Families of Algebraic Varieties Parametrized by Topological Spaces

We study families of algebraic varieties parametrized by topological spaces and generalize some classical results such as Hilbert Nullstellensatz and primary decomposition of commutative rings. We show that there is an equivalence between the category of bivariant coherent sheaves and the category of sheaves of finitely generated modules.     

Countably generated Hilbert modules, the Kasparov Stabilisation Theorem, and frames in Hilbert modules

We consider a class of countably generated Hilbert modules in which the generators are multipliers of the module, and prove a version of the Kasparov Stabilisation Theorem for these modules. We then extend recent work of Frank and Larson on frames in Hilbert modules.

     

Repeated-root cyclic and negacyclic codes over a finite chain ring

We show that repeated-root cyclic codes over a finite chain ring are in general not principally generated. Repeated-root negacyclic codes are principally generated if the ring is a Galois ring with characteristic a power of 2. For any other finite chain ring they are in general not principally generated. We also prove results on the structure, cardinality and Hamming distance of repeated-root cyclic and negacyclic codes over a finite chain ring.     

Dilations,Product Systems,and Weak Dilations

We generalize Bhat's construction of product systems of Hilbert spaces from E₀-semigroups on B(H) for some Hilbert space H to the construction of product systems of Hilbert modules from E₀-semigroups on B^a(E) for some Hilbert module E. As a byproduct we find the representation theory for B^a(E) if E has a unit vector. We establish a necessary and sufficient criterion for the conditional expectation generated by the unit vector to define a weak dilation of a CP-semigroup in the sense of [1]. Finally, we also show that white noises on general von Neumann algebras in the sense of [2] can be extended to white noises on a Hilbert module.     

Tensor product varieties,perverse sheaves,and stability conditions

The space spanned by the class of simple perverse sheaves in Zheng (2008) without localization is isomorphic to the tensor product of a Verma module with a tensor product of irreducible integrable highest weight modules of the quantum enveloping algebra associated with a graph. Under the isomorphism, the simple perverse sheaves get identified with the canonical basis elements of the tensor product module. The two stability conditions coincide with the localization process in Zheng (2008), by using supports and singular supports of complexes of sheaves, respectively.     

Quasi-Free Resolutions Of Hilbert Modules

The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra $\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and $\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.     

Biliaison Classes of Rank Two Reflexive Sheaves on P3

A biliaison theory for rank two reflexive sheaves on projective 3 - space is established, In analogy with the existing theory for curves and, more generally, for 2 - codimensional subschemes of projective n - space. By definition two such sheaves are in the same biliaison class if their first cohomology modules are isomorphic. A parametrization of these classes is given. It is then shown that any class not corresponding to the zero module has a structure described by the Lazarsfeld - Rao property. In particular, it is shown that vector bundles are precisely the minimal elements (in the of the LR - property) in their biliaison classes.     

Geometric two‐dimensional idèles with cycle module coefficients

We give a theory of idèles with coefficients for smooth surfaces over a field. It is an analogue of Beilinson/Huber's theory of higher adèles, but handling cycle module sheaves instead of quasi‐coherent ones. We prove that they give a flasque resolution of the cycle module sheaves in the Zariski topology. As a technical ingredient we show the Gersten property for cycle modules on equicharacteristic complete regular local rings, which might be of independent interest.     

Modules de cycles et motifs mixtes

For a perfect field k, we give a relation between the category of homotopy invariant sheaves with transfers defined by Voevodsky and the category of cycle modules defined by Rost. More precisely, the category of cycle modules over k is equivalent to the category obtained from the homotopy invariant sheaves with transfers by formally inverting the sheaf represented by $\text{G}{}_{\mathrm{m}}$ with its canonical structure of a presheaf with transfers. This gives a canonical monoidal structure on the category of cycle modules over k, and shows that it is Abelian. To cite this article: F. Déglise, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Abelianness of the “missing part” from a sheaf category to a module category

This paper investigates the structure of the"missing part"from the category of coherent sheaves over a weighted projective line of weight type(2,2,n)to the category of finitely generated right modules on the associated canonical algebra.By constructing a t-structure in the stable category of the vector bundle category,we show that the"missing part"is equivalent to the heart of the t-structure,hence it is abelian.Moreover,it is equivalent to the category of finitely generated modules on the path algebra of type An-1.     

