Manuel Norman

Bio: Manuel Norman is an academic researcher. The author has contributed to research in topics: Cohomology & Partially ordered set. The author has an hindex of 3, co-authored 5 publications receiving 13 citations.

Two cohomology theories for structured spaces

Abstract: In [1] we defined a new kind of space called 'structured space' which locally resembles, near each of its points, some algebraic structure. We noted in the conclusion of the cited paper that the maps $f_s$ and $h$, which are of great importance in the theory of structured spaces, have some connections with the notions of presheaves (and hence also sheaves) and vector bundles. There are well known cohomology theories involving such objects; this suggests the possibility of the existence of (co)homology theories for structured spaces which are somehow related to $f_s$ and $h$. In this paper we indeed develop two cohomology theories for structured spaces: one of them arises from $f_s$, while the other one arises from $h$. In order to do this, we first develop a more general cohomology theory (called rectangular cohomology in the finite case, and square cohomology in the infinite case), which can actually be applied also in many other situations, and then we obtain the cohomology theories for structured spaces as simple consequences of this theory.

Co)homology theories for structured spaces arising from their corresponding poset

Abstract: In [1] we introduced the notion of 'structured space', i.e. a space which locally resembles various algebraic structures. In [2] and [3] we studied some cohomology theories related to these space. In this paper we continue in this direction: while in [2] we mainly focused on cohomologies arising from $f_s$ and $h$, and in [3] we considered cohomologies for generalisations of objects which involved structured spaces, here we deal with (co)homologies coming from the poset associated to a structured space via an equivalence relation defined at the end of Section 4 in [1]. More precisely, we will show that various (co)homologies for posets can also be applied (under some assumptions) to structured spaces.

Structured spaces: categories, sheaves, bundles, schemes and cohomology theories

Abstract: In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is to define categories of structured spaces, (pre)sheaves with values in such categories, and generalised notions of vector bundles, ringed spaces and schemes. Then, we will construct (using some techniques and also the general method in Section 2 of [2]) various cohomology theories for these objects.

A (co)homology theory for some preordered topological spaces

Abstract: The aim of this short note is to develop a (co)homology theory for topological spaces together with the specialisation preorder. A known way to construct such a (co)homology is to define a partial order on the topological space starting from the preorder, and then to consider some (co)homology for the obtained poset; however, we will prove that every topological space with the above preorder consists of two disjoint parts (one called 'poset part', and the other one called 'complementary part', which is not a poset in general): this suggests an improvement of the previous method that also takes into account the poset part, and this is indeed what we will study here.

Conical calculus on schemes and perfectoid spaces via stratification

Abstract: In this paper we show that, besides the usual calculus involving Kahler differentials, it is also possible to define conical calculus on schemes and perfectoid spaces; this can be done via a stratification process. Following some ideas from [1-2], we consider some natural stratifications of these spaces and then we build upon the work of Ayala, Francis, and Tanaka [3] (see also [4-5] and [18]); using their definitions of derivatives, smoothness and vector fields for stratified spaces, and thanks to some particular methods, we are able to transport these concepts to schemes and perfectoid spaces. This also allows us to define conical differential forms and the conical de Rham complex. At the end, we compare this approach with the usual one, noting that it is a useful \textit{addition} to Kahler method.

Arrangements Of Hyperplanes

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Extensions and cohomology of monoids with coefficients in semimodules

Abstract: A natural generalization of the notion of group extensions was introduced by Redei [86] for semigroups and called Scheier extension of semigroups These extensions have been investigated in [86,100,36] and other works without using homological methods The extensions of semigroups with respect to the factorizations of Redei [86], Rees [87] and Preston [83] have been described by homological methods in [39–41,43]In particular in [39] the functors Ext n are defined for Schreier extensions of commutative semigroups with cancellation

Co)homology theories for structured spaces arising from their corresponding poset

Abstract: In [1] we introduced the notion of 'structured space', i.e. a space which locally resembles various algebraic structures. In [2] and [3] we studied some cohomology theories related to these space. In this paper we continue in this direction: while in [2] we mainly focused on cohomologies arising from $f_s$ and $h$, and in [3] we considered cohomologies for generalisations of objects which involved structured spaces, here we deal with (co)homologies coming from the poset associated to a structured space via an equivalence relation defined at the end of Section 4 in [1]. More precisely, we will show that various (co)homologies for posets can also be applied (under some assumptions) to structured spaces.

Structured spaces: categories, sheaves, bundles, schemes and cohomology theories

Abstract: In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is to define categories of structured spaces, (pre)sheaves with values in such categories, and generalised notions of vector bundles, ringed spaces and schemes. Then, we will construct (using some techniques and also the general method in Section 2 of [2]) various cohomology theories for these objects.