Search results for: morphism

Authors: Ying Yi Wu

Abstract:

In this paper, we present a proof of the fact that a finite morphism is proper. We show that a finite morphism is universally closed and of finite type, which are the conditions for properness. Our proof is based on the theory of schemes and involves the use of the projection formula and the base change theorem. We first show that a finite morphism is of finite type and then proceed to show that it is universally closed. We use the fact that a finite morphism is also an affine morphism, which allows us to use the theory of coherent sheaves and their modules. We then show that the map induced by a finite morphism is flat and that the module it induces is of finite type. We use these facts to show that a finite morphism is universally closed. Our proof is constructive, and we provide details for each step of the argument.

Keywords: finite, morphism, schemes, projection.

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Authors: P. G. Romeo

Abstract:

A Lie group is a highly sophisticated structure which is a smooth manifold whose underlying set of elements is equipped with the structure of a group such that the group multiplication and inverse-assigning functions are smooth. This structure was introduced by the Norwegian mathematician So- phus Lie who founded the theory of continuous groups. The Lie groups are well developed and have wide applications in areas including Mathematical Physics. There are several advances and generalizations for Lie groups and Lie groupoids is one such which is termed as a "many-object generalization" of Lie groups. A groupoid is a category whose morphisms are all invertible, obviously, every group is a groupoid but not conversely. Definition 1. A Lie groupoid G ⇒ M is a groupoid G on a base M together with smooth structures on G and M such that the maps α, β: G → M are surjective submertions, the object inclusion map x '→ 1x, M → G is smooth, and the partial multiplication G ∗ G → G is smooth. A bundle is a triple (E, p, B) where E, B are topological spaces p: E → B is a map. Space B is called the base space and space E is called total space and map p is the projection of the bundle. For each b ∈ B, the space p−1(b) is called the fibre of the bundle over b ∈ B. Intuitively a bundle is regarded as a union of fibres p−1(b) for b ∈ B parametrized by B and ’glued together’ by the topology of the space E. A cross-section of a bundle (E, p, B) is a map s: B → E such that ps = 1B. Example 1. Given any space B, a product bundle over B with fibre F is (B × F, p, B) where p is the projection on the first factor. Definition 2. A principal bundle P (M, G, π) consists of a manifold P, a Lie group G, and a free right action of G on P denoted (u, g) '→ ug, such that the orbits of the action coincide with the fibres of the surjective submersion π : P → M, and such that M is covered by the domains of local sections σ: U → P, U ⊆ M, of π. Definition 3. A Lie group bundle, or LGB, is a smooth fibre bundle (K, q, M ) in which each fibre (Km = q−1(m), and the fibre type G, has a Lie group structure, and for which there is an atlas {ψi: Ui × G → KUi } such that each {ψi,m : G → Km}, is an isomorphism of Lie groups. A morphism of LGB from (K, q, M ) to (K′, q′, M′) is a morphism (F, f ) of fibre bundles such that each Fm: Km → K′ is a morphism of Lie groups. In this paper, we will be discussing the Lie groupoid bundles. Here it is seen that to a Lie groupoid Ω on base B there is associated a collection of principal bundles Ωx(B, Ωx), all of which are mutually isomorphic and conversely, associated to any principal bundle P (B, G, p) there is a groupoid called the Ehresmann groupoid which is easily seen to be Lie. Further, some interesting properties of the category of Lie groupoids and bundles will be explored.

Keywords: groupoid, lie group, lie groupoid, bundle

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Authors: Yousuf Alkhezi

Abstract:

Since most of the literature on algebra does not make much deal with the special case of mapping cone. This paper is an alternative way to examine the special tensor product and mapping cone. Also, we show that the isomorphism that implies the mapping cone commutes with the tensor product for the ordinary tensor product no longer holds for the pinched tensor product. However, we show there is a morphism. We will introduce an alternative way of mapping cone. We are looking for more properties which is our future project. Also, we want to apply these new properties in some application. Many results and examples with classical algorithms will be provided.

Keywords: complex, tensor product, pinched tensore product, mapping cone

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Authors: Nisreen Agha

Abstract:

Investigation of representative core samples under the petrological microscope reveals common petrographic and mineralogical characteristics with distinct faunal assemblages, allowing establishing of the microfacies associations and deducing the paleoenvironmental trends of the Paleocene Farrud and Mabruk rock units. Recognition of the early and post-diagenetic processes, particularly dolomitization and micritization, as well as dissolution and precipitation of spary drusy calcite as a new morphism process affecting the reservoir rocks, is established. The microfacies trends detected from the investigation of 46 core samples from Farrud member (Lower Paleocene) representing six wells; QQQ1-11, GG1-11, LLL1-11, RRR1-11, RRR40-11, and RRR45-11 indicate that the deposition was started within the realm of shallow supratidal and intertidal subenvironments followed by deeper environments of the shelf bays with maximum sea level during inner shelf environment where fossiliferous bioclastic packstone dominated. The microfacies associations determined in 8 core samples from two wells LLL1and RRR40 representing Mabruk Member (Upper Paleocene), indicate paleoenvironmental trends marked by sea level fluctuations accompanied with a relatively marine shelf bay conditions intervened with short-lived shallow intertidal and supratidal warm coastal sedimentation. As a result, dolostone, evaporitic dismicrites, and gypsiferous dolostone of supratidal characters were deposited. They reflect rapid oscillation of the sea level marked by drop land-ward shift of the sea shore deposition prevailed by supratidal gypsiferous dolostone and numerous ferruginous materials as clouds straining many parts of dolomite and surrounded the micritized fossils. This situation ends the deposition of the Farrud Member in most of the studied wells. On the other hand, the facies in the northern part of the Concession -11 field indicates deposition in a deeper marine setting than in the southern facies.

Keywords: Farrud and Mabruk members, paleocene, microfacies associations, diagenesis, sea level oscillation, depositional environments

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