\documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage{amsmath,amssymb} \usepackage{longtable,booktabs,array} % Compatibility with Pandoc's captionless tables on older longtable versions. \ifcsname c@none\endcsname\else\newcounter{none}\fi \usepackage{graphicx} \usepackage[margin=25mm]{geometry} \usepackage{hyperref} \hypersetup{colorlinks=true,linkcolor=blue,urlcolor=blue} \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \providecommand{\pandocbounded}[1]{#1} \setlength{\emergencystretch}{3em} \setcounter{secnumdepth}{-1} \begin{document} \section{The right adjoint of derived pushforward}\label{the-right-adjoint-of-derived-pushforward} \emph{Written by GPT-6.1 Sol (OpenAI), in Codex at Ultra, October 2026. Self-checked by the writing AI. Original text: public domain (CC0). The compact-generation, finite-factor, duality and Nagata compactification arguments follow constructions of the Stacks Project, cited in Section 8 and Appendix N. See the \href{licensing.html}{course notice}.} The trace in Serre duality sends top cohomology of a canonical sheaf to the ground field. Its derived formulation is a counit: a map from the pushforward of a distinguished complex to the original complex on the base. This viewpoint puts finite maps, closed immersions and projective space into one construction. It also explains why a singular scheme may need a complex with several nonzero cohomology sheaves. We use cohomological grading: \(H^i(C[r])=H^{i+r}(C)\). Write \(D_{\mathrm{QCoh}}(X)\) for the full subcategory of \(D(\mathcal O_X)\) whose cohomology sheaves are quasi-coherent. A perfect complex is locally represented by a bounded complex of finite locally free sheaves. All schemes in Sections 1--2 are quasi-compact and quasi-separated. No separation assumption is silently added to that convention. The bounded-below injective machinery is taught in \href{../derived-categories-and-sheaf-operations/injective-modules-and-bounded-below-derived-functors.html}{Injective modules and bounded-below derived functors, Theorem 2.3, Lemma 1.1 and Theorem 4.1}. The exact earlier programme providers for the unbounded operations are identified in Section 1 and at their uses below. Our two preceding lessons supply the classical projective duality proofs. Appendix N writes the Noetherian Nagata construction. Section 8 says which of its prerequisites are not proved in these lessons. \subsection{1. Representing maps out of pushforward}\label{representing-maps-out-of-pushforward} For a morphism \(f:X\to Y\), its derived pushforward has a right adjoint \[ a_f:D_{\mathrm{QCoh}}(Y)\longrightarrow D_{\mathrm{QCoh}}(X), \qquad \operatorname{Hom}_X(L,a_fK) \simeq\operatorname{Hom}_Y(Rf_*L,K). \tag{1} \] The bijection is natural in both variables. We prove existence before using it: the scheme arguments below construct a perfect compact generator, the exact earlier programme theorem supplies coproduct preservation by pushforward, and the categorical proof constructs the representing object. \subsubsection{The scheme input, including quasi-separated schemes}\label{the-scheme-input-including-quasi-separated-schemes} We first prove the input needed for representability. The following earlier programme results are used at their full stated generality: \href{../derived-categories-and-sheaf-operations/quasi-coherent-sheaves-and-concentrated-maps.html}{Quasi-coherent sheaves and concentrated scheme maps}, Theorem 4.2 (unbounded affine comparison, qcqs pushforward, coproducts and finite cohomological amplitude), Theorem 5.1 (unbounded quasi-coherent projection formula), and Lemma 1.4 (finite covers and quasi-separated intersections); \href{../derived-categories-and-sheaf-operations/perfect-complexes-and-duals.html}{Perfect complexes and duals}, Propositions 2.1--2.2, Theorem 3.2, Corollary 3.3 and Theorem 4.1 (finite approximation, the Tor-amplitude criterion, permanence and perfect duality); and \href{../derived-categories-and-sheaf-operations/sections-with-support-and-the-localization-triangle.html}{Sections with support and the localization triangle}, Theorem 2.1. In particular, the scheme input is not replaced by a separated-scheme assertion. \textbf{Lemma 1.A (compactness and a finite Hom window).