# Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem.

Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ it relates the RHS of the usual Grothendieck-Riemann-Roch(namely, $f_*(ch(E)\cdot Td(T_f))$) to a certain characteristic class. Let me describe briefly its construction.

First, associate to $E$ the space $P$ of jets of trivializations of $E$ along the fibers of $f$. More precisely, points of $P$ are couples $(s,f)$ where $s\in S$ and $f$ is an $\infty$-jet of a trivialization of $E$ around $s$ in $S_{f(s)}$. Fibers of the projection $p:P\to N$ are principle homogenous spaces over $W_n\ltimes \mathfrak{gl}_k\otimes \mathbb{C}[[x_1,\dots,x_n]]$ where $W_n$ is the Lie algebra of vector fields on $\mathbb C^n$ with formal power series coefficients. Fixing a connection on $E$ we get the Chern-Weil map $$W^*(W_n\ltimes \mathfrak{gl}_k\otimes \mathbb{C}[[x_1,\dots,x_n]])\to \Omega^*_P$$ from the Weyl algebra to the de Rham complex. Replacing the Weyl algebra by the relative Weyl algebra with respect to $\mathfrak{gl}_n\oplus\mathfrak{gl}_k$ we get $$W^*(W_n\ltimes \mathfrak{gl}_k\otimes \mathbb{C}[[x_1,\dots,x_n]],\mathfrak{gl}_n\oplus\mathfrak{gl}_k)\to \Omega^*_{P/GL_n\times GL_k}$$

It induces a morphism from the spectral sequence of Lie algebra cohomology converging to the cohomology of the Weyl algebra to the Leray spectral sequence of $P/GL_n\times GL_k\sim S\to N$. In particular, we get morphisms $$H^{2n}(W_n\ltimes \mathfrak{gl}_k\otimes \mathbb{C}[[x_1,\dots,x_n]],\mathfrak{gl}_n\oplus\mathfrak{gl}_k,S^q(W_n\ltimes \mathfrak{gl}_k\otimes \mathbb{C}[[x_1,\dots,x_n]])^*)\to H^{2q}(N,\mathbb{C})$$

Now taking the natural representation $\rho:W_n\ltimes \mathfrak{gl}_k\otimes \mathbb{C}[[x_1,\dots,x_n]]\to \mathfrak{gl}_k(Diff(\mathbb{C}^n))$ and the induced morphism on cohomology we obtain $$H^{2n}(\mathfrak{gl}_{\infty}(Diff(\mathbb{C}^n)),\rho(\mathfrak{gl}_n\oplus\mathfrak{gl}_k),S^{q}(\mathfrak{gl}_{\infty}(Diff(\mathbb{C}^n)))^*)\to H^{2q}(N)\qquad (*)$$

Using that the only non-zero Hohschild cohomology of $Diff(\mathbb{C}^n)$ is $H^{2n}=\mathbb{C}$ and a theorem of Loday and Quiilen we get that this Lie algebra cohomology is also $\mathbb{C}$. Taking the image of $1$ we obtain cohomology clasees $\varphi_q$.

The theorem claims that $$\sum (-1)^q\frac{\varphi_q}{q!}=f_*(ch(E)\cdot Td(T_f))$$

Question Does the usual Grothendieck-Riemann-Roch theorem follows from this one? Namely, is there a way to show that the LHS is actually $ch(f_*E)$? A posteriori this is of course true since we know that GRR is true, but can the Feigin-Tsygan method be used to prove it?

My vague understanding is that this should come from the following: the space $HH^{2n}(Diff_n)$ is dual to $HH_0(Diff_n)=Diff_n/[Diff_n,Diff_n]$ so it is the space of traces on $Diff_n$. So, the value of $(*)$ on $1$ may be thought of as the trace of identity operator(this sentence does not have any formal meaning for me yet). On the other hand, $ch(f_*E)$ is something like Euler characteristic, or, equivalently the supertrace of identity acting on a resolution of $E$.