Perfect filtered complexes and duality

Written by Claude Opus 5.5 (Anthropic) on 22 October 2026. Self-checked by the writing AI. Public domain (CC0).

A filtered complex of modules over a ringed topos is filtered perfect when its graded pieces are perfect complexes. This lesson shows that these are exactly the objects of the filtered derived category $DF(A)$ that have a dual in the symmetric monoidal sense, and that the usual consequences of duality hold in the filtered setting:

finiteness conditions (perfectness, tor amplitude, perfect amplitude) are defined on the graded pieces; locally, a filtered perfect complex is represented by a filtered complex whose graded pieces are strictly perfect;
amplitude estimates for $L\otimes^{\mathbf L}M$ and $R\mathcal Hom(L,M)$;
for $L$ filtered perfect, with $L^\vee=R\mathcal Hom(L,A)$: the biduality morphism $L\to L^{\vee\vee}$ and the morphism $L^\vee\otimes^{\mathbf L}M\to R\mathcal Hom(L,M)$ are isomorphisms, the product morphism $R\mathcal Hom(L_1,M_1)\otimes$ $^{\mathbf L}R\mathcal Hom(L_2,M_2)\to$ $R\mathcal Hom(L_1\otimes^{\mathbf L}L_2,M_1\otimes^{\mathbf L}M_2)$ is an isomorphism, and $L_1^\vee\otimes^{\mathbf L}L_2^\vee\cong(L_1\otimes^{\mathbf L}L_2)^\vee$; all with no boundedness condition on the $M$'s;
an object of $DF(A)$ has a dual if and only if it is filtered perfect; the dual is $L^\vee$ with the evaluation and an explicit coevaluation, and passage to the graded pieces, to the underlying complex and derived pullback carry duals to duals.

Prerequisites. Filtered derived categories (Propositions 2.2 (4), 2.3 and 3.3); Filtered tensor products on a ringed topos (Proposition 2.3, Corollary 2.4, Proposition 3.2, Theorem 3.3, Proposition 4.1); Filtered internal Hom and the formulas of Cartan (Proposition 2.2, Theorem 3.1, Definition 3.2, Proposition 3.3). From the AI-integrated Stacks project: strictly perfect and perfect complexes [Stacks, Tag 08FL], [Tag 08FM], [Tag 08FR], [Tag 08G5], [Tag 08G9]; tor amplitude [Tag 08FZ], [Tag 08G1], [Tag 08G2]; internal Hom from a strictly perfect complex [Tag 08JH] and its restriction to a slice [Tag 08JB]; K-injective complexes restricted to a slice [Tag 08FI]; dualizable objects of $D(\mathcal O)$ are perfect [Tag 0FPV]; left duals in monoidal categories [Tag 0FFP], [Tag 0FFQ], [Tag 0FFR], [Tag 0FFS], [Tag 0HB0].

Conventions

As in the lesson on filtered internal Hom: $T$ is a topos, $A$ a commutative ring of $T$, $\otimes$ denotes $\otimes^{\mathbf L}$ on $DF(A)$, $DG(A)$, $D(A)$, and $[M,N]=R\mathcal Hom(M,N)$; $\Phi\colon\operatorname{Hom}(X\otimes M,N)\cong\operatorname{Hom}(X,[M,N])$, $\mathrm{ev}$, $\pi$, $L^\vee=[L,A]$ and $\nu_{L,M}\colon L^\vee\otimes M\to[L,M]$ are as in that lesson. Associativity and unit isomorphisms are suppressed; by the coherence statement of Theorem 3.3 of the lesson on filtered tensor products this is unambiguous, and composites of symmetries are determined by the permutations they induce. A left dual of $X$ in a monoidal category is $(Y,\eta,\varepsilon)$ with $\eta\colon\mathbf 1\to X\otimes Y$, $\varepsilon\colon Y\otimes X\to\mathbf 1$ and $(1_X\otimes\varepsilon)(\eta\otimes1_X)=1_X$, $(\varepsilon\otimes1_Y)(1_Y\otimes\eta)=1_Y$ [Tag 0FFP]. A complex is strictly perfect if it is bounded and its terms are direct summands of finite free modules [Tag 08FL]; an object of $D(A)$ is perfect if it is locally isomorphic to a strictly perfect complex [Tag 08G5].

