Galois Cohomology and Hochschild Cohomology over a Base Topos.

基拓扑上的伽罗瓦上同调和 Hochschild 上同调。

基本信息

  • 批准号:
    2611023
  • 负责人:
  • 金额:
    --
  • 依托单位:
  • 依托单位国家:
    英国
  • 项目类别:
    Studentship
  • 财政年份:
    2021
  • 资助国家:
    英国
  • 起止时间:
    2021 至 无数据
  • 项目状态:
    未结题

项目摘要

Difference algebra A difference ring is a commutative ring with identity, equipped with an endomorphism. Naturally, we define morphisms which commute with the ring structure and the endomorphism, and consider the category of difference rings; we also define difference modules such that the module structure commutes with the endomorphism, with analogous modules, and consider the category of difference modules. Previously, model-theoretic approaches have been successful in establishing the analogues of commutative algebra, Galois theory and Boolean logic in the difference case. However, in the cohomological case, these methods have proved unsuccessful, and the aim of Dr Tomasic's project is to pursue a continuation of Hakim's monograph from a category-theoretic and topos-theoretic viewpoint. This has the potential to connect numerous cohomology theories for algebras, relative schemes and relative algebraic groups. Difference cohomology Cohomological algebra in the context of ring modules relies on the simple fact that given two objects of R-mod for a given ring R, the hom-set is itself an object in R-mod. It is here that the difference analogue falls short; in general, given two difference sets, their hom-set is a bare set, that is, it is not necessarily endowed with a difference endomorphism. By the same token, given two difference R-modules, their hom-set is not in general a difference R-module, and therefore the obvious analogy of cohomology in the category of difference R-modules is impossible. Enriched category theory Dr Tomasic instead considers difference algebra within an enriched category, by defining the internal hom-object of any two difference sets via the direct sum of one of the objects with the difference set of natural numbers, equipped with the expected shift endomorphism. The categories of difference objects defined above are then the underlying categories of these enriched categories. Not only does this produce a hom-set that is itself a difference object, but it obtains the "currying" isomorphism of hom-sets involving the tensor of two objects that is needed to develop a cohomological framework. Topos theory Dr Tomasic shows that such an enriched category is equivalent to the category of sets with an action of the monoid of natural numbers under addition, which is a Grothendieck topos, known as the classifying topos of the natural numbers. We therefore seek to develop our cohomological theories over an arbitrary base topos. By working over the classifying topos of a group or even a groupoid, one can then develop the corresponding group equivariant algebraic geometry. Therefore, in particular the topos of difference sets yields a respective difference algebraic geometry. Galois cohomology and Hochschild cohomology The specific topic I am researching is to develop Galois cohomology and Hochschild cohomology over an arbitrary base topos, with attention to development over the topos of difference sets. This research will be a natural continuation of my Master's thesis, which was an exposition of a graded Hochschild cohomological vanishing result in an upcoming paper by Dr Ambrus Pal. In particular, I will explore the Galois cohomology and Hochschild cohomology of difference algebras. This topic was agreed upon with Dr Tomasic, and it will form an intrinsic part of the programme. Dr Tomasic's project has already established the foundations of enriched/internal homological algebra, and further exploration of application to both Galois cohomology and Hochschild cohomology is a compelling continuation of this work.
差环是一个有单位元的交换环,它具有一个自同态。自然地,我们定义与环结构和自同态交换的态射,并考虑差环的范畴;我们还定义差模,使得模结构与自同态交换,与类似的模交换,并考虑差模的范畴。在此之前,模型论的方法已经成功地建立了交换代数,伽罗瓦理论和布尔逻辑在不同情况下的类似物。然而,在上同调的情况下,这些方法都被证明是不成功的,而Tommy博士的项目的目的是从范畴论和拓扑论的观点继续哈基姆的专著。这有可能连接代数,相对计划和相对代数群的众多上同调理论。差上同调环模下的上同调代数依赖于一个简单的事实,即给定一个环R的两个R-mod对象,其hom-set本身也是R-mod中的一个对象。在这里,差的类比福尔斯不够的;一般来说,给定两个差集,它们的hom-set是一个裸集,也就是说,它不一定被赋予差自同态。同样,给定两个差R-模,它们的hom-set一般不是差R-模,因此在差R-模范畴中上同调的明显类比是不可能的。丰富范畴理论Tomorrow博士考虑的是丰富范畴内的差代数,通过定义任何两个差集的内部hom-object,通过其中一个对象与自然数的差集的直接和,配备了预期的移位自同态。然后,上面定义的差异对象的类别是这些丰富类别的基础类别。这不仅产生了一个本身就是差对象的hom-set,而且还获得了包含两个对象张量的hom-set的“currying”同构,这是开发上同调框架所需要的。拓扑理论托姆斯基博士表明,这样一个丰富的范畴是等价的范畴集的作用下,自然数的幺半群加法,这是一个格罗滕迪克拓扑,被称为分类拓扑的自然数。因此,我们寻求在任意基拓扑上发展我们的上同调理论。通过研究一个群甚至一个广群的分类拓扑,人们可以发展相应的群等变代数几何。因此,特别是拓扑的差集产生相应的差代数几何。伽罗瓦上同调和Hochschild上同调我正在研究的具体课题是在任意基拓扑上发展伽罗瓦上同调和Hochschild上同调,并注意在差集拓扑上的发展。这项研究将是一个自然的延续我的硕士论文,这是一个分级Hochschild上同调消失的结果在即将到来的论文由Ambrus博士的阐述。特别是,我将探讨差代数的伽罗瓦上同调和Hochschild上同调。这一主题是与汤姆博士商定的,它将成为该计划的一个固有部分。Tommy博士的项目已经建立了丰富/内部同调代数的基础,进一步探索Galois上同调和Hochschild上同调的应用是这项工作令人信服的延续。