} Every perfect object \(P\) on a qcqs scheme \(X\) is compact in \(D_{\mathrm{QCoh}}(X)\). If \(P^\vee\) has Tor amplitude \([a,b]\) and quasi-coherent cohomological dimension of \(X\) is at most \(d\), then \[ \operatorname{Hom}(P,E)=0 \quad\text{if}\quad H^q(E)=0\quad(-b-d\leq q\leq-a). \tag{1.A} \] In particular \(\operatorname{Ext}^q(P,F)=0\) for every quasi-coherent sheaf \(F\) and \(q>b+d\), uniformly in \(F\). \textbf{Proof.} Perfect duality identifies the Hom group with \(H^0R\Gamma(X,P^\vee\otimes^{\mathbf L}E)\). Tensor preserves sums, and the earlier unbounded pushforward theorem says that derived sections preserve sums on quasi-coherent complexes. Exactness of sums then proves compactness. The same theorem says that degree zero of derived sections depends only on degrees \([-d,0]\) of its input. Tensoring by an object of amplitude \([a,b]\) makes those degrees depend only on degrees \([-b-d,-a]\) of \(E\). To see this without an unbounded spectral-sequence convergence assumption, truncate \(E\) above and below that interval: the omitted lower part tensors into degrees at most \(-d-1\), and the upper part into degrees at least 1. The finite-window pushforward bound kills both contributions to degree zero. The remaining finite truncation has zero cohomology under the hypothesis. For \(E=F[q]\), its only cohomology degree is \(-q\); if \(q>b+d\), it lies below that window. \(\square\) \textbf{Lemma 1.B (affine denominators in derived maps).} Let \(V=\operatorname{Spec}A\), \(W=\bigcup_{i=1}^rD(f_i)\), and let \(P\) be perfect on \(V\). Every derived map \(P|_W\to E|_W\), with \(E\in D_{\mathrm{QCoh}}(V)\), is represented by \[ P\xleftarrow{\beta}P\otimes^{\mathbf L}I\xrightarrow{\gamma}E, \tag{1.B} \] where \(I\) is perfect, has a map \(I\to\mathcal O_V\), and this map, hence \(\beta\), is an isomorphism on \(W\). \textbf{Proof.} Let \(K_e\) be the Koszul complex on \(f_1^e,\ldots,f_r^e\), in degrees \(-r,\ldots,0\), with degree-zero term \(A\), and put \(I_e=\operatorname{Cone}(A\to K_e)[-1]\). Thus \(I_e\to A\to K_e\) is a triangle; \(K_e|_W=0\) because one of its entries is a unit locally. The transition \(K_{e+1}\to K_e\) is the identity in degree zero and multiplies the exterior basis indexed by \(S\) by \(\prod_{i\in S}f_i\). It induces \(I_{e+1}\to I_e\). For every module complex \(M\), there is a chain isomorphism \[ \underset e{\operatorname{colim}}\operatorname{Hom}^{\bullet}_A(I_e,M) \simeq\operatorname{Tot}\check C^{\bullet}(D(f_1),\ldots,D(f_r);\widetilde M). \tag{1.C} \] Indeed, cancel the identity pair coming from \(A\to K_e\) in the cone. The remaining dual terms are indexed by nonempty subsets \(S\), in Čech degree \(|S|-1\). Their transition maps multiply by \(\prod_{i\in S}f_i\), so their direct limits are \(M_{\prod_{i\in S}f_i}\). The Koszul incidence differential becomes the alternating restriction differential; choose the usual increasing exterior order on both sides. There are only \(r\) columns, so this identification holds also for unbounded \(M\). Affine intersections and the earlier affine comparison identify this finite Čech total complex with \(R\Gamma(W,\widetilde M)\). Filtered module colimits are exact, hence \[ \underset e{\operatorname{colim}}\operatorname{Hom}_{D(A)}(I_e,M) =H^0(W,\widetilde M). \tag{1.D} \] Use \(M=R\operatorname{Hom}_A(P,E)\). A bounded finite-projective model of \(P\) makes its internal Hom quasi-coherent. Tensor--Hom adjunction turns a representative \(I_e\to M\) of the specified class into \(P\otimes I_e\to E\), proving (1.B). The existence of the bounded finite-projective affine model is proved in the next paragraph, independently of this use of (1.B). Here are the finite-model facts used here and below. If a quasi-coherent complex on an affine is pseudo-coherent, it has a bounded-above finite-free resolution. More generally, if \(E\) is pseudo-coherent on a quasi-compact open \(W\subset V\), such a complex on \(V\) can represent \(E\) after restriction. First extend \(E\) as \(Rj_*E\) and use the earlier affine equivalence to obtain an \(A\)-complex \(M\). Choose a finite principal cover of \(W\). Start above the common upper cohomology bound with the zero complex, and descend. If a finite-free tail \(F^{\ge n}\to M\) is an isomorphism above \(n\) and surjective in degree \(n\) on every member of the cover, its cone has finite top cohomology in degree \(n-1\) on every member, by the finite-approximation criterion. Choose finitely many generators there. Represent each by a localized cycle of the cone; multiply by a sufficiently high power of that principal denominator to obtain an actual global cycle. This is possible because a localized zero differential is killed by some power of the denominator. The finitely many resulting global cycles span each localized top-cohomology module: multiplication by the denominator is a unit on that member. Attach one free generator in degree \(n-1\) for each cycle. Its two components in \(M^{n-1}\oplus F^n\) specify the comparison and the differential, with the cone sign; the cycle equation proves both \(d^2=0\) and compatibility with \(M\). The new tail is an isomorphism above \(n-1\) and surjective there. Continuing downward gives the required bounded-above finite-free complex, since each fixed degree stops changing. On each principal open its map is a quasi-isomorphism in every degree. For \(W=V\), this is the asserted affine resolution. If the complex is also perfect, its Tor amplitude has a common finite bound. Cut this free resolution sufficiently far below that bound. Its bottom syzygy is flat by the bottom-syzygy argument in the earlier perfectness theorem; two preceding free terms give a finite presentation of it. A flat finitely presented module is finite projective, as proved in that lesson, Lemma 1.2. Replacing the bottom of the resolution by that syzygy gives a bounded finite-projective model. This also establishes the fact used before (1.C), without circular reliance on the denominator lemma. \(\square\) \textbf{Lemma 1.C (lifting a perfect object and its map).} Let \(U\subset X\) be quasi-compact open in a qcqs scheme. Given perfect \(P\) on \(U\), quasi-coherent \(E\) on \(X\), and \(\alpha:P\to E|_U\), there is perfect \(R\) on \(X\) and \(R\to E\) such that \(R|_U\) is a finite sum of shifts of \(P\), containing \(P\) as a summand on which the map is \(\alpha\). If \(P\) is supported on \(T\cap U\), where \(T\) is closed with quasi-compact complement, \(R\) can be chosen supported on \(T\). \textbf{Proof.} First let \(X=\operatorname{Spec}A\). Apply the last paragraph of Lemma 1.B to represent \(P\) on \(U\) by a bounded-above finite-free complex \(F\) on \(X\). Cut it at degree \(s\) far below its Tor-amplitude bound. On \(U\), \(N=\ker(F^s\to F^{s+1})\) is finite locally free, by the same flat-syzygy and finite-presentation argument. The finite tail \(Q=F^{\ge s}\) has a triangle \[ Q|_U\longrightarrow P\longrightarrow N[1-s]\longrightarrow Q|_U[1]. \] For sufficiently negative \(s\), the middle arrow is zero by Lemma 1.A's uniform Ext bound, applied on \(U\). Thus \(Q|_U\simeq P\oplus N[-s]\). The endomorphism of this restriction which is zero on \(P\) and the identity on \(N[-s]\) lifts, by Lemma 1.B, to a map between perfect objects on \(X\), after replacing its domain by \(Q\otimes I\). Its cone restricts to \(P\oplus P[1]\). This proves affine extension up to a summand. For the support assertion, write \(T=V(g_1,\ldots,g_l)\) on this affine. On the qcqs open \(U\), some power of each \(g_i\) acts by zero on \(P\) in the derived category. Indeed the perfect-Hom complex \(P^\vee\otimes P\) is quasi-coherent. The earlier concentrated-map Theorem 6.2, applied to the flat principal localization of \(A\), identifies its derived-section complex localized at \(g_i\) with its derived sections on \(U\cap D(g_i)\). There \(P=0\), so the identity class becomes zero after localization. A zero localized element of the Hom module is killed by some power of \(g_i\), which proves the assertion globally on \(U\). Tensor the affine extension with the Koszul complexes