1. $K^0(X)$ and $K_0(X)$. In the previous note, we already define the Grothendieck $K^0$-group as a group of isomorphism classes of vector bundles on $X$ modulo the group generated by elements of the form $[E]-[E']-[E'']$ where $0\to E'\to E\to E''\to 0$ is a short exact sequence of vector bundle on $X$. With the multiplication as tensor product with respect to $\mathscr{O}_X$, $K^0(X)$ forms a ring.
The Grothendieck group $K_0(X)$ is defined similarly, but for coherent sheaves on $X$. When $X$ is smooth, we have the ismorphism between $K^0(X)$ and $K_0(X)$. First, if $E$ is any vector bundle on $X$, then $[E]\mapsto [\text{sheaf of sections of }E]$ gives a well-defined map from $K^0(X)\to K_0(X)$. Conversely, let $F$ be any coherent sheaf on a smooth variety $X$ of dimension $n$, then there exists the following resolution of $F$
$$0\to E_n\to E_{n-1}\to ...\to E_1\to E_0\to F\to 0$$
where $E_i$ are vector bundles on $X$. This yields the map $[F]\mapsto \sum_{i\ge 0}(-1)^i[E_i]$ from $K_0(X)$ to $K^0(X)$. This map is well-defined since if we have another coherent sheaf $F''$ with $f: F\to F''$ is an onto sheaf map with kernel $F'$ (i.e. $[F]=[F']+[F'']\in K^0(X)$) and another resolution of $F''$
$$0\to E''_n\to E''_{n-1}\to ...\to E''_1\to E''_0\to F''\to 0$$
where $E''_i$ are also vector bundles on $X$, then there exists $f_i: E_i\to E''_i$ are onto sheaf maps with kernels $E'_i$, and due to these information, we obtain the following resolution for $F'$
$$0\to E'_n\to E'_{n-1}\to ...\to E'_1\to E'_0\to 0$$
And $E'_i$ are vector bundles. From this, one can see
$$\sum_{i\ge 0}(-1)^i [E_i]=\sum_{i\ge 0}(-1)^i([E'_i]+[E''_i])$$
i.e., the map from $K_0(X)$ to $K^0(X)$ above is well-defined. The two maps are inverse of each other, and hence, we obtain the isomorphism between $K^0(X)$ and $K_0(X)$ when $X$ is a smooth variety. So, for smooth variety we can identify $K_0(X)$ and $K^0(X)$, and we can denote them both as $K(X)$.
2. Proper morphism and map between $K$-groups. Assume that $f: X\to Y$ is proper morphism between two varieties $X, Y$, then this will induce the map between $f_!: K_0(X)\to K_0(Y)$ defined as $[F]\mapsto \sum_{i\ge0}(-1)^i[R^if_*F]$, where $R^if_*F(U):=H^i(f^{-1}(U), F)$, which is called the higher direct image of $F$. Note that this map is well-defined, since it is a basic fact that if $F$ is coherent then $R^if_*$ is also coherent. And if we have the short exact sequence of coherent sheaves on $X$
$$0\to F'\to F\to F''\to 0$$
It then yields the long exact sequence
$$0\to R^0f_*F'\to R^0f_*F\to R^0f_*F''\to R^1f_*F'\to R_1f_*F\to R^1f_*F'' \to ... $$
Hence $\sum_{i\ge 0}(-1)^i [R^if_*F]=\sum_{i\ge 0}(-1)^i([R^if_*F']+[R^if_*F''])$, and $f_!: K_0(X)\to K_0(Y)$ is well-defined.
Example 2.1. Let $Y=\{pt\}$ be a point, then it can be seen that $K_0(Y)\cong K^0(Y)$. And one can see, any vector bundle over $Y$ are just vector spaces over $k$. And any vector space over $k$ are isomorphic iff they have the same dimensions. And if $V, V',V''$ are vector spaces over $k$ such that $\dim V'+\dim V''=\dim V$, then $[V]=[V']+[V'']$. This yields $K^0(Y)\cong \mathbb{Z}$ via the dimension map. In this case, for any smooth projective variety $X$ with $f: X\to \{pt\}$ will induce the push-forward map $f_!: K(X)\to \mathbb{Z}$ defined by $[E]\mapsto \sum_{i\ge 0}h^i(X, E)$, where $E$ is a vector bundle over $X$. And the later sum is just the Euler characteristic of $X$ with respect to vector bundle $E$. So in particular, if $X\to \{pt\}$ is a morphism from a smooth projective variety $X$ then $f_!([E])=\chi(X, E)$, for any vector bundle $E$ on $X$.
3. The theorem of Grothendieck-Riemann-Roch. We also recall that in the previous post, we already if $f: X\to Y$ is a proper morphism, then the proper push-forward $f_*: A(X)\to A(Y)$ is well-defined.
Theorem 3.2 (Grothendieck-Riemann-Roch). Let $f: X\to Y$ be proper morphism between smooth varieties. Then for all $E\in K(X)$, we have
$$f_*(ch(E)Td(X))=ch(f_!E)Td(X)$$
This means the following diagram commutes
Due to Example 2.1, when $Y=\{pt\}$, $K_0(Y)\cong\mathbb{Z}$, and when $X$ is a smooth variety, we can see $f_!(-)\equiv \chi(X, -)$, and by our discussion in the previous note $f_*\equiv \int_X$. We then obtain the theorem of HRR
$$\chi(X,E)=\int_X ch(E)Td(X)$$
Next parts will be devoted for applications of GRR.
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