We do the minimal amount of work to define the second adams operator. Let $X$ be a scheme. Recall that $\textit{Vect}(X)$ denotes the category of finite locally free $\mathcal{O}_ X$-modules. Moreover, recall that we have constructed a zeroth $K$-group $K_0(\textit{Vect}(X))$ associated to this category in Derived Categories of Schemes, Section 36.38. Finally, $K_0(\textit{Vect}(X))$ is a ring, see Derived Categories of Schemes, Remark 36.38.6.
Lemma 42.56.1. Let $X$ be a scheme. There is a ring map which sends $[\mathcal{L}]$ to $[\mathcal{L}^{\otimes 2}]$ when $\mathcal{L}$ is invertible and is compatible with pullbacks.
\[ \psi ^2 : K_0(\textit{Vect}(X)) \longrightarrow K_0(\textit{Vect}(X)) \]
Proof. Let $X$ be a scheme. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. We will consider the element
of $K_0(\textit{Vect}(X))$.
Let $X$ be a scheme and consider a short exact sequence
of finite locally free $\mathcal{O}_ X$-modules. Let us think of this as a filtration on $\mathcal{F}$ with $2$ steps. The induced filtration on $\text{Sym}^2(\mathcal{F})$ has $3$ steps with graded pieces $\text{Sym}^2(\mathcal{E})$, $\mathcal{E} \otimes \mathcal{F}$, and $\text{Sym}^2(\mathcal{G})$. Hence
In exactly the same manner one shows that
Thus we see that $\psi ^2(\mathcal{F}) = \psi ^2(\mathcal{E}) + \psi ^2(\mathcal{G})$. We conclude that we obtain a well defined additive map $\psi ^2 : K_0(\textit{Vect}(X)) \to K_0(\textit{Vect}(X))$.
It is clear that this map commutes with pullbacks.
We still have to show that $\psi ^2$ is a ring map. Let $X$ be a scheme and let $\mathcal{E}$ and $\mathcal{F}$ be finite locally free $\mathcal{O}_ X$-modules. Observe that there is a short exact sequence
where the first map sends $(e \wedge e') \otimes (f \wedge f')$ to $(e \otimes f)(e' \otimes f') - (e' \otimes f)(e \otimes f')$ and the second map sends $(e \otimes f) (e' \otimes f')$ to $ee' \otimes ff'$. Similarly, there is a short exact sequence
where the first map sends $e e' \otimes f \wedge f'$ to $(e \otimes f) \wedge (e' \otimes f') + (e' \otimes f) \wedge (e \otimes f')$ and the second map sends $(e \otimes f) \wedge (e' \otimes f')$ to $(e \wedge e') \otimes (f f')$. As above this proves the map $\psi ^2$ is multiplicative. Since it is clear that $\psi ^2(1) = 1$ this concludes the proof. $\square$
Remark 42.56.2. Let $X$ be a scheme such that $2$ is invertible on $X$. Then the Adams operator $\psi ^2$ can be defined on the $K$-group $K_0(X) = K_0(D_{perf}(\mathcal{O}_ X))$ (Derived Categories of Schemes, Definition 36.38.2) in a straightforward manner. Namely, given a perfect complex $L$ on $X$ we get an action of the group $\{ \pm 1\} $ on $L \otimes ^\mathbf {L} L$ by switching the factors. Then we can set where $(-)^+$ denotes taking invariants and $(-)^-$ denotes taking anti-invariants (suitably defined). Using exactness of taking invariants and anti-invariants one can argue similarly to the proof of Lemma 42.56.1 to show that this is well defined. When $2$ is not invertible on $X$ the situation is a good deal more complicated and another approach has to be used.
\[ \psi ^2(L) = [(L \otimes ^\mathbf {L} L)^+] - [(L \otimes ^\mathbf {L} L)^-] \]
Lemma 42.56.3. Let $X$ be a scheme. There is a ring map $\psi ^{-1} : K_0(\textit{Vect}(X)) \to K_0(\textit{Vect}(X))$ which sends $[\mathcal{E}]$ to $[\mathcal{E}^\vee ]$ when $\mathcal{E}$ is finite locally free and is compatible with pullbacks.