on those powers. It is then supported on \(T\); its restriction is a finite sum of positive shifts of \(P\), containing \(P\), since each cone of a zero multiplication splits as the object plus its shift. Lemma 1.B now lifts the map from that summand to \(E|_U\), using a further perfect factor \(I\) which is the unit on \(U\). This factor preserves the support. This proves the entire affine assertion, with or without \(T\). For general \(X\), add one affine \(V\) at a time to \(U\). In the step \(X=U\cup V\), the intersection \(W\) is quasi-compact. The affine construction extends \(P|_W\) to perfect \(Q\) on \(V\), with \(Q|_W\) a finite sum \(P|_W\oplus\bigoplus_{t>0}P|_W[t]^{m_t}\), and lifts the map to \(E|_V\). On \(U\) take the same finite sum of shifts of \(P\), with map \(\alpha\) on its first summand and zero on the rest. These objects and maps agree on \(W\) and glue. For completeness, this derived gluing has an explicit cone construction. With open direct-image functors denoted by \(j_*\), take the homotopy fibre of \[ Rj_{U*}P_U\oplus Rj_{V*}Q \longrightarrow Rj_{W*}(P_U|_W), \] whose components are restriction and negative restriction, the latter composed with the chosen identification. Its restriction to \(U\) is \(P_U\): the extra two copies coming from \(W\) cancel by the identity; likewise its restriction to \(V\) is \(Q\). The Mayer--Vietoris triangle for \(E\) gives the map of fibres to \(E\); at the complex level the chosen equality of derived maps is represented by a homotopy on \(W\), which supplies that fibre map. Thus its restrictions are the specified maps. The fibre has quasi-coherent cohomology by the earlier qcqs pushforward theorem and is perfect locally on this cover. If both local objects are supported on \(T\), so is the fibre. There are finitely many added affines; induction proves the assertion. \(\square\) \textbf{Lemma 1.D (the affine support detector).} On \(V=\operatorname{Spec}A\), let \(Z=V(f_1,\ldots,f_r)\) and \(j:V\setminus Z\hookrightarrow V\). For the Koszul complex \(K=K(f_1,\ldots,f_r)\), \[ \operatorname{Hom}(K,E[q])=0\ (q\in\mathbb Z) \quad\Longrightarrow\quad E\simeq Rj_*(E|_{V\setminus Z}). \tag{1.E} \] In particular \(K\) detects zero objects with cohomology supported on \(Z\). \textbf{Proof.} Every \(K_e\) in Lemma 1.B is obtained by finite sums, cones and shifts from \(K\). For one entry, the cone triangles for the composable multiplications \(f^a,f^b\) give a triangle connecting \(K(f^a)\), \(K(f^{a+b})\) and \(K(f^b)\); the cone matrix or octahedral axiom gives this triangle. Tensor it with the remaining entries and induct on each exponent. Thus the asserted Hom vanishing holds for all \(K_e\) and shifts. The triangle \(I_e\to A\to K_e\) consequently identifies \(\operatorname{Hom}(A,E[q])\) with \(\operatorname{Hom}(I_e,E[q])\). Taking the direct limit and using (1.D) identifies the map with restriction \(H^q(V,E)\to H^q(V\setminus Z,E)\). These are isomorphisms in every degree. Affine unbounded comparison detects the cone of \(E\to Rj_*E|_{V\setminus Z}\), proving (1.E). If \(E\) is supported on \(Z\), its restriction is zero. \(\square\) \textbf{Theorem 1.E (qcqs compact generation).} The category \(D_{\mathrm{QCoh}}(X)\) has a single perfect compact generator. More generally, if \(T\subset X\) is closed with quasi-compact complement, its full subcategory of supported objects has a perfect compact generator, which is compact also in the ambient category. \textbf{Proof.} Induct on a finite affine cover. On an affine, the generator is \(\mathcal O_X\); for the supported category use Lemma 1.D's Koszul complex. Suppose \(X=U\cup V\), with \(U\) already covered and \(V\) affine. Lift a perfect generator \(P\) of \(U\) to \(Q\) on \(X\) by Lemma 1.C. Let \(Z=X\setminus U\), a closed subset contained in \(V\) whose complement in \(V\) is quasi-compact. Choose a Koszul detector \(K\) for \(Z\) on \(V\). Its extension \(K'\) to \(X\) is perfect: it is \(K\) on \(V\) and zero on \(X\setminus Z\). More explicitly, exact open extension by zero and derived open pushforward agree for this complex. Their