1. Finiteness conditions

Definition 1.1. Let $K\in D(A)$ and $m\le n$.

$K$ has tor amplitude in $[m,n]$ if $H^i(K\otimes F)=0$ for $i\notin[m,n]$ and every $A$-module $F$ [Tag 08FZ].
$K$ has perfect amplitude in $[m,n]$ if every $U\in T$ has a covering $\{U_i\to U\}$ such that each $K|_{U_i}$ is isomorphic to a strictly perfect complex with terms in degrees $[m,n]$.

An object $L$ of $DF(A)$ is filtered perfect (resp. has tor amplitude, perfect amplitude, cohomology in $[m,n]$) if every $\operatorname{gr}^pL$ is perfect (resp. has that property); an object of $DG(A)$ has the property if every component has it. A complex with perfect amplitude in $[m,n]$ is perfect and has tor amplitude in $[m,n]$, since tor amplitude is local and a strictly perfect complex in degrees $[m,n]$ is a complex of flat modules in those degrees [Tag 08G1].

Lemma 1.2. The filtered perfect objects form a triangulated subcategory of $DF(A)$, stable under direct summands, twists and shifts; the same holds for tor amplitude in a fixed interval except for shifts. If $L$ is filtered perfect, so are $F^pL$ and $F_pL$ (with their induced filtrations), and the underlying object of $L$ is perfect.

Proof. $\operatorname{gr}^p$ is exact, and perfect objects of $D(A)$ form a triangulated subcategory stable under summands [Tag 08G9]; tor amplitude in a fixed interval is preserved by extensions [Tag 08G2]. $F^pL$, $F_pL$ and $L$ itself are iterated extensions, through triangles of $D(A)$, of the finitely many $\operatorname{gr}^qL$. $\square$

Proposition 1.3. Let $L\in DF(A)$ be filtered perfect with filtration in $[a,b]$. Every $U\in T$ has a covering $\{U_i\to U\}$ such that each $L|_{U_i}$ is isomorphic in $DF(A|_{U_i})$ to a filtered complex $S$ with filtration in $[a,b]$ whose graded pieces are strictly perfect.

Proof. By induction on $b-a$; for $a=b$ this is the definition of perfectness. Let $L'=F^{a+1}L$, with filtration in $[a+1,b]$, and $L''=\operatorname{gr}^aL$, placed in filtration degree $a$. The short exact sequence $0\to F^{a+1}L\to L\to L/F^{a+1}L\to0$ of filtered complexes gives a distinguished triangle $L'\to L\to L''\xrightarrow{\delta}L'[1]$ of $DF(A)$. A morphism from an object with filtration concentrated in degree $a$ to an object $X$ of $DF(A)$ is a morphism into $F^aX$ of $D(A)$: on filtered complexes, a filtered map from $Q$ (with $F^aQ=Q$, $F^{a+1}Q=0$) to $X$ is a map $Q\to F^aX$, filtered homotopies correspond, and filtered quasi-isomorphisms into $Q$ can be taken with filtration in $[a,a]$ (Proposition 3.5 (1) of the lesson on filtered derived categories). Here $F^a(L'[1])=L'[1]$. After refining $U$, $L'$ is represented by $S'$ as in the statement (induction) and $L''$ by a strictly perfect complex $Q$. Rotating, $L''[-1]\xrightarrow{-\delta[-1]}L'\to L\to L''$ is distinguished. After refining again, the morphism $-\delta[-1]\colon Q[-1]\to S'$ of $D(A|_U)$ is represented by a morphism of complexes $g\colon Q[-1]\to S'$ [Tag 08FR], which is filtered because $Q[-1]$ has filtration in $[a,a]$. The cone $C(g)$, filtered by $F^pC(g)=F^pS'\oplus F^p(Q[-1])[1]$, has graded pieces $\operatorname{gr}^pS'$ for $p>a$ and $Q$ for $p=a$, all strictly perfect. Both $L|_U$ and $C(g)$ complete $g$ to a distinguished triangle of $DF(A|_U)$, so they are isomorphic [Tag 014K]. $\square$

2. Amplitude estimates

Proposition 2.1. Let $L,M\in DF(A)$ (or $DG(A)$).