Proof. The only thing to check is that taking duals is compatible with short exact sequences and with pullbacks. This is clear. $\square$
Remark 42.56.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. The Chern class map defines a canonical map by sending a generator $[\mathcal{E}]$ on the left hand side to $c(\mathcal{E}) = 1 + c_1(\mathcal{E}) + c_2(\mathcal{E}) + \ldots $ and extending multiplicatively. Thus $-[\mathcal{E}]$ is sent to the formal inverse $c(\mathcal{E})^{-1}$ which is why we have the infinite product on the right hand side. This is well defined by Lemma 42.40.3.
\[ c : K_0(\textit{Vect}(X)) \longrightarrow \prod \nolimits _{i \geq 0} A^ i(X) \]
Remark 42.56.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. The Chern character map defines a canonical ring map by sending a generator $[\mathcal{E}]$ on the left hand side to $ch(\mathcal{E})$ and extending additively. This is well defined by Lemma 42.45.2 and a ring homomorphism by Lemma 42.45.3.
\[ ch : K_0(\textit{Vect}(X)) \longrightarrow \prod \nolimits _{i \geq 0} A^ i(X) \otimes \mathbf{Q} \]
Lemma 42.56.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. If $\psi ^2$ is as in Lemma 42.56.1 and $c$ and $ch$ are as in Remarks 42.56.4 and 42.56.5 then we have $c_ i(\psi ^2(\alpha )) = 2^ i c_ i(\alpha )$ and $ch_ i(\psi ^2(\alpha )) = 2^ i ch_ i(\alpha )$ for all $\alpha \in K_0(\textit{Vect}(X))$.
Proof. Observe that the map $\prod _{i \geq 0} A^ i(X) \to \prod _{i \geq 0} A^ i(X)$ multiplying by $2^ i$ on $A^ i(X)$ is a ring map. Hence, since $\psi ^2$ is also a ring map, it suffices to prove the formulas for additive generators of $K_0(\textit{Vect}(X))$. Thus we may assume $\alpha = [\mathcal{E}]$ for some finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$. By construction of the Chern classes of $\mathcal{E}$ we immediately reduce to the case where $\mathcal{E}$ has constant rank $r$, see Remark 42.38.10. In this case, we can choose a projective smooth morphism $p : P \to X$ such that restriction $A^*(X) \to A^*(P)$ is injective and such that $p^*\mathcal{E}$ has a finite filtration whose graded parts are invertible $\mathcal{O}_ P$-modules $\mathcal{L}_ j$, see Lemma 42.43.1. Then $[p^*\mathcal{E}] = \sum [\mathcal{L}_ j]$ and hence $\psi ^2([p^\mathcal {E}]) = \sum [\mathcal{L}_ j^{\otimes 2}]$ by definition of $\psi ^2$. Setting $x_ j = c_1(\mathcal{L}_ j)$ we have
in $\prod A^ i(P)$ and we have
in $\prod A^ i(P)$. From these formulas the desired result follows. $\square$
Remark 42.56.7. Let $X$ be a locally Noetherian scheme. Let $Z \subset X$ be a closed subscheme. Consider the strictly full, saturated, triangulated subcategory consisting of perfect complexes of $\mathcal{O}_ X$-modules whose cohomology sheaves are settheoretically supported on $Z$. Denote $\textit{Coh}_ Z(X) \subset \textit{Coh}(X)$ the Serre subcategory of coherent $\mathcal{O}_ X$-modules whose set theoretic support is contained in $Z$. Observe that given $E \in D_{Z, perf}(\mathcal{O}_ X)$ Zariski locally on $X$ only a finite number of the cohomology sheaves $H^ i(E)$ are nonzero (and they are all settheoretically supported on $Z$). Hence we can define (equality by Lemma 42.23.6) by the rule This works because given a distinguished triangle in $D_{Z, perf}(\mathcal{O}_ X)$ we have a long exact sequence of cohomology sheaves.
