group SL1(D)

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL₁(A). For a field extension K of F, we study the first Galois Cohomology group H ¹(K,SL₁(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.     

Algebraic K-Theory In Low Degree and The Novikov Assembly Map

We prove that the Novikov assembly map for a group factorizes,in ‘low homological degree’, through the algebraicK-theory of its integral group ring. In homological degree 2,this answers a question posed by N. Higson and P. Julg. As adirect application, we prove that if is torsion-free and satisfiesthe Baum-Connes conjecture, then the homology group H₁(; Z)injects in and in , for any ring A such that . If moreover B is of dimension lessthan or equal to 4, then we show that H₂(; Z) injects in and in , where A is as before, and ₂ is generated by the Steinberg symbols{,}, for . 2000 Mathematical Subject Classification: primary 19D55, 19Kxx,58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.     

Chern characters associated with almost commuting algebras

A trace formula related to -almost commuting subalgebras and is established. By means of this formula, homomorphisms from to and from to are established. An index map from to is also given.

     

Tate''s Conjecture, Algebraic Cycles and Rational K-Theory in Characteristic p

The purpose of this article is to discuss conjectures on motives, algebraic cycles and K-theory of smooth projective varieties over finite fields. We give a characterization of Tate's conjecture in terms of motives and their Frobenius endomorphism. This is used to prove that if Tate's conjecture holds and rational and numerical equivalence over finite fields agree, then higher rational K-groups of smooth projective varieties over finite fields vanish (Parshin's conjecture). Parshin's conjecture in turn implies a conjecture of Beilinson and Kahn giving bounds on rational K-groups of fields in finite characteristic. We derive further consequences from this result.