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Seminar Cartan (phần 2) - Đại số phân bậc vi phân

Trong phần này, mình (em) sẽ trình bày lại những chi tiết trong bài nói của Cartan, DGA-algebres et DGA-modules. Giả sử $\Lambda$ là một vành giao hoán có đơn vị.Khái niệm đại số phân bậcĐịnh nghĩa. Một $\Lambda$-đại số phân bậc là một $\Lambda$-đại số và các module con $A_k (k \geq 0)$ sao cho $1 \in A_0$. Mỗi phần tử $x$ trong $A_k$ được gọi là thuần nhất bậc $k$, ta kí hiệu $\left |x \right| = k$ và chỉ dùng kí hiệu này khi ám chỉ phần tử thuần nhất.
Đại số $A$ được gọi là phản giao hoán nếu $yx = (-1)^{\left |x \right| \left |y \right|}xy$. Một vi phân trên $A$ là một ánh xạ $d: A \to A$ thỏa mãn$$\begin{cases} d^2 =0 \\ d(A_k) \subset A_{k-1} \\ d(xy)=(dx)y + (-1)^{\left |x \right|}x(dy).\end{cases}$$ Khi $(A,d)$ là một đại số phân bậc vi phân (đspbvp), xét nhóm đồng điều$$H_k(A):= \frac{\mathrm{Ker}(d:A_k \to A_{k-1})}{\mathrm{Im}(d:A_{k+1} \to A_k)}, H_{\star}(A):= \bigoplus H_k(A).$$ $\Lambda$-module $H_{\star}(A)$ thừa hưởng một cấu trúc nhân tự nhiên$$[x][y]:=[xy] \ \forall …

Seminar Cartan (phần 1) - Xây dựng không gian Eilenberg-MacLane

Flat base change and the semicontinuity theorem

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The goal of this blog is to introduct multiple fundamental results on cohomology of fibre of flat morphisms. This is actually an old topic for those who are familiar with stuffs like sheaf cohomology or Cech cohomology. So why do I decide to write this? Relearning. Currently, despite having a lack of time, I still try to relearn Hartshorne, simultaneously, Étale cohomology. It should be noted that one does not need to finish Hartshorne to study étale theory, a basic understanding of scheme is sufficient. The étale morphisms is defined as flat + unramified, which play the role of a local isomorphism in algebraic geometry in comparison with its corresponding version in complex geometry. Therefore, it is understandable to study flat morphisms carefully and how it is related to cohomology. All things represented here are taken either in Hartshorne or QingLiu, but I present them in a much more flexible way knowledge and clarify technical or ambiguous point if necessary. A par…

Étale cohomlogy of the spectrum of a field as Galois cohomology

This blog post is devoted to give a quick introduction to étale cohomology of $\mathrm{Spec}(k)$ where $k$ is a field. There are three things to do here: define étale morphism, sites and sheaves on sites, finally, étale cohomology. Our goal is to prove the following isomorphism$$H^i_{ét}(\mathrm{Spec}(k),F) = H^i(\mathrm{Gal}(k^s/k),\underset{\longrightarrow}{\lim}F(\mathrm{spec}(k')) \ \forall i \geq 0$$ where $k^s$ is the separable closure of $k$, the limit is taken over all finite extensions of $k$ in $k^s$. The left-side is étale cohomology while the right side refers to group cohomology. The readers is assumed to be familiar with the notions of absolute Galois (i.e. Krull topology), sheaf cohomology, derived functor. Sites and sheaves on sites might be new but at this point, the philosophy is essentally as same as in the case of topological spaces. In this short blog, I would like to stress on ideas rather on techniques, such as the following definitions of being smoothSmooth…

A lemma on the vanishing of L^2 cohomology of manifolds with a weighted Poincaré inequality

A paper of Jiuru Zhou deals with several vanishing results for $L^2$-harmonic forms. One of the key lemma is stated as following. Lemma. If an $n$-dimensional complete Riemannian manifold $M$ satisfies a weighted $p$-Poincaré inequality with weight function $\rho \in C^{\infty}(M)$, that is, $$\int_M \rho(x)\left|\alpha \right|^2 dV \leq \int(\left|d\alpha \right|^2 + \left|\delta \alpha \right|^2)dV \ \forall \alpha \in \Omega^p_c(M)$$ provided that the weighted function $\rho(x) > 0$ then$$\mathcal{H}^p(L^2(M)) = 0.$$ This lemma is quite interesting and simple since it requires no curvature conditions. Proof. For each $\omega \in \mathcal{H}^p(L^2(M))$, we have $$d\omega = 0, \delta \omega =0, \int_M \left| \omega \right|^2 dV < \infty.$$ We could choose a cut-off function$$\phi = \begin{cases} 1 & \text{on} \ B(R), \\ 0 & \text{on} \ M - B(2R) \end{cases}$$ such that $\left| \nabla \phi \right|^2 \leq C/R^2$ on $B(2R)/B(R)$ (this can be always done, think of the case …

Workshop 'The Mordell-Weil theorem' and something I have learnt

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I write this post to summarize something I was particularly interested in during the Mordell-Weil theorem workshopin Tuan Chau, Quang Ninh held by the Institute of Mathematics two days ago. The workshop provides the most fundamental concepts of arithmetic of ellitpic curves over number fields and ends up with the proof of the famous Mordell-Weil theorem asserting that the group of rational points on an elliptic curves over number field is finitely generated. The so-called method Galois-descent was presented which allows us compute the Mordell-Weil group in certain 'good' cases. All follows Silverman's text on elliptic curves.
There are totally 10 talks, starting with my talk about algebraic varieties, actually, recently I am quite swamped in other researches on complex geometry so I just chose the most intelligible one, that you could say my talk is superfluous. After my talk, there are two talks are about basic properties of curves such as Riemann-Roch, isogenies, ...Some …

Computational examples with Chern characteristic classes

This blog post is where I shall list my favourite examples about the applications of characteristic class, particularly about Euler class and Chern classes. Rather than stressing theoretical aspects, I would like to provide practical examples based on my own experience, most of them could be found in standard textbook or math-stackexchange but I do not really remember the source, just type everything. I would also update the post whenever I see a new useful exampleSo I may assume you guys are familiar with the definitions of either Euler or Chern classes. Usually, there are two kinds of definitions. The first one is from differential geometry point of view, via Chern-Simon theory and the other one is given by Grothendieck, more ad hoc, of course, which presents a list of axioms. It is encouraged everyone should know both approachs. The second seems to be easier at first but gradually become quite difficult for newbie to figure and does any concrete example. Characteristic classes, ver…