项目成果

期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

其他文献

吉治仁志 他: "トランスジェニックマウスによるTIMP-1の線維化促進機序"最新医学. 55. 1781-1787 (2000)
Hitoshi Yoshiji 等:“转基因小鼠中 TIMP-1 的促纤维化机制”现代医学 55. 1781-1787 (2000)。
  • DOI:
  • 发表时间:
  • 期刊:
  • 影响因子:
    0
  • 作者:
  • 通讯作者:
LiDAR Implementations for Autonomous Vehicle Applications
  • DOI:
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    0
  • 作者:
  • 通讯作者:
生命分子工学・海洋生命工学研究室
生物分子工程/海洋生物技术实验室
  • DOI:
  • 发表时间:
  • 期刊:
  • 影响因子:
    0
  • 作者:
  • 通讯作者:
吉治仁志 他: "イラスト医学&サイエンスシリーズ血管の分子医学"羊土社(渋谷正史編). 125 (2000)
Hitoshi Yoshiji 等人:“血管医学与科学系列分子医学图解”Yodosha(涉谷正志编辑)125(2000)。
  • DOI:
  • 发表时间:
  • 期刊:
  • 影响因子:
    0
  • 作者:
  • 通讯作者:
Effect of manidipine hydrochloride,a calcium antagonist,on isoproterenol-induced left ventricular hypertrophy: "Yoshiyama,M.,Takeuchi,K.,Kim,S.,Hanatani,A.,Omura,T.,Toda,I.,Akioka,K.,Teragaki,M.,Iwao,H.and Yoshikawa,J." Jpn Circ J. 62(1). 47-52 (1998)
钙拮抗剂盐酸马尼地平对异丙肾上腺素引起的左心室肥厚的影响:“Yoshiyama,M.,Takeuchi,K.,Kim,S.,Hanatani,A.,Omura,T.,Toda,I.,Akioka,
  • DOI:
  • 发表时间:
  • 期刊:
  • 影响因子:
    0
  • 作者:
  • 通讯作者:

的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

{{ truncateString('', 18)}}的其他基金

An implantable biosensor microsystem for real-time measurement of circulating biomarkers
用于实时测量循环生物标志物的植入式生物传感器微系统
  • 批准号:
    2901954
  • 财政年份:
    2028
  • 资助金额:
    --
  • 项目类别:
    Studentship
Exploiting the polysaccharide breakdown capacity of the human gut microbiome to develop environmentally sustainable dishwashing solutions
利用人类肠道微生物群的多糖分解能力来开发环境可持续的洗碗解决方案
  • 批准号:
    2896097
  • 财政年份:
    2027
  • 资助金额:
    --
  • 项目类别:
    Studentship
A Robot that Swims Through Granular Materials
可以在颗粒材料中游动的机器人
  • 批准号:
    2780268
  • 财政年份:
    2027
  • 资助金额:
    --
  • 项目类别:
    Studentship
Likelihood and impact of severe space weather events on the resilience of nuclear power and safeguards monitoring.
严重空间天气事件对核电和保障监督的恢复力的可能性和影响。
  • 批准号:
    2908918
  • 财政年份:
    2027
  • 资助金额:
    --
  • 项目类别:
    Studentship
Proton, alpha and gamma irradiation assisted stress corrosion cracking: understanding the fuel-stainless steel interface
质子、α 和 γ 辐照辅助应力腐蚀开裂:了解燃料-不锈钢界面
  • 批准号:
    2908693
  • 财政年份:
    2027
  • 资助金额:
    --
  • 项目类别:
    Studentship
Field Assisted Sintering of Nuclear Fuel Simulants
核燃料模拟物的现场辅助烧结
  • 批准号:
    2908917
  • 财政年份:
    2027
  • 资助金额:
    --
  • 项目类别:
    Studentship
Assessment of new fatigue capable titanium alloys for aerospace applications
评估用于航空航天应用的新型抗疲劳钛合金
  • 批准号:
    2879438
  • 财政年份:
    2027
  • 资助金额:
    --
  • 项目类别:
    Studentship
Developing a 3D printed skin model using a Dextran - Collagen hydrogel to analyse the cellular and epigenetic effects of interleukin-17 inhibitors in
使用右旋糖酐-胶原蛋白水凝胶开发 3D 打印皮肤模型,以分析白细胞介素 17 抑制剂的细胞和表观遗传效应
  • 批准号:
    2890513
  • 财政年份:
    2027
  • 资助金额:
    --
  • 项目类别:
    Studentship
CDT year 1 so TBC in Oct 2024
CDT 第 1 年,预计 2024 年 10 月
  • 批准号:
    2879865
  • 财政年份:
    2027
  • 资助金额:
    --
  • 项目类别:
    Studentship
Understanding the interplay between the gut microbiome, behavior and urbanisation in wild birds
了解野生鸟类肠道微生物组、行为和城市化之间的相互作用
  • 批准号:
    2876993
  • 财政年份:
    2027
  • 资助金额:
    --
  • 项目类别:
    Studentship

相似海外基金

CAREER: Elliptic cohomology and quantum field theory
职业:椭圆上同调和量子场论
  • 批准号:
    2340239
  • 财政年份:
    2024
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Symplectic cohomology and quantum cohomology of Fano manifolds
Fano流形的辛上同调和量子上同调
  • 批准号:
    2306204
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Cohomology theories for algebraic varieties
代数簇的上同调理论
  • 批准号:
    2883661
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Studentship
LEAPS-MPS: Quantum Field Theories and Elliptic Cohomology
LEAPS-MPS:量子场论和椭圆上同调
  • 批准号:
    2316646
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Cohomology of arithmetic groups in GL(2) over definite quaternion algebras
GL(2) 定四元数代数上算术群的上同调
  • 批准号:
    2884658
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Studentship
Koszul duality and the singularity category for the enhanced group cohomology ring
增强群上同调环的 Koszul 对偶性和奇点范畴
  • 批准号:
    EP/W036320/1
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Research Grant
Dual complexes and weight filtrations: Applications to cohomology of moduli spaces and invariants of singularities
对偶复形和权重过滤:模空间上同调和奇点不变量的应用
  • 批准号:
    2302475
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Matrix Approximations, Stability of Groups and Cohomology Invariants
矩阵近似、群稳定性和上同调不变量
  • 批准号:
    2247334
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Research on commutative rings via etale cohomology theory
基于etale上同调理论的交换环研究
  • 批准号:
    23K03077
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Topological Hopf Algebras and Their cyclic cohomology
拓扑 Hopf 代数及其循环上同调
  • 批准号:
    RGPIN-2018-04039
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Discovery Grants Program - Individual
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了