natural comparison is an isomorphism on \(V\) and on \(X\setminus Z\), where both vanish, so it is an isomorphism globally. In particular \[ \operatorname{Hom}_X(K',E[q])=\operatorname{Hom}_V(K,E|_V[q]). \] If all maps from shifts of \(Q\oplus K'\) to \(E\) vanish, Lemma 1.D on \(V\) says that the unit \(E\to Rj_{U*}E|_U\) is an isomorphism there. It is already an isomorphism on \(U\), so its cone vanishes on a cover and it is an isomorphism globally. Now \[ \operatorname{Hom}_X(Q,E[q])= \operatorname{Hom}_U(Q|_U,E|_U[q]) \] contains the group with \(P\) as a summand. The generator on \(U\) forces \(E|_U=0\), whence \(E=0\). For the supported category, lift its generator on \(U\) with support in \(T\) using Lemma 1.C, and take on \(V\) the Koszul complex for \(Z\cap T\). Its complement is the union of the two quasi-compact opens \(V\setminus Z\) and \(V\setminus T\), so such a finite Koszul complex exists. For \(E\) supported on \(T\), its restriction to \(V\setminus(Z\cap T)\) is supported on \(T\cap U\cap V\), a closed subset of that open contained in \(U\cap V\). The same extension-by-zero argument identifies its derived direct image from this larger open with its derived direct image from \(U\cap V\). Lemma 1.D therefore again says \(E\simeq Rj_{U*}E|_U\). The preceding argument with the supported generator proves detection. Every constructed generator is perfect, and Lemma 1.A proves compactness, both in the supported subcategory and in the ambient category. \(\square\) \subsubsection{The representability argument}\label{the-representability-argument} \textbf{Lemma 1.F (cellular generation).} Let \(\mathcal T\) be a locally small triangulated category with coproducts, and let a set \(\mathcal G\) of compact objects, closed under shifts, detect zero objects. Every object belongs to the smallest triangulated subcategory containing \(\mathcal G\) and closed under coproducts. \textbf{Proof.} For \(Y\), take \(X_1=\bigoplus_{(G,u:G\to Y)}G\) with evaluation \(v_1:X_1\to Y\). Given \(v_n\), let \(B_n\) be the coproduct of all maps \(k:G\to X_n\) with \(v_nk=0\), and complete evaluation to a triangle \(B_n\to X_n\xrightarrow{a_n}X_{n+1}\to B_n[1]\). Exactness of \(\operatorname{Hom}(-,Y)\) gives \(v_{n+1}\) with \(v_{n+1}a_n=v_n\). Define \(X\) by the telescope triangle \[ \bigoplus_nX_n\xrightarrow{1-a}\bigoplus_nX_n \longrightarrow X\longrightarrow\bigoplus_nX_n[1]. \tag{1.F} \] The compatible \(v_n\)'s lift to \(v:X\to Y\). For compact \(G\), \[ \operatorname{Hom}(G,X)=\underset n{\operatorname{colim}}\operatorname{Hom}(G,X_n). \] Apply \(\operatorname{Hom}(G,-)\) to (1.F): on the direct sums of these groups, \(1-a\) is injective, since its first coordinate in a kernel element vanishes, then its second, and so on; every element has finite support. Its cokernel is the defining direct-limit quotient. The same assertion holds after shifts. Surjectivity of \(\operatorname{Hom}(G,v)\) holds at stage 1. Every kernel element at any stage is killed at the next by the defining triangle. Hence these maps are bijective for all \(G\). The cone is undetected by \(\mathcal G\), so is zero. Every \(X_n\) and its telescope were built using the allowed operations, proving the lemma. \(\square\) \textbf{Theorem 1.G (Brown representability and adjoints).} A contravariant cohomological functor \(H:\mathcal T\to\mathrm{Ab}\) which sends coproducts to products, with \(\mathcal T\) as in Lemma 1.F, is represented by an object of \(\mathcal T\). Consequently every coproduct-preserving exact functor from \(\mathcal T\) to a locally small triangulated category has a right adjoint. \textbf{Proof.