On divisorial filtrations on sheaves

In this paper, we generalize the notion of Poincaré series of a multi-index divisorial filtration corresponding to a collection of sigma-processes to the case of an arbitrary locally-free sheaf on the space of blow-ups of the complex plane ?². For an arbitrary sheaf, we establish a representation of the series in terms of topological invariants of the sheaf. In particular, for the sheaf of functions, this representation coincides with the Poincaré series obtained by Gusein—Zade and Delgado.     

Recognition in a Distributed System

A general design for a distributed system is presented which characterizes system recognition of external events as a sheaf in a Grothendieck topos on a category of diagrams. The information that the system may derive from this sheaf involves transitions through a state space constructed from a collection of topoi interconnected by geometric morphisms. Similarities of this design with natural neural systems leads to the premise that cognitive processes involve the construction and manipulation of sheaf-like structures.     

Flat and Cotorsion Quasi-Coherent Sheaves. Applications

The study of flat covers and cotorsion envelopes has turned out to be very useful since their existence was proved in [3] for the category of R-modules. The problem is even more interesting in categories of sheaves on a topological space, because these categories do not have enough projectives. But the existence of flat covers and cotorsion envelopes allow us to form flat and cotorsion resolutions to compute cohomology.     

Prüfer Sheaves and Generic Sheaves Over the Weighted Projective Lines and Elliptic Curves

In the present paper, we introduce the concepts of Prüfer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Prüfer sheaves and adic sheaves can characterize the category of coherent sheaves. Moreover, we describe the relationship between Prüfer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Prüfer sheaves.     

Stone duality for spectral sheaves and the patch monad

We establish a duality between global sheaves on spectral spaces and right distributive bands. This is a sheaf-theoretical extension of classical Stone duality between spectral spaces and bounded distributive lattices.The topology of a spectral space admits a refinement, the so-called patch topology, giving rise to a patch monad on sheaves over a fixed spectral space. Under the duality just mentioned the algebras of this patch monad are shown to correspond to distributive skew lattices.     

The Bloch-Ogus spectral sequence of a real algebraic variety

For ann-dimensional nonsingular real varietyX, we study the local-global spectral sequence

On the rigidity of super-Grassmannians

The rigidity of the complex super-Grassmannian Gr _m|n,k|l with 0<k<m, 0<l<n, supposing that (k, l) (1,n–1), (m–1, 1), (1,n–2), (m–2, 1), (2,n–1), (m–1, 2), is proved.Partially supported by Erwin Schrödinger International Institute for Mathematical Physics (Vienna, Austria)     

On normal crossings in a Banach space

We show that if A is a closed complex analytic subset of a Banach space X with an unconditional basis (e.g., X=?₂) that has only normal crossings for singularities, then the structure and ideal sheaves of A are cohesive sheaves over X, and consequently, they are acyclic over any pseudoconvex open subset of X.     

Complex Lie algebras corresponding to weighted projective lines

The aim of this paper is to give an alternative proof of Kac’s theorem for weighted projective lines over the complex field. The geometric realization of complex Lie algebras arising from derived categories is essentially used.     

On an Algebraic Extension of A(E)

An algebraic extension of the algebra A(E), where E is a compactum in with nonempty connected interior, leads to a Banach algebra B of functions that are holomorphic on some analytic set K^{o 2} with boundary bK and continuous up to bK. The singular points of the spectrum of B and their defects are investigated. For the case in which B is a uniform algebra, the depth of B in the algebra C(bK) is estimated. In particular, conditions under which B is maximal on bK are obtained.