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数学代写|代数拓扑代考Algebraic Topology代考|MA753 Statement of HKR Theorem

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数学代写|代数拓扑代考Algebraic Topology代考|Statement of HKR Theorem

If $X$ is a space, we denote $\widetilde{H}_*(X)$ its reduced homology coalgebra. Recall that a space is formal if its cochain algebra is quasi-isomorphic to its cohomology as a CDGA. This includes all spheres, suspensions, Lie group or Kähler varieties.

Theorem 1.6.1 Assume $X$ is a formal space of finite type in each degree. And let $(\operatorname{Sym}(V), d) \stackrel{\cong}{\longrightarrow} A$ be a cofibrant resolution ${ }^{33}$ of $A$. There are natural $($ in $A, M)$ equivalences
respectively in CDGA and in $\mathbf{C H}_X(A)$-Mod. Further, if $f: X \rightarrow Y$ is $a$ formal $\operatorname{map}^{34}$ we have a commutative diagrams (respectively in CDGA and $\left.\mathbf{C H}_X(A)-\operatorname{Mod}\right):$

数学代写|代数拓扑代考Algebraic Topology代考|HKR Isomorphism and Hodge Decomposition

We now relate the HKR isomorphisms from Sect. 1.6.1 with the Hodge filtrations on the various (co)chains functors.

Recall from Example 1.4.6 that $\operatorname{Sym}(V \oplus V[d])$ and $\operatorname{Sym}\left(V \oplus V^{\vee}[d]\right)$ are endowed with canonical dg-multiplicative- $\gamma$-ring with zero multiplication structure for which $V$ is of pure weight 0 and $V[d]$ or $V^{\vee}[d]$ are of pure weight 1 . More generally if $U$ and $W$ are graded modules and $d$ is a differential on $\operatorname{Sym}(U \oplus W)$

such that $d(W) \subset W \otimes \operatorname{Sym}(U)$, then $(\operatorname{Sym}(U \oplus W), d)$ has a dg-multiplicative$\gamma$-ring with zero multiplication structure for which $U$ is on weight 0 and $W$ in weight 1.

Corollary 1.6.8 Assume $X$ is a formal space of finite type in each degree. And let $(\operatorname{Sym}(V), d) \stackrel{\cong}{\longrightarrow} A$ be a cofibrant resolution of $A$.
The HKR quasi-isomorphisms yields natural (in $A$ and $M$ ) equivalences
1.
$H K R: \mathbf{C H}{S^d \times X}(A) \stackrel{\cong}{\longrightarrow} \operatorname{Sym}\left(\left(V \otimes H(X)\right) \oplus\left(V \otimes H_(X)\right)[d]\right), \quad(1.100)$
$H K R: \mathbf{C H}{S^d \wedge X}(A) \stackrel{\cong}{\longrightarrow} \operatorname{Sym}\left(V \oplus\left(V \otimes \widetilde{H}(X)\right)[d]\right)$ of $d g$-multiplicative $\gamma$-ring with trivial multiplication, 2. \begin{aligned} &\mathbf{C H}{S^d \times X}(A, M) \ &\stackrel{\cong}{\stackrel{\cong}{\longrightarrow}} M \underset{\operatorname{Sym}(V)}{\otimes} \operatorname{Sym}\left(\left(V \otimes H(X)\right) \oplus\left(V \otimes H_*(X)\right)[d]\right), \ & \end{aligned}
of $d g-\gamma$-ring with trivial multiplication in $\mathbf{C H}{S^d \times X}(A)-\mathrm{Mod}$ and $\mathrm{CH}{S^d \wedge X}(A)$-Mod respectively

1. as well as
\begin{aligned} &\mathbf{C H}^{S^d \times X}(A, M) \cong \underset{\operatorname{Sym}(V)}{M} \otimes \operatorname{Sym}\left(\left(V^{\vee} \otimes \widetilde{H}^(X) \oplus V\right) \oplus\left(V^{\vee} \otimes H^(X)\right)[-d]\right),(1.104) \ &\mathbf{C H}^{S^d \wedge X}(A, M) \cong M_{\operatorname{Sym}(V)}^{\otimes} \operatorname{Sym}\left(V \oplus\left(V^{\vee} \otimes \widetilde{H}^*(X)\right)[-d]\right) \ & \end{aligned}

数学代写|代数石扑代考Algebraic Topology代考|HKR Isomorphism and Hodge Decomposition

$H K R: \mathbf{C H} S^d \wedge X(A) \stackrel{\cong}{\longrightarrow} \operatorname{Sym}(V \oplus(V \otimes \widetilde{H}(X))[d])$ 的 $d g$-乘刧 $\gamma$-环与平凡的乘法，2。

1. $$\left.\left.\mathbf{C H}^{S^{d^d} \times}(A, M) \cong \underset{\operatorname{Sym}(V)}{M} \otimes \operatorname{Sym}\left(\left(V^{\vee} \otimes \widetilde{H}^{(} X\right) \oplus V\right) \oplus\left(V^{\vee} \otimes H^{(} X\right)\right)[-d]\right),(1.104) \quad \mathbf{C H}^{S^d \wedge X}(A, M) \cong M_{\mathrm{Sym}(V)}^{\otimes} \operatorname{Sym}\left(V \oplus \left(V ^ { \vee } \otimes \widetilde { H } ^ { * } \left($$

MATLAB代写

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