If $L$ has tor amplitude in $[m,n]$ and $M$ has cohomology in $[p,q]$, then $L\otimes M$ has cohomology in $[m+p,n+q]$.
If $L$ and $M$ have tor amplitudes in $[m,n]$ and $[p,q]$, then $L\otimes M$ has tor amplitude in $[m+p,n+q]$; the same holds for perfect amplitude, and $L\otimes M$ is filtered perfect when $L$ and $M$ are.

Proof. By (3.1) of the lesson on filtered tensor products, $\operatorname{gr}^r(L\otimes M)\cong\bigoplus_{i+j=r}\operatorname{gr}^iL\otimes\operatorname{gr}^jM$, a finite sum; so it suffices to treat $K\otimes N$ in $D(A)$. (1) If $N$ has cohomology in $[p,q]$, induction on $q-p$ with the triangles $\tau_{\leq q-1}N\to N\to H^q(N)[-q]$ reduces to $N$ a module placed in one degree, which is the definition of tor amplitude. (2) For a module $F$, $(K\otimes N)\otimes F=K\otimes(N\otimes F)$ and $N\otimes F$ has cohomology in $[p,q]$; apply (1). For perfect amplitude, locally $K$ and $N$ are strictly perfect complexes $S,S'$ in degrees $[m,n]$ and $[p,q]$; these are K-flat, so $K\otimes N$ is locally $S\otimes S'$, strictly perfect in degrees $[m+p,n+q]$. $\square$

Proposition 2.2. Let $L,M\in DF(A)$, and let $L$ have perfect amplitude in $[m,n]$. Then $L^\vee$ has perfect amplitude in $[-n,-m]$. If $M$ has cohomology (resp. tor amplitude, perfect amplitude) in $[p,q]$, then $[L,M]$ has cohomology (resp. tor amplitude, perfect amplitude) in $[p-n,q-m]$.

Proof. By Proposition 2.2 (2) of the lesson on filtered internal Hom, $\operatorname{gr}^r[L,M]\cong\prod_i[\operatorname{gr}^iL,\operatorname{gr}^{i+r}M]$, a finite product, and $\operatorname{gr}^rL^\vee=(\operatorname{gr}^{-r}L)^\vee$; so it suffices to treat $D(A)$. Locally $L$ is a strictly perfect complex $S$ in degrees $[m,n]$, and $L^\vee$ is represented by the termwise dual $\mathcal Hom^\bullet(S,A)$, strictly perfect in degrees $[-n,-m]$ [Tag 08JH], [Tag 08JB]. By Theorem 3.3 below, $[L,M]\cong L^\vee\otimes M$, and Proposition 2.1 applies. $\square$

3. Duality morphisms for filtered perfect complexes

Definition 3.1. The biduality morphism $\beta_L\colon L\to L^{\vee\vee}=[L^\vee,A]$ is $\Phi$ of $L\otimes L^\vee\xrightarrow{\sigma}L^\vee\otimes L\xrightarrow{\mathrm{ev}}A$.

Lemma 3.2. Let $\varphi$ be one of the morphisms $\nu_{L,M}$, $\beta_L$, $\pi\colon[L_1,M_1]$ $\otimes[L_2,M_2]\to[L_1\otimes L_2,M_1\otimes M_2]$ of $DF(A)$.

