Serre-duality for Tails

A version of Serre-duality is proved for Artin's non-commutative projective schemes.

     

A hydrodynamic exercise in free probability: Setting up free Euler equations

For the free probability analogue of Euclidean space endowed with the Gaussian measure we apply the approach of Arnold to derive Euler equations for a Lie algebra of non-commutative vector fields which preserve a certain trace. We extend the equations to vector fields satisfying non-commutative smoothness requirements. We introduce a cyclic vorticity and show that it satisfies vorticity equations and that it produces a family of conserved quantities.     

Towards the multiplicative behavior of the K-theoretical McKay correspondence

When the quotient of a symplectic vector space by the action of a finite subgroup of symplectic automorphisms admits as a crepant projective resolution of singularities the Hilbert scheme of regular orbits of Nakamura, then there is a natural isomorphism between the Grothendieck group of this resolution and the representation ring of the group, given by the Bridgeland-King-Reid map. However, this isomorphism is not compatible with the ring structures. For the Hilbert scheme of points on the affine plane, we study the multiplicative behavior of this map.     

Euler characteristics of general linear sections and polynomial Chern classes

We obtain a precise relation between the Chern–Schwartz–MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a hypersurface, this leads to simple proofs of formulas of Dimca–Papadima and Huh for the degrees of the polar map of a homogeneous polynomial, extending these formula to any algebraically closed field of characteristic \(0\), and proving a conjecture of Dolgachev on ‘homaloidal’ polynomials in the same context. We generalize these formulas to subschemes of higher codimension in projective space. We also describe a simple approach to a theory of ‘polynomial Chern classes’ for varieties endowed with a morphism to projective space, recovering properties analogous to the Deligne–Grothendieck axioms from basic properties of the Euler characteristic. We prove that the polynomial Chern class defines homomorphisms from suitable relative Grothendieck rings of varieties to \(\mathbb{Z }[t]\).     

The Grothendieck group of a cluster category

For the cluster category of a hereditary or a canonical algebra, or equivalently for the cluster category of the hereditary category of coherent sheaves on a weighted projective line, we study the Grothendieck group with respect to an admissible triangulated structure.     

A characterization theorem for geometric logic

We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories which are geometric in terms of their models in Grothendieck toposes, solving a problem posed by Ieke Moerdijk in 1989.     

Fields and forms on ρ-algebras

In this paper we introduce non-commutative fields and forms on a new kind of non-commutative algebras: ρ-algebras. We also define the Frölicher-Nijenhuis bracket in the non-commutative geometry on ρ-algebras.     

(Quasi-)Poisson enveloping algebras

We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.     

A Tannakian classification of torsors on the projective line

In this small note, we present a Tannakian proof of the theorem of Grothendieck–Harder on the classification of torsors under a reductive group on the projective line over a field.     

Grothendieck Property for the Symmetric Projective Tensor Product

For a Banach space $E$, we give sufficient conditions for the Grothendieck property of $⨶_{n,s,\pi}E$, the symmetric projective tensor product of $E.$ Moreover, if $E^∗$ has the bounded compact approximation property, then these sufficient conditions are also necessary.     

Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as ``Gromov invariants') on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one.

     

BIK+-逻辑与非可换模糊逻辑

引入BIK -逻辑的概念,证明了BIK -逻辑的可靠性定理(基于BCC-代数)。同时,研究了BIK -逻辑与非可换模糊逻辑的关系,说明了各种源于模糊逻辑的代数结构之间的内在联系,并用一个图示表达了这些关系。     

非交换群上一类代数的不可约表示

在一类无限维非交换Hopf代数上,借助其Hopf理想,构造出商Hopf代数,讨论了此商代数上的有限维不可约模,得出此非平凡不可约模的维数一定是2.     

Non-commutative Hopf algebra of formal diffeomorphisms

This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double-tensor algebra on a bi-algebra.     

Localization on certain Grothendieck categories

Using categorical techniques we obtain some results on localization and colocalization theory in Grothendieck categories with a set of small projective generators. In particular, we give a sufficient condition for such category to be semiartinian. For semiartinian Grothendieck categories where every simple object has a projective cover, we obtain that every localizing subcategory is a TTF-class. In addition, some applications to semiperfect categories are obtained.     

Riemann-Roch for tensor powers

Mapping a locally free module on a scheme to its l-th tensor power gives rise to a natural map from the Grothendieck group of all locally free modules to the Grothendieck group of all locally free representations of the l-th symmetric group. In this paper, we prove some formulas of Riemann-Roch type for the behavior of this tensor power operation with respect to the push-forward homomorphism associated with a projective morphism between schemes. We furthermore establish analogous formulas for higher K-groups. Received February 13, 1998; in final form January 12, 1999     

关于Grothendieck群的几点注记

作者在本文中围绕Ｇｒｏｔｈｅｎｄｉｅｃｋ群对几个问题进行了讨论，主要结果有：１．一切有限生成Ｒ－模的同构类作成一个集合；２．在任意由Ｒ－模作成的集合中稳定同的关系是合同关系；３．Ｇｒｏｔｈｅｎｄｉｅｃｋ群的同构不变性成立。     

Numerical equivalence defined on Chow groups of Noetherian local rings

In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional -vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendiecks standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equivalence can be proven. Using the theory of numerical equivalence, the notion of numerically Roberts rings is defined. It is proved that a Cohen–Macaulay local ring of positive characteristic is a numerically Roberts ring if and only if the Hilbert–Kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength. Numerically Roberts rings satisfy the vanishing property of intersection multiplicities. We shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized Chern characters.     

Properties (u) and (V*) of Pelczynski in symmetric spaces of <Emphasis Type="Italic">τ</Emphasis>-measurable operators

It is shown that order continuity of the norm and weak sequential completeness in non-commutative strongly symmetric spaces of τ-measurable operators are respectively equivalent to properties (u) and (V ^*) of Pelczynski. In addition, it is shown that each strongly symmetric space with separable (Banach) bidual is necessarily reflexive. These results are non-commutative analogues of well-known characterisations in the setting of Banach lattices.     

On the local Picard group

In his book SGA2, A. Grothendieck developed Lefschetz theorems for the Picard group, the aim being to compare the Picard group of a projective variety with the one of a hyperplane section. An intermediate object is the Picard group of the formal completion along the hyperplane section. Here we proceed similarly but in the local complex analytic context. The use of the exponential sequence leads to analytic as well as topological depth conditions.