} Take \(X_1=\bigoplus_{(G,b\in H(G))}G\). The product property gives \(b_1\in H(X_1)\) with component \(b\) on its labelled summand. An element \(b_n\in H(X_n)\) defines \(\eta_n(u)=H(u)b_n\). Let \(B_n\) be the coproduct of all \(k:G\to X_n\) in this transformation's kernel and take the same triangle as in Lemma 1.F. Its restriction of \(b_n\) to \(H(B_n)\) is zero; the cohomological exact sequence therefore gives \(b_{n+1}\) restricting to \(b_n\). The telescope and product property give an exact sequence \[ H(X)\longrightarrow\prod_nH(X_n) \xrightarrow{1-a^*}\prod_nH(X_n). \] Choose \(b\in H(X)\) over the compatible family and define \(\eta_Y(u)=H(u)b\). For every \(G\), compactness and the telescope calculation show that \(\eta_G\) is surjective by stage 1 and injective because every zero class is killed at the next stage. The objects \(Y\) on which \(\eta_{Y[q]}\) is bijective for all \(q\) form a triangulated subcategory, by natural long exact sequences and the five lemma. It is closed under coproducts, since both functors send them to products. Lemma 1.F says this subcategory is all of \(\mathcal T\), proving representability. For an exact coproduct-preserving \(F\), apply this to \(H_K(L)=\operatorname{Hom}(FL,K)\). A map \(K\to K'\) induces a unique map of representing objects; uniqueness preserves identities and compositions and makes the bijections natural in both variables. This is the required adjoint. \(\square\) Apply Theorems 1.E and 1.G to \(F=Rf_*\), whose exactness and preservation of coproducts are the earlier concentrated-map Theorem 4.2. Every morphism between qcqs schemes is concentrated. Choose a finite affine cover of \(X\) whose members map into affine opens of \(Y\). For an affine open \(V\subset Y\), its intersection with each of those target affines is quasi-compact by quasi-separatedness and has a finite principal cover there. Its inverse image in the corresponding source affine is therefore a finite union of principal opens. The finite union over the source cover proves quasi-compactness of \(f^{-1}V\). Quasi-separatedness follows from quasi-compact intersections in \(X\), equivalently the affine-cover criterion in that earlier lesson, Lemma 1.4. This establishes (1) for arbitrary unbounded quasi-coherent complexes. The same argument applied to the inclusion \(D_{\mathrm{QCoh}}(Y)\hookrightarrow D(\mathcal O_Y)\), which preserves coproducts since quasi-coherence and cohomology commute with sums, constructs \(DQ_Y\). Its counit is universal among maps from quasi-coherent complexes. This proves the coherator used below, in precisely the required unbounded category. The \textbf{trace} is the counit \[ \operatorname{Tr}_f(K):Rf_*a_fK\longrightarrow K, \tag{2} \] corresponding to the identity of \(a_fK\). Thus the transpose of \(u:L\to a_fK\) is precisely \(\operatorname{Tr}_f(K)\circ Rf_*u\). This definition specifies the normalization of every computation below. \textbf{Proposition 1.1.} The functor \(a_f\) is exact, commutes with shifts, and sends bounded-below objects to bounded-below objects. For composable morphisms \(X\xrightarrow fY\xrightarrow gZ\), there is a canonical isomorphism \[ a_{gf}\simeq a_fa_g. \tag{3} \] Under it, the composite trace is \(\operatorname{Tr}_g\circ Rg_*\operatorname{Tr}_f\). \textbf{Proof.} Shifting both sides of (1) gives the shift isomorphism. Right adjoints of exact triangulated functors are exact: given a triangle on the target, transpose its first two maps, complete to a triangle, and test the third map by the adjunction and the long exact Hom sequences. The five lemma gives the required isomorphism with the third adjoint object. This is the usual derived adjunction construction, with its compatible shift maps. Choose \(d\geq0\) such that \(Rf_*D^{\leq b}_{\mathrm{QCoh}}(X)\subset D^{\leq b+d}_{\mathrm{QCoh}}(Y)\) for every \(b\). The uniform bound, including for unbounded complexes, is proved in the earlier concentrated-map lesson, Proposition 3.3 and Theorem 4.2. On each member of a finite affine cover of \(Y\), use its finite-cover bound on the inverse image and take the maximum; thus a single \(d\) works on \(Y\). If \(K\in D^{\geq c}(Y)\), set \(T=\tau_{\leq c-d-1}a_fK\). Then \(Rf_*T\in D^{\leq c-1}(Y)\), so \[ \operatorname{Hom}_X(T,a_fK)=\operatorname{Hom}_Y(Rf_*T,K)=0. \] In particular the truncation map \(T\to a_fK\) is zero. On every cohomology sheaf in degrees at most \(c-d-1\) that map is the identity identification, so those sheaves vanish. Hence \(a_fK\in D^{\geq c-d}(X)\). Finally, transpose twice: \[ \operatorname{Hom}_X(L,a_fa_gK) \simeq\operatorname{Hom}_Y(Rf_*L,a_gK) \simeq\operatorname{Hom}_Z(Rg_*Rf_*L,K). \] Using the canonical composition of derived pushforwards, Yoneda proves (3). The displayed rule for transposes proves the trace formula. Repeating the argument three times shows associativity: both comparison maps induce the same bijection on every Hom set. \(\square\) The bound need not be zero. For projective \(n\)-space, the adjoint of the field occurs in degree \(-n\), as Section 4 will show. There are now three different operations to keep track of. Pullback is the left adjoint of pushforward; it transports sections and tensors them with the structure sheaf upstairs. The functor \(a_f\) is its right adjoint and represents maps from a pushforward into a prescribed target. The compactification construction \(f^!\), introduced later, uses the right adjoint of a proper map followed by restriction. Finite maps make the second operation concrete as coinduction; open immersions will show why the second and third operations require separate names. \subsection{2. What sheafified duality says}\label{what-sheafified-duality-says} Distinguish internal \(R\mathcal H om_X\), which is a complex of sheaves, from global \(R\operatorname{Hom}_X\). Evaluation and the counit give a canonical morphism \[ Rf_*R\mathcal H om_X(L,a_fK) \longrightarrow R\mathcal H om_Y(Rf_*L,K). \tag{4} \] One construction first maps into \(R\mathcal H om_Y(Rf_*L,Rf_*a_fK)\), using evaluation and the projection-formula comparison, and then applies (2). Internal Hom of arbitrary unbounded quasi-coherent complexes need not have quasi-coherent cohomology. This prevents us from treating (4) as an isomorphism merely by applying (1). Let \(DQ_Y:D(\mathcal O_Y)\to D_{\mathrm{QCoh}}(Y)\) be the right adjoint of the inclusion, constructed after Theorem 1.G. This is the adjoint of the full quasi-coherent-cohomology subcategory in the unbounded category, not merely the derived sheaf-level coherator. \textbf{Proposition 2.1.} Applying \(DQ_Y\) to (4) gives an isomorphism. Taking global sections of (4) gives \[ R\operatorname{Hom}_X(L,a_fK) \simeq R\operatorname{Hom}_Y(Rf_*L,K). \tag{5} \] \textbf{Proof.} Test (4) against \(M\in D_{\mathrm{QCoh}}(Y)\). The tensor--Hom and pullback--pushforward adjunctions identify its source Hom group with \[ \operatorname{Hom}_X(Lf^*M\otimes^{\mathbf L}L,a_fK) \simeq \operatorname{Hom}_Y(Rf_*(Lf^*M\otimes^{\mathbf L}L),K). \] The unbounded quasi-coherent projection formula identifies the last group with \[ \operatorname{Hom}_Y(M\otimes^{\mathbf L}Rf_*L,K) \simeq \operatorname{Hom}_Y(M,R\mathcal H om_Y(Rf_*L,K)). \] The projection formula holds for every qcqs morphism and both indicated quasi-coherent complexes, without a perfectness condition on \(M\); the exact earlier programme proof is the concentrated-map lesson, Theorem 5.1. The identifications just made are induced by evaluation and the counit, hence by (4). The defining adjunction of \(DQ_Y\) and Yoneda prove the first assertion. Testing with \(M=\mathcal O_Y[-r]\) proves that (4) induces an isomorphism on every global hypercohomology group. This gives (5), since global derived Hom is the derived global sections of internal Hom. \(\square\) There is a second, distinct locality issue. If \(V\subset Y\) is quasi-compact open, \(U=f^{-1}(V)\), and \(f_V:U\to V\), there is a canonical comparison \[ (a_fK)|_U\longrightarrow a_{f_V}(K|_V). \tag{6} \] It comes from \(K\to Rj_*(K|_V)\) and the two open adjunctions. The following proof establishes it for proper maps over a Noetherian base and explains exactly what is additionally needed over an arbitrary base. \textbf{Lemma 2.2 (finite factorization and killing a supported map).} If \(C\) is compact and \(E\) is built by the cellular construction of Lemma 1.F from compact objects \(G\), every map \(C\to E\) factors through a finite sequence of sums, shifts and cones of the \(G\)'s. Consequently, if \(T\subset Y\) is closed with quasi-compact complement \(V\), \(P\) is pseudo-coherent, and \(Q\in D^+_{\mathrm{QCoh}}(Y)\) is supported on \(T\), every map \(P\to Q\) is killed after precomposition with \[ I\otimes^{\mathbf L}P\longrightarrow P, \tag{2.A} \] where \(I\to\mathcal O_Y\) is a perfect map which restricts to the identity isomorphism on \(V\). \textbf{Proof.