Hilben C*-模之间的同构定理

获得了Hilbert C^*-模之间的两个同构定理．作为推论，证明了C^*-代数上每个可数生成的Hilbert C^*-模均稳定酉等价于A。     

Bezout's theorem and Cohen-Macaulay modules

We define very proper intersections of modules and projective subschemes. It turns out that equidimensional locally Cohen-Macaulay modules intersect very properly if and only if they intersect properly. We prove a Bezout theorem for modules which meet very properly. Furthermore, we show for equidimensional subschemes X and Y: If they intersect properly in an arithmetically Cohen-Macaulay subscheme of positive dimension then X and Y are arithmetically Cohen-Macaulay. The module version of this result implies splitting criteria for reflexive sheaves. Received August 26, 1999 / Published online March 12, 2001     

On Gorenstein injective discrete modules over profinite groups

Gorenstein homological algebra was introduced in categories of modules. But it has proved to be a fruitful way to study various other categories such as categories of complexes and of sheaves. In this paper, the research of relative homological algebra in categories of discrete modules over profinite groups is initiated. This seems appropriate since (in some sense) the subject of Gorenstein homological algebra had its beginning with Tate homology and cohomology over finite groups. We prove that if the profinite group has virtually finite cohomological dimension then every discrete module has a Gorenstein injective envelope, a Gorenstein injective cover and we study various cohomological dimensions relative to Gorenstein injective discrete modules. We also study the connection between relative and Tate cohomology theories.     

Compact operators on Hilbert modules

We prove that an adjointable contraction acting on a countably generated Hilbert module over a separable unital C*-algebra is compact if and only if the set of its second contractive perturbations is separable.

     

Indexed algebras associated to a log structure and a theorem of p-descent on log schemes

After discussing gradings by sheaves of degrees, we associate to any log scheme a canonical invertible sheaf endowed with a certain multiplicative structure, which we call its associated graded algebra. In the relative case we construct a canonical connection on this algebra. In the log smooth case over a base of positive characteristic p, we study integrable and p-integrable graded modules over this algebra, and establish a Cartier type p-descent theorem, generalizing previous results of Ogus. We apply it to give an alternate proof of a result of Tsuji on closed forms fixed by the Cartier operator Received: 15 January 1999 / Revised version: 30 September 1999     

On M-coidempotent elements and fully coidempotent modules

Abstract

In this article, we introduce the notion of M-coidempotent elements of a ring and investigate their connections with fully coidempotent modules, fully copure modules and vn-regular modules where M is a module. We prove that if M is a finitely cogenerated module, then M is fully copure if and only if M is semisimple. We prove that if M is a Noetherian module or M is a finitely cogenerated module, then M is fully coidempotent if and only if M is a vn-regular module. Finally, we give a characterization of semisimple Artinian modules via weak idempotents.     

主拟-Baer模

In this paper, we give the equivalent characterizations of principally quasi-Baer modules, and show that any direct summand of a principally quasi-Baer module inherits the property and any finite direct sum of mutually subisomorphic principally quasi-Baer modules is also principally quasi-Baer. Moreover, we prove that left principally quasi-Baer rings have Morita invariant property. Connections between Richart modules and principally quasi-Baer modules are investigated.     

Comparing Castelnuovo-Mumford regularity and extended degree: The borderline cases

Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, and Vasconcelos shows that both can be compared in the case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained if we choose as extended degree the homological or the smallest extended degree. The corresponding algebras are characterized in three ways: by relations among the algebra generators, by using generic initial ideals, and by their Hilbert series.