Smooth and proper categories#
Let $A \in \mathsf{CAlg}_ A(\mathsf{LinCat})$. (The most important case is $A = \mathsf{Mod}_ \Lambda$). We can then look at modules over $A$, $$ \mathsf{LinCat}_ A = \mathsf{LMod}_ A(\mathsf{LinCat}). $$ This is now a closed symmetric monoidal category with internal Hom given by $$ \mathsf{Fun}_ A^L(M, N) $$ being $A$-linear continuous functors.
These correspond to diagrams $$ \begin{CD} A \otimes M @>{\id \otimes F}>> A \otimes N \br @V{\mathrm{act}_ M}VV @V{\mathrm{act}_ N}VV \br M @>{F}>> N. \end{CD} $$ If $F^R$ is continuous, then we have a map $$ \mathrm{act}_ M \circ (\id_A \otimes F^R) \to F^R \circ \mathrm{act}_ N. $$ If this map is an isomorphism, then $F^R$ will be $A$-linear as well.
Theorem 1. If $A$ is compactly generated, $1_A$ is compact, and every compact object is dualizable, then $F^R$ is automatically $A$-linear.
Definition 2. Let $\mathcal{C}$ be a dualizable object in $\mathsf{LinCat}_ A$ so that we have $$ u_{\mathcal{C}/A} \colon A \to \mathcal{C} \otimes_A \mathcal{C}^\vee, \quad e_{\mathcal{C}/A} \colon \mathcal{C}^\vee \otimes_A \mathcal{C} \to A. $$ We say that $\mathcal{C}$ is smooth if $u_{\mathcal{C}/A}$ admits an $A$-linear right adjoint, and proper if $e_{\mathcal{C}/A}$ admits an $A$-linear left adjoint. We say that $\mathcal{C}$ is 2-dualizable if $\mathcal{C}$ is smooth and proper.
This is really a strong condition. If $A = \mathsf{Mod}_ \Lambda$ and $X / \Lambda$ is smooth proper, then $\mathcal{C} = \mathsf{QCoh}(X)$ is 2-dualizable.
Lemma 3. Suppose $F \colon M \to N$ a functor between dualizable categories admits an $A$-linear right adjoint $F^R$. Then $F^\circ = (F^R)^\vee \colon M^\vee \to N^\vee$ admits an $A$-linear right adjoint $F^\vee$.
In the above setting, if $M = N = \mathcal{C}$ is 2-dualizable, then $$ \tr(\mathcal{C}, F) \in A $$ is dualizable because it is the composition $$ A \xrightarrow{u_ \mathcal{C}} \mathcal{C} \otimes_A \mathcal{C}^\vee \xrightarrow{F \otimes \id} \mathcal{C} \otimes_A \mathcal{C}^\vee \simeq \mathcal{C}^\vee \otimes_A \mathcal{C} \xrightarrow{e_\mathcal{C}} A. $$
Lemma 4. Let $F \colon C \to D$ be an $A$-linear functor between dualizable $A$-modules, with $A$-linear right adjoint $F^R$. This induces natural transformations $$ (F \otimes F^\circ) \circ u_\mathcal{C} \Rightarrow u_\mathcal{D}, \quad e_\mathcal{C} \Rightarrow e_\mathcal{D} \circ (F \otimes F^\circ). $$ Let $\phi_\mathcal{C} \colon \mathcal{C} \to \mathcal{C}$ and $\phi_\mathcal{D} \colon \mathcal{D} \to \mathcal{D}$ be right adjointable $A$-linear functors. Let $$ \eta \colon F \circ \phi_\mathcal{C} \Rightarrow \phi_\mathcal{D} \circ F. $$ Then $\tr(F, \eta)$ is the composite $$ \begin{align} \tr(\mathcal{C}, \phi_\mathcal{C}) &= e_\mathcal{C}((\phi_\mathcal{C} \otimes \id_{\mathcal{C}^\vee}) (u_\mathcal{C})) \to e_\mathcal{D}((F \phi_\mathcal{C} \otimes F^\circ)(u_\mathcal{C})) \br &\to e_\mathcal{D}((\phi_\mathcal{D} \circ F \otimes F^\circ)(u_\mathcal{C})) \to e_\mathcal{D}(\phi_\mathcal{D} \otimes \id_{\mathcal{D}^\vee})(u_\mathcal{D}) = \tr(\mathcal{D}, \phi_\mathcal{D}). \end{align} $$
The trace formula#
Using this, we will prove the following.
Theorem 5. Let $\mathcal{C}$ be 2-dualizable over $A$, and let $\phi_1, \phi_2 \colon \mathcal{C} \to \mathcal{C}$ be right adjointable $A$-linear endomorhisms. Let $$ \eta \colon \phi_1 \circ \phi_2 \simeq \phi_2 \circ \phi_1 $$ be an isomorphism. Then $$ \tr(\tr(\mathcal{C}/A, \phi_1), \tr(\phi_2, \eta^{-1})) = \tr(\tr(\mathcal{C}/A, \phi_2), \tr(\phi_1, \eta)) $$ as elements of $\End 1_A$.
The proof will be a giant diagram.
Example 6. Let $A = \mathsf{Mod}_ k$ and let $C = \mathsf{QCoh}(X)$ for $X$ smooth and proper over $k$. If we take $\phi_1 = f^\ast$ for some $f \colon X \to X$ and $\phi_2 = \id$, this gives some Lefschetz trace formula. If we take $\phi_2 = - \otimes \mathscr{E}$ instead, we get some Atiyah–Bott formula.
For $C$ an $A$-linear category, recall that we have an equivalence $$ C \simeq \mathsf{Fun}_ A^L(A, C); \quad c \mapsto - \otimes c. $$
Definition 7. We say $c$ is $A$-compact if $F_c$ admits an $A$-linear right adjoint. We say that $c$ is $A$-admissible if $F_c$ admits an $A$-linear left adjoint.
Example 8. For $A = \mathsf{Mod}_ \Lambda$, let $C$ be a compactly generated $\Lambda$-linear category. Then the functor $F_c$ looks like $$ F_c(M) = M \otimes c, $$ defined so that $$ \Hom(M, \Hom(c, d)) = \Hom(M \otimes c, d). $$ Then $F_c^R = \Hom(c, -)$ is continuous if and only if $c$ is compact. On the other hand, we need to have $$ \Hom(F_c^L(d), \Lambda) = \Hom(d, c). $$ So for $d$ compact, $F_c^L(d)$ needs to be a perfect $\Lambda$-module. In fact, $F_c^L$ exists if and only if $\Hom(d, c)$ is perfect for every $d \in C^\omega$.
Let $C$ be dualizable in $\mathsf{LinCat}_ \Lambda$. Suppose $c \in C$ is $A$-compact, so that $F_c \colon A \to C$ is right adjointable. Let $\Phi \colon C \to C$ be an $A$-linear map, and suppose we have $$ \eta \colon c \to \Phi(c). $$ This induces $$ \eta \colon F_c \circ \id_A \Rightarrow \Phi \circ F_c. $$ From this we can define $$ \operatorname{ch}(c, \eta) = \tr(F_c, \eta) \colon 1_A \to \tr(C, \Phi). $$ This is called the twisted Chern character. For $\Phi = \id_C$ and $\eta = \id_c$, we get the Chern character $$ \operatorname{ch}(c) \in \tr(C, \id_C). $$ If $A = \mathsf{Mod}_ \Lambda$, this lives in degree $0$.
Example 9. Let $C = \mathsf{QCoh}(X)$ and let $X$ be smooth. We saw that $\mathscr{F} \in \mathsf{QCoh}(X)$ is $A$-compact if and only if $\mathscr{F}$ is perfect. Consider $\Phi = \id_C$. Then we have $$ \tr(C, \id_C) = \bigoplus_{i,j} H^i(X, \Omega_X^j). $$ Then its $0$th piece is $$ H^0 \tr(C, \id_C) = \bigoplus_ i H^i(X, \Omega^i). $$
Theorem 10. Let $A = \mathsf{Mod}_ \Lambda$ and let $C$ be a compactly generated $A$-linear category.
- The Chern character factors as $$ K_0(C^\omega) \to H^0 \tr(C, \id_C). $$
- Let $F \colon C \to D$ be right adjointable. Then the diagram $$ \begin{CD} K_0(C^\omega) @>{\mathrm{ch}}>> H^0 \tr(C, \id_C) \br @VVV @VVV \br K_0(D^\omega) @>{\mathrm{ch}}>> H^0 \tr(D, \id_D) \end{CD} $$ commutes.
Recovering classical Grothendieck–Riemann–Roch requires a bit more work.
Localizations#
Definition 11. Let $A \in \mathsf{Alg}(\mathsf{LinCat})$. A sequence of $A$-modules $$ M \xrightarrow{F} C \xrightarrow{G} N $$ is a localization sequence if
- $F^R$ and $G^R$ exists as $A$-linear functors,
- the unit can counit maps $$ \id_M \xrightarrow{\simeq} F^R \circ F, \quad G \circ G^R \xrightarrow{\simeq} \id_N $$ are $A$-linear,
- $G \circ F = 0$,
- for every $c \in C$, the sequence $$ F \circ F^R(c) \to c \to G^R \circ G(c) $$ is a cofiber sequence.
Definition 12. Let $$ M \xrightarrow{F} C \xrightarrow{G} N $$ be a localization sequence as before. If $G^R$ is also right adjointable, then we say that $(M, G^R(N))$ form a semi-orthogonal decomposition.
Proposition 13. Suppose $A \in \mathsf{CAlg}(\mathsf{LinCat})$ and assume $M, N, C$ are all dualizable. Then $$ (F \otimes F^\circ) u_M \to u_C \to (G^R \otimes (G^\circ)^R) u_N $$ is a coiber sequence in $C \otimes_A C^\vee$.