} Compactness first factors a map to the telescope through some stage \(E_n\), by the calculation in Lemma 1.F. We prove the finite factorization for that stage by induction. Stage 1 is a coproduct, so compactness gives a finite subcoproduct. In the stage triangle \(B\to E_{n-1}\to E_n\to B[1]\), the composite \(C\to B[1]\) factors through a finite subcoproduct \(B_0[1]\). Complete it to \(B_0\to C'\to C\to B_0[1]\). The triangle axiom gives a compatible map to the stage triangle. The object \(C'\) is compact, because compact objects are closed under cones: apply the two Hom long exact sequences to a coproduct comparison and use the five lemma. By induction, \(C'\to E_{n-1}\) factors through a finite object \(A\). Take the cone of \(B_0\to C'\to A\); triangle morphisms give a factorization of the specified composite to \(B[1]\) through this finite cone. Its map to \(E_n\) may differ from the original by a map coming from \(E_{n-1}\). Apply the induction hypothesis once more to that difference, and add its finite factor as a summand. This gives the exact original map, not just its composite into \(B[1]\). It proves finite factorization. Now put \(H=R\mathcal H om_Y(P,Q)\). It has quasi-coherent cohomology: on an affine, resolve \(P\) by a bounded-above finite-free complex using Lemma 1.B, and represent \(Q\) by a bounded-below module complex. In each fixed Hom degree only finitely many pairs of terms occur. Thus localization commutes with this Hom complex and its cohomology. It is supported on \(T\), since internal derived Hom restricts to the corresponding internal Hom and \(Q|_V=0\). The given map is a class \(\mathcal O_Y\to H\). Use Theorem 1.E's supported perfect generator to build \(H\) cellularly within the supported category. That telescope is also a telescope in the ambient category, since the inclusion preserves coproducts and triangles. The ambient compact object \(\mathcal O_Y\) therefore factors its map through a perfect \(A\) supported on \(T\), by the finite argument just proved; its use does not require \(\mathcal O_Y\) to belong to the supported category. Complete \(\mathcal O_Y\to A\) to \(I\to\mathcal O_Y\to A\to I[1]\). Then \(I\) is perfect, is the unit on \(V\), and \(I\to\mathcal O_Y\to H\) is zero. Tensor--Hom adjunction says exactly that (2.A) kills the original map. \(\square\) \textbf{Theorem 2.3 (proper restriction, with a verified Noetherian case).} If \(f\) is proper and \(Y\) is Noetherian, (6), for its quasi-compact open \(V\), is an isomorphism for every \(K\in D^+_{\mathrm{QCoh}}(Y)\). More generally, the same proof applies whenever \(Rf_*P\) is pseudo-coherent for every perfect \(P\) on \(X\). \textbf{Proof.} First we show that \(a_f\) takes bounded-below objects supported on \(Y\setminus V\) to objects supported on \(X\setminus U\). Otherwise a perfect generator on \(U\) gives a nonzero map \(P_U\to(a_fQ)|_U\). Lift this map by Lemma 1.C to a map \(P\to a_fQ\), with \(P\) perfect on \(X\), whose restriction includes that nonzero map. Its transpose is \(Rf_*P\to Q\). The source is pseudo-coherent under the stated general hypothesis. In the Noetherian case this hypothesis follows from \href{proper-morphisms-and-coherent-direct-images.md}{Proper morphisms and coherent direct images}, Theorem 4.1: the finite truncation filtration of \(P\) has coherent cohomology sheaves, each with coherent higher direct images; the cohomological-dimension bound makes the output bounded. On a Noetherian affine, a bounded-above complex with finite cohomology has a bounded-above finite-free resolution by successively resolving its top cohomology and the next cone, so it is pseudo-coherent. Lemma 2.2 supplies \(I\to\mathcal O_Y\), the unit on \(V\), killing the transpose after tensoring. The earlier unbounded projection formula identifies the new source with \(Rf_*(Lf^*I\otimes P)\). Transposing back shows that \[ Lf^*I\otimes P\longrightarrow P\longrightarrow a_fQ