\[ D_{Z, perf}(\mathcal{O}_ X) \subset D(\mathcal{O}_ X) \]
\[ K_0(D_{Z, perf}(\mathcal{O}_ X)) \longrightarrow K_0(\textit{Coh}_ Z(X)) = K'_0(Z) \]
\[ E \longmapsto [\bigoplus \nolimits _{i \in \mathbf{Z}} H^{2i}(E)] - [\bigoplus \nolimits _{i \in \mathbf{Z}} H^{2i + 1}(E)] \]
Remark 42.56.8. Let $X$, $Z$, $D_{Z, perf}(\mathcal{O}_ X)$ be as in Remark 42.56.7. Assume $X$ is regular. Then there is a canonical map defined as follows. For any coherent $\mathcal{O}_ Z$-module $\mathcal{F}$ denote $\mathcal{F}[0]$ the object of $D(\mathcal{O}_ X)$ which has $\mathcal{F}$ in degree $0$ and is zero in other degrees. Then $\mathcal{F}[0]$ is a perfect complex on $X$ by Derived Categories of Schemes, Lemma 36.11.8. Hence $\mathcal{F}[0]$ is an object of $D_{Z, perf}(\mathcal{O}_ X)$. On the other hand, given a short exact sequence $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ of coherent $\mathcal{O}_ Z$-modules we obtain a distinguished triangle $\mathcal{F}[0] \to \mathcal{F}'[0] \to \mathcal{F}''[0] \to \mathcal{F}[1]$, see Derived Categories, Section 13.12. This shows that we obtain a map $K_0(\textit{Coh}(Z)) \to K_0(D_{Z, perf}(\mathcal{O}_ X))$ by sending $[\mathcal{F}]$ to $[\mathcal{F}[0]]$ with apologies for the horrendous notation.
\[ K_0(\textit{Coh}(Z)) \longrightarrow K_0(D_{Z, perf}(\mathcal{O}_ X)) \]
Lemma 42.56.9. Let $X$ be a Noetherian regular scheme. Let $Z \subset X$ be a closed subscheme. The maps constructed in Remarks 42.56.7 and 42.56.8 are mutually inverse and we get $K'_0(Z) = K_0(D_{Z, perf}(\mathcal{O}_ X))$.
Proof. Clearly the composition
is the identity map. Thus it suffices to show the first arrow is surjective. Let $E$ be an object of $D_{Z, perf}(\mathcal{O}_ X)$. Recall that $D_{perf}(\mathcal{O}_ X) = D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ by Derived Categories of Schemes, Lemma 36.11.8. Hence the cohomologies $H^ i(E)$ are coherent, can be viewed as objects of $D_{Z, perf}(\mathcal{O}_ X)$, and only a finite number are nonzero. Using the distinguished triangles of canonical truncations the reader sees that
in $K_0(D_{Z, perf}(\mathcal{O}_ X))$. Then it suffices to show that $[\mathcal{F}[0]]$ is in the image of the map for any coherent $\mathcal{O}_ X$-module set theoretically supported on $Z$. Since we can find a finite filtration on $\mathcal{F}$ whose subquotients are $\mathcal{O}_ Z$-modules, the proof is complete. $\square$
Remark 42.56.10. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme and let $D_{Z, perf}(\mathcal{O}_ X)$ be as in Remark 42.56.7. If $X$ is quasi-compact (or more generally the irreducible components of $X$ are quasi-compact), then the localized Chern classes define a canonical map by sending a generator $[E]$ on the left hand side to and extending multiplicatively (with product on the right hand side as in Remark 42.34.7). The quasi-compactness condition on $X$ guarantees that the localized chern classes are defined (Situation 42.50.1 and Definition 42.50.3) and that these localized chern classes convert distinguished triangles into the corresponding products in the bivariant chow rings (Lemma 42.52.4).
\[ c(Z \to X, -) : K_0(D_{Z, perf}(\mathcal{O}_ X)) \longrightarrow A^0(X) \times \prod \nolimits _{i \geq 1} A^ i(Z \to X) \]
\[ c(Z \to X, E) = 1 + c_1(Z \to X, E) + c_2(Z \to X, E) + \ldots \]
Remark 42.56.11. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme and let $D_{Z, perf}(\mathcal{O}_ X)$ be as in Remark 42.56.7. If the irreducible components of $X$ are quasi-compact, then the localized Chern character defines a canonical additive and multiplicative map by sending a generator $[E]$ on the left hand side to $ch(Z \to X, E)$ and extending additively. Namely, the condition on the irreducible components of $X$ guarantees that the localized chern character is defined (Situation 42.50.1 and Definition 42.50.3) and that these localized chern characters convert distinguished triangles into the corresponding sums in the bivariant chow rings (Lemma 42.52.5). The multiplication on $K_0(D_{Z, perf}(X))$ is defined using derived tensor product (Derived Categories of Schemes, Remark 36.38.9) hence $ch(Z \to X, \alpha \beta ) = ch(Z \to X, \alpha ) ch(Z \to X, \beta )$ by Lemma 42.52.6. If $X$ is quasi-compact, then the map $ch(Z \to X, -)$ has image contained in $A^*(Z \to X) \otimes \mathbf{Q}$; we omit the details.
\[ ch(Z \to X, -) : K_0(D_{Z, perf}(\mathcal{O}_ X)) \longrightarrow \prod \nolimits _{i \geq 0} A^ i(Z \to X) \otimes \mathbf{Q} \]
Remark 42.56.12. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ and assume $X$ is quasi-compact (or more generally the irreducible components of $X$ are quasi-compact). With $Z = X$ and notation as in Remarks 42.56.10 and 42.56.11 we have $D_{Z, perf}(\mathcal{O}_ X) = D_{perf}(\mathcal{O}_ X)$ and we see that see Derived Categories of Schemes, Definition 36.38.2. Hence we get as a special case of Remarks 42.56.10 and 42.56.11. Of course, instead we could have just directly used Definition 42.46.3 and Lemmas 42.46.7 and 42.46.11 to construct these maps (as this immediately seen to produce the same classes). Recall that there is a canonical map $K_0(\textit{Vect}(X)) \to K_0(X)$ which sends a finite locally free module to itself viewed as a perfect complex (placed in degree $0$), see Derived Categories of Schemes, Section 36.38. Then the diagram commutes where the south-east arrow is the one constructed in Remark 42.56.4. Similarly, the diagram commutes where the south-east arrow is the one constructed in Remark 42.56.5.
\[ K_0(D_{Z, perf}(\mathcal{O}_ X)) = K_0(D_{perf}(\mathcal{O}_ X)) = K_0(X) \]
\[ c : K_0(X) \to \prod A^ i(X) \quad \text{and}\quad ch : K_0(X) \to \prod A^ i(X) \otimes \mathbf{Q} \]
\[ \xymatrix{ K_0((\textit{Vect}(X)) \ar[rd]_ c \ar[rr] & & K_0(D_{perf}(\mathcal{O}_ X)) = K_0(X) \ar[ld]^ c \\ & \prod A^ i(X) } \]
\[ \xymatrix{ K_0((\textit{Vect}(X)) \ar[rd]_{ch} \ar[rr] & & K_0(D_{perf}(\mathcal{O}_ X)) = K_0(X) \ar[ld]^{ch} \\ & \prod A^ i(X) \otimes \mathbf{Q} } \]
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