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# 数学代写|黎曼几何代写Riemannian Geometry代考|MATH213B The Leray–Serre Spectral Sequence

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## 数学代写|黎曼几何代写Riemannian Geometry代考|The Leray–Serre Spectral Sequence

Theorem 4.66 (The Leray-Serre Spectral Sequence) Suppose we are given
(1) a map $\iota: B \longrightarrow A$ which is a differential fibration (see Definition 4.61);
(2) a flat left A-module $E$, with a flat left connection $\nabla_{E}: E \rightarrow \Omega_{A}^{1} \otimes_{A} E$;
(3) the exterior algebra $\Omega_{B}$ has each $\Omega_{B}^{p}$ flat as a right $B$ module.
There is a spectral sequence converging to $\mathrm{H}\left(A, E, \nabla_{E}\right)$ with second page position $(p, q)$ being $\mathrm{H}^{p}\left(B, \hat{\mathrm{H}}^{q}(M), \nabla_{q}\right)$ where $\hat{\mathrm{H}}^{q}(M)$ is the cohomology of
$$\cdots \stackrel{\mathrm{d}}{\longrightarrow} M_{0, q} \stackrel{\mathrm{d}}{\longrightarrow} M_{0, q+1} \stackrel{\mathrm{d}}{\longrightarrow} \cdots$$

with
$$M_{0, q}=\frac{\Omega_{A}^{q} \otimes_{A} E}{\iota \Omega_{B}^{1} \wedge \Omega^{q-1} A \otimes_{A} E}, \quad \mathrm{~d}[x \otimes e]{0, q}=\left[\mathrm{d} x \otimes e+(-1)^{q} x \wedge \nabla{E} e\right]{0, q+1}$$ and $\nabla{q}: \hat{\mathrm{H}}^{q}(M) \rightarrow \Omega_{B}^{1} \otimes_{B} \hat{\mathrm{H}}^{q}(M)$ defined in Lemma 4.65.
Proof The first part of the proof is given in Lemma 4.64. Now we need to calculate the cohomology of
$$\mathrm{d}: \Omega_{B}^{p} \otimes_{B} \hat{\mathrm{H}}^{q}(M) \longrightarrow \Omega_{B}^{p+1} \otimes_{B} \hat{\mathrm{H}}^{q}(M) .$$
If $\xi \in \Omega_{B}^{p}, \eta \in \Omega_{A}^{q}$ and $e \in E$ then $\xi \otimes\langle\eta \otimes e\rangle_{0, q}$ corresponds to $\iota \xi \wedge \eta \otimes e$ and applying $\mathrm{d}$ to the latter gives
$$\iota \mathrm{d} \xi \wedge \eta \otimes e+(-1)^{p} \iota \xi \wedge \mathrm{d} \eta \otimes e+(-1)^{p+q} \iota \xi \wedge \eta \wedge \nabla_{E} e .$$

## 数学代写|黎曼几何代写Riemannian Geometry代考|Correspondences, Bimodules and Positive Maps

In classical geometry a vector field is both a section of the tangent bundle and a bundle map from the cotangent bundle to the trivial bundle. Geometry is full of things which can be considered both as objects and as maps, and sometimes this can be used to generalise the idea of a mapping. Such is the case for the idea of a correspondence between spaces in topology and algebraic geometry. Roughly speaking (omitting much detail and generalisation), a correspondence between $X$ and $Y$ is a subset $\mathcal{C} \subseteq X \times Y$. In terms of our algebraic picture, the projection to the first coordinate $\pi_{1}: \mathcal{C} \rightarrow X$ gives a map of functions $\pi_{1}^{*}: C(X) \rightarrow C(\mathcal{C})$ and by using this we can regard functions on $\mathcal{C}$ as a module over $C(X)$. Projection to the second coordinate $\pi_{2}: \mathcal{C} \rightarrow Y$ allows us to similarly regard functions on $\mathcal{C}$ as a module over $C(Y)$. These actions commute and $C(\mathcal{C})$ becomes a $C(X)-C(Y)$ bimodule. Tensoring with $C(\mathcal{C})$ gives a functor from $C(Y)$-modules to $C(X)$ modules. This point of view includes the idea of viewing a function $f: X \rightarrow Y$ as a graph ${(x, f(x)) \in X \times Y: x \in X}$, in which case the functor is the pull back. In the noncommutative case we can still consider a $B-A$ bimodule $M$ as a kind of generalised morphism between algebras $A, B$. If we have an actual algebra map $\varphi: A \rightarrow B$ then we construct a $B-A$ bimodule $B_{\varphi}$ by $B_{\varphi}=B$ as a left $B$-module, and right $A$-action given by $b . a=b \varphi(a)$. We have already used this for twisted homology in $\S 3.3 .5$ (albeit the twist in ${ }_{\varsigma} A$ was on the other side) and for the inverse image sheaf in Proposition 4.46. Thus bimodules can be constructed from algebra maps. But we are not limited to this case and can think of a general bimodule in the same spirit as a functor between the algebra representation categories. Bimodules can also be given differentiability properties, as we will see.

Another motivation comes from quantum mechanics, where a measurement on a system gives a projection to an eigenspace of the measurement operator and can be expressed as a completely positive map. We shall focus on the KSGNS construction, which deals with completely positive maps and links them to bimodules.

# 黎曼几何代写

## 数学代写|黎曼几何代写Riemannian Geometry代考|The Leray-Serre Spectral Sequence

(1) 一个映射 $\iota: B \longrightarrow A$ 这是一种微分纤维化（见定义 4.61) ;
(2)一个扁平的左A模块 $E$, 有一个扁平的左连接 $\nabla_{E}: E \rightarrow \Omega_{A}^{1} \otimes_{A} E$;
(3) 外代数 $\Omega_{B}$ 有每个 $\Omega_{B}^{p}$ 平权 $B$ 模块。

$$\cdots \stackrel{\mathrm{d}}{\rightarrow} M_{0, q} \stackrel{\mathrm{d}}{\longrightarrow} M_{0, q+1} \stackrel{\mathrm{d}}{\longrightarrow} \cdots$$

$$M_{0, q}=\frac{\Omega_{A}^{q} \otimes_{A} E}{\iota \Omega_{B}^{1} \wedge \Omega^{q-1} A \otimes_{A} E}, \quad \mathrm{~d}[x \otimes e] 0, q=\left[\mathrm{d} x \otimes e+(-1)^{q} x \wedge \nabla E e\right] 0, q+1$$

$$\mathrm{d}: \Omega_{B}^{p} \otimes_{B} \hat{\mathrm{H}}^{q}(M) \longrightarrow \Omega_{B}^{p+1} \otimes_{B} \hat{\mathrm{H}}^{q}(M) .$$

$$\iota \mathrm{d} \xi \wedge \eta \otimes e+(-1)^{p} \iota \xi \wedge \mathrm{d} \eta \otimes e+(-1)^{p+q} \iota \xi \wedge \eta \wedge \nabla_{E} e .$$

## 数学代写|黎曼几何代写Riemannian Geometry代考|Correspondences, Bimodules and Positive Maps

KSGNS 构建，它处理完全正映射并将它们链接到双模块。

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