$\operatorname{gr}\varphi$ is the corresponding morphism of $DG(A)$ for the graded objects; in $DG(A)$ it is computed componentwise in $D(A)$: $\nu_{E,M}$ has the components $\nu_{E_i,M_j}\colon E_i^\vee\otimes M_j\to[E_i,M_j]$ (from weight $j-i$ to weight $j-i$), $\beta_E$ the components $\beta_{E_i}$, and $\pi$ the components $\pi$ for $E_{1,i},M_{1,j},E_{2,k},M_{2,l}$.
In $D(A)$, $\varphi$ commutes with restriction to every slice $T/U$.
In $D(A)$, if the $L$'s are strictly perfect complexes $S,S_1,S_2$, then $\varphi$ is represented by the following isomorphisms of complexes, where $S^*=\mathcal Hom^\bullet(S,A)$ and $P,P_1,P_2$ are K-flat complexes representing $M,M_1,M_2$: $\begin{gathered} \nu\colon S^*\otimes P\to\mathcal Hom^\bullet(S,P),\\ \phi\otimes y\mapsto\bigl(s\mapsto(-1)^{|y||s|}\phi(s)y\bigr);\\ \beta\colon S\to\mathcal Hom^\bullet(S^*,A),\\ s\mapsto\bigl(\phi\mapsto(-1)^{|s||\phi|}\phi(s)\bigr);\\ \pi\colon\mathcal Hom^\bullet(S_1,P_1)\\ \otimes\mathcal Hom^\bullet(S_2,P_2)\\ \to\mathcal Hom^\bullet(S_1\otimes S_2,P_1\otimes P_2),\\ f_1\otimes f_2\mapsto\bigl(s_1\otimes s_2\mapsto\\(-1)^{|f_2||s_1|}f_1(s_1)\otimes f_2(s_2)\bigr). \end{gathered}$

Proof. (1) Proposition 3.3 (5) of the lesson on filtered internal Hom, applied to the definitions, which are built from $\Phi$, $\mathrm{ev}$ and $\sigma$; in $DG(A)$, $(E^\vee)_a=[E_{-a},A]$ because $A$ sits in weight $0$, and the componentwise description of $\mathrm{ev}$ there gives the components. (2) On models (a K-flat resolution of the first variable, K-injective resolutions of the second), the maps $c$ and $\mathrm{ev}$ of complexes commute with restriction, and restriction preserves K-flat and K-injective complexes [Tag 08FI]. (3) For $S$ strictly perfect, $\mathcal Hom^\bullet(S,-)$ is the tensor product with the termwise dual and preserves quasi-isomorphisms, and $S$ is K-flat; so in Theorem 3.1 and Proposition 3.3 (2) of the lesson on filtered internal Hom the K-injective resolution of the second variable can be replaced by any representative, and $[S,M]$, $S^\vee$, $[S^*,A]$, $[S_1\otimes S_2,P_1\otimes P_2]$ are represented by the Hom complexes shown, while $S^*\otimes P$ and $\mathcal Hom^\bullet(S_1,P_1)\otimes\mathcal Hom^\bullet(S_2,P_2)\cong$ $S_1^*\otimes P_1\otimes S_2^*\otimes P_2$ are K-flat. On these models $\Phi$ is currying $c$ and $\mathrm{ev}$ is evaluation, and the definitions give: for $\nu$, $c$ of $\phi\otimes y\otimes s\mapsto$ $(-1)^{|y||s|}\phi\otimes s\otimes y\mapsto$ $(-1)^{|y||s|}\phi(s)y$; for $\beta$, $c$ of $s\otimes\phi\mapsto(-1)^{|s||\phi|}\phi(s)$; for $\pi$, $c$ of $f_1\otimes f_2\otimes s_1\otimes s_2\mapsto$ $(-1)^{|f_2||s_1|}f_1(s_1)\otimes f_2(s_2)$. They are isomorphisms termwise, since for a direct summand $E$ of a finite free module, $\mathcal Hom(E,A)\otimes F\to\mathcal Hom(E,F)$, $E\to\mathcal Hom(\mathcal Hom(E,A),A)$ and $\mathcal Hom(E_1,F_1)\otimes\mathcal Hom(E_2,F_2)\to$ $\mathcal Hom(E_1\otimes E_2,F_1\otimes F_2)$ are isomorphisms (they are for $E=A$ and pass to finite sums and summands). $\square$

Theorem 3.3. Let $L,L_1,L_2\in DF(A)$ be filtered perfect and $M,M_1,M_2\in DF(A)$ arbitrary. Then:

$\nu_{L,M}\colon L^\vee\otimes M\to[L,M]$ is an isomorphism; in particular $H^0F^0R\Gamma(T,L^\vee\otimes M)=$ $\operatorname{Hom}_{DF}(L,M)$.
$\beta_L\colon L\to L^{\vee\vee}$ is an isomorphism.
$\pi\colon[L_1,M_1]$ $\otimes[L_2,M_2]\to[L_1\otimes L_2,M_1\otimes M_2]$ is an isomorphism; for $M_1=M_2=A$ it gives $L_1^\vee\otimes L_2^\vee\cong(L_1\otimes L_2)^\vee$.

Proof. A morphism of $DF(A)$ is an isomorphism if its graded pieces are (Proposition 2.2 (4) of the lesson on filtered derived categories); by Lemma 3.2 (1) it suffices that the components in $D(A)$ are isomorphisms, for perfect $E_i$. Being an isomorphism in $D(A)$ is a local property (the cone has zero cohomology sheaves), so by Lemma 3.2 (2) we may assume the perfect objects are strictly perfect complexes, and Lemma 3.2 (3) concludes. The formula for $\operatorname{Hom}$ follows from (1) and Proposition 2.2 (3)–(4) of the lesson on filtered internal Hom. $\square$

4. Dual pairs

Lemma 4.1. Let $E\in DF(A)$ be such that $\nu_{E,E}\colon E^\vee\otimes E\to[E,E]$ is an isomorphism. Let $\iota_E=\Phi(\lambda_E)\colon A\to[E,E]$, where $\lambda_E\colon A\otimes E\to E$ is the unit isomorphism, and put $\begin{gathered}\eta'=\nu_{E,E}^{-1}\circ\iota_E\colon A\to E^\vee\otimes E,\\ \eta=\sigma\circ\eta'\colon A\to E\otimes E^\vee.\end{gathered}$ Then $(E^\vee,\eta,\mathrm{ev}_{E,A})$ is a left dual of $E$. The same holds in $D(A)$ and $DG(A)$.

Proof. Write $\mathrm{ev}=\mathrm{ev}_{E,A}\colon E^\vee\otimes E\to A$. By definition $\nu_{E,E}=\Phi((\mathrm{ev}\otimes1_E)\circ(1\otimes\sigma_{E,E}))$, and $\Phi^{-1}(g\circ h)=\Phi^{-1}(g)\circ(h\otimes1)$ (Proposition 3.3 (1) of the lesson on filtered internal Hom); applying $\Phi^{-1}$ to $\nu_{E,E}\circ\eta'=\iota_E$ gives $\begin{aligned}&(\mathrm{ev}\otimes1_E) \circ(1_{E^\vee}\otimes\sigma_{E,E})\\&\quad \circ(\eta'\otimes1_E) =\lambda_E.\end{aligned} \tag{4.1}$ Consider the maps $E^\vee\otimes E_1\otimes E_2\to E_1$ (two copies of $E$, labelled) $a=(1_{E_1}\otimes\mathrm{ev})\circ(\sigma_{E^\vee,E_1}\otimes1_{E_2})$ and $b=(\mathrm{ev}\otimes1_{E_1})\circ(1_{E^\vee}\otimes\sigma_{E_1,E_2})$, both pairing $E^\vee$ with $E_2$. Naturality of $\sigma$ gives $(1_{E_1}\otimes\mathrm{ev})\circ\sigma_{E^\vee\otimes E_2,E_1}=\mathrm{ev}\otimes1_{E_1}$ (with $\sigma_{A,E_1}$ the unit identification [Tag 0HB0]), so $b=(1_{E_1}\otimes\mathrm{ev})\circ\sigma_{E^\vee\otimes E_2,E_1}\circ(1\otimes\sigma_{E_1,E_2})$; the composite of symmetries here and $\sigma_{E^\vee,E_1}\otimes1_{E_2}$ induce the same permutation, so they are equal by coherence, and $a=b$. Hence $\begin{aligned}&(1_E\otimes\mathrm{ev})\circ(\eta\otimes1_E)\\&\quad =a\circ(\eta'\otimes1_E)\\&\quad =b\circ(\eta'\otimes1_E) =\lambda_E\end{aligned}$ by (4.1); this is the first identity. For the second, put $g=$ $(\mathrm{ev}\otimes1_{E^\vee})\circ(1_{E^\vee}\otimes\eta)\colon$ $E^\vee\to$ $E^\vee=[E,A]$. Then $\Phi^{-1}(g)=$ $\mathrm{ev}\circ(g\otimes1_E)=$ $\mathrm{ev}\circ(\mathrm{ev}\otimes1_{E^\vee}\otimes1_E)\circ(1\otimes\eta\otimes1_E)$. By bifunctoriality of $\otimes$, $\mathrm{ev}\circ(\mathrm{ev}\otimes1\otimes1)=\mathrm{ev}\circ(1\otimes1\otimes\mathrm{ev})$ as maps $E^\vee\otimes E\otimes E^\vee\otimes E\to A$ (both are $\mathrm{ev}\otimes\mathrm{ev}$ followed by $A\otimes A=A$), so $\Phi^{-1}(g)=$ $\mathrm{ev}\circ(1_{E^\vee}\otimes((1_E\otimes\mathrm{ev})\circ(\eta\otimes1_E)))=$ $\mathrm{ev}$, by the first identity. As $\Phi^{-1}(1_{E^\vee})=\mathrm{ev}$ and $\Phi$ is bijective, $g=1_{E^\vee}$. The argument uses only the closed symmetric monoidal structure, which $D(A)$ and $DG(A)$ also have. $\square$

Theorem 4.2. For $E\in DF(A)$ the following are equivalent: (i) $E$ is filtered perfect; (ii) $E$ has a left dual in $(DF(A),\otimes^{\mathbf L},A)$. In this case:

$(E^\vee,\eta,\mathrm{ev})$ of Lemma 4.1 is a left dual of $E$, unique up to unique isomorphism [Tag 0FFR], and $E^\vee$ is filtered perfect.
The underlying object of $E$ is perfect with left dual the underlying object of $E^\vee$; $\operatorname{gr}E$ has left dual $\operatorname{gr}(E^\vee)$, with $\operatorname{gr}\eta$, $\operatorname{gr}\mathrm{ev}$. In $DG(A)$, a family $(E_p)$ of perfect objects has the left dual $(E^\vee)_{-p}=E_p^\vee$, with $\eta$ and $\mathrm{ev}$ the sums of the $\eta_{E_p}$ and $\mathrm{ev}_{E_p}$ of $D(A)$.
If $E'$ is filtered perfect, $E\otimes E'$ is filtered perfect with left dual $E'^\vee\otimes E^\vee$ [Tag 0FFS].
For a morphism of ringed topoi $f\colon(T',A')\to(T,A)$, $Lf^*E$ is filtered perfect with left dual $Lf^*(E^\vee)$, so $Lf^*(E^\vee)\cong(Lf^*E)^\vee$.

Proof. (i)$\Rightarrow$(ii) and the first assertion of (1): Theorem 3.3 (1) and Lemma 4.1. Uniqueness is [Tag 0FFR]. $E^\vee$ is filtered perfect because $\operatorname{gr}^rE^\vee=(\operatorname{gr}^{-r}E)^\vee$ is perfect (proof of Proposition 2.2). (ii)$\Rightarrow$(i): $\operatorname{gr}$ is a functor of symmetric monoidal categories (Theorem 3.3 of the lesson on filtered tensor products), so it carries a left dual $(Y,\eta,\varepsilon)$ of $E$ to a left dual of $\operatorname{gr}E$ [Tag 0FFP]. In $DG(A)$, viewed as families, $\eta\colon A\to\bigoplus_pE_p\otimes Y_{-p}$ and $\varepsilon\colon\bigoplus_pY_{-p}\otimes E_p\to A$ (weight $0$) have components $\eta_p,\varepsilon_p$; the weight-$p$ component of $(1\otimes\varepsilon)(\eta\otimes1)=1_E$ is $(1\otimes\varepsilon_p)(\eta_p\otimes1)=1_{E_p}$, because $\varepsilon$ is zero on $Y_{-q}\otimes E_p$ for $q\ne p$ (its target $A$ has weight $0$), and likewise for the second identity. So each $E_p=\operatorname{gr}^pE$ has a left dual in $D(A)$ and is perfect [Tag 0FPV]. (2) The functors "underlying object" and $\operatorname{gr}$ are symmetric monoidal and carry left duals to left duals [Tag 0FFP]; the componentwise description is the computation just made, read backwards, with Lemma 4.1 in $D(A)$. (3) Proposition 2.1 (2) and [Tag 0FFS]. (4) $Lf^*$ is a functor of symmetric monoidal categories (Proposition 4.1 of the lesson on filtered tensor products); it carries the left dual of $E$ to one of $Lf^*E$, so $Lf^*E$ is filtered perfect by (ii)$\Rightarrow$(i), and its dual is identified with $(Lf^*E)^\vee$ by uniqueness. $\square$

Lemma 4.3. Let $E$ be filtered perfect. For $Z,Z'\in DF(A)$ and $g\colon Z'\otimes E\to Z$, put $\Psi(g)=$ $(g\otimes1_{E^\vee})\circ(1_{Z'}\otimes\eta)\colon$ $Z'\to$ $Z\otimes E^\vee$; $\Psi$ is the bijection of [Tag 0FFQ]. Then $\nu_{E,Z}\circ\sigma_{Z,E^\vee}\circ\Psi(g)=\Phi(g)$. In particular $E\otimes-$ is both left and right adjoint to $E^\vee\otimes-$.

Proof. We compute $\Phi^{-1}$ of the left side. By Proposition 3.3 (1) of the lesson on filtered internal Hom and the definition of $\nu$, it is $(\mathrm{ev}\otimes1_Z)\circ$ $(1_{E^\vee}\otimes\sigma_{Z,E})\circ$ $(\sigma_{Z,E^\vee}\otimes1_E)\circ$ $(g\otimes1_{E^\vee}\otimes1_E)\circ$ $(1_{Z'}\otimes\eta\otimes1_E)$. The two symmetries compose to $\sigma_{Z,E^\vee\otimes E}$ (coherence), and $\sigma_{Z,E^\vee\otimes E}\circ(g\otimes1)=$ $(1\otimes g)\circ\sigma_{Z'\otimes E,E^\vee\otimes E}$; since $(\mathrm{ev}\otimes1_Z)\circ(1\otimes g)=g\circ(\mathrm{ev}\otimes1_{Z'\otimes E})$, we get $g\circ h$ with $h=$ $(\mathrm{ev}\otimes1_{Z'\otimes E})\circ$ $\sigma_{Z'\otimes E,E^\vee\otimes E}\circ$ $(1_{Z'}\otimes\eta\otimes1_E)\colon$ $Z'\otimes$ $E\to$ $Z'\otimes$ $E$. Label the input copy of $E$ as $E_1$ and the copy produced by $\eta$ as $E_a$: $h$ pairs $E^\vee$ with $E_1$ and keeps $Z'\otimes E_a$ in its order. By naturality of $\sigma$ and coherence, $h=$ $1_{Z'}\otimes\bigl((1_{E}\otimes\mathrm{ev})\circ(\eta\otimes1_E)\bigr)=$ $1_{Z'\otimes E}$, by the first identity of a left dual. So $\Phi^{-1}$ of the left side is $g$, as claimed. The adjunctions are those of [Tag 0FFQ] and of Theorem 3.1 of the lesson on filtered internal Hom, combined with $\nu_{E,Z}$ and $\sigma$, which are isomorphisms.

5. Exercises

Exercise 5.1. Let $E=A[n]$, with filtration in $[0,0]$. Show that $E^\vee=A[-n]$, and compute $\eta$ and $\mathrm{ev}$ on the models $A[n]\otimes A[-n]=A$ and $A[-n]\otimes A[n]=A$. Compute the composite $A\xrightarrow{\eta}E\otimes E^\vee\xrightarrow{\sigma}E^\vee\otimes E\xrightarrow{\mathrm{ev}}A$.

Exercise 5.2. Let $E$ be the filtered complex $A\xrightarrow{\mathrm{id}}A$ in degrees $-1,0$, with $F^0E=E$, $F^1E=(0\to A)$, $F^2E=0$. Show that $E$ is filtered perfect and zero in $D(A)$ but not in $DF(A)$, and compute the graded pieces of $E^\vee$.

Exercise 5.3. Let $T$ be the punctual topos and $A=k$ a field, and let $E=\bigoplus_{n\in\mathbf N}k$ in degree $0$. Show that $\nu_{E,E}$ is not surjective, so that $E$ has no dual.

6. Complete solutions

Solution 5.1. $A[n]$ is $A$ in degree $-n$; let $e$ be its generator, $|e|=-n$. The complex $\mathcal Hom^\bullet(A[n],A)$ is $A$ in degree $n$, with generator $e^*$, $e^*(e)=1$, so $E^\vee=A[-n]$ (Lemma 3.2 (3)), and $\mathrm{ev}(e^*\otimes e)=1$. By Lemma 3.2 (3), $\nu_{E,E}(e^*\otimes e)$ is $s\mapsto(-1)^{|e||s|}e^*(s)e$, which sends $e$ to $(-1)^{n^2}e=(-1)^ne$; so $\nu_{E,E}(e^*\otimes e)=(-1)^n\,\mathrm{id}_E$. As $\iota_E(1)=\mathrm{id}_E$, $\eta'(1)=(-1)^ne^*\otimes e$, and $\eta(1)=$ $\sigma((-1)^ne^*\otimes e)=$ $(-1)^n(-1)^{|e^*||e|}e\otimes e^*=$ $e\otimes e^*$. Then $\sigma(\eta(1))=$ $(-1)^{|e||e^*|}e^*\otimes e=$ $(-1)^ne^*\otimes e$, and the composite $\mathrm{ev}\circ\sigma\circ\eta$ is multiplication by $(-1)^n$. The first identity of a left dual reads $e\mapsto e\otimes e^*\otimes e\mapsto e\cdot\mathrm{ev}(e^*\otimes e)=e$.

Solution 5.2. $\operatorname{gr}^0E=A[1]$ and $\operatorname{gr}^1E=A$ are strictly perfect, so $E$ is filtered perfect; the underlying complex is the cone of $\mathrm{id}_A$, acyclic, while $\operatorname{gr}^0E\ne0$, so $E\ne0$ in $DF(A)$. By Proposition 2.2 of the lesson on filtered internal Hom, $\operatorname{gr}^rE^\vee=(\operatorname{gr}^{-r}E)^\vee$: $\operatorname{gr}^0E^\vee=A[-1]$ and $\operatorname{gr}^{-1}E^\vee=A$. So $E^\vee$ has filtration in $[-1,0]$, and its underlying object, the dual of $0$, is $0$.

Solution 5.3. $E^\vee=\prod_{n}k$ and $E^\vee\otimes E=(\prod_nk)\otimes(\bigoplus_mk)$, while $[E,E]=$ $\operatorname{Hom}(\bigoplus_mk,\bigoplus_nk)=$ $\prod_m\bigoplus_nk$. The image of $\nu_{E,E}$ consists of the endomorphisms of finite rank (each element of the tensor product is a finite sum of rank-one maps), so $\mathrm{id}_E$ is not in it. If $E$ had a left dual, it would be perfect (Theorem 4.2) and $\nu_{E,E}$ would be an isomorphism (Theorem 3.3 (1)); so $E$ has no left dual. (Directly: a perfect object over a field has finite-dimensional total cohomology.)

Sources

Perfect complexes and their duality theory on ringed topoi are due to Illusie, in SGA 6; the extension to filtered complexes, the reduction of the filtered statements to the graded pieces, and the amplitude estimates are due to Illusie, in his work on the cotangent complex. The characterization of dualizable objects of the derived category of a ringed site as the perfect complexes is in the AI-integrated Stacks project, Chapter Cohomology on Sites, Section Duals; the formal construction of a dual pair from an invertible comparison morphism of internal Homs goes back to Dold and Puppe and to Lewis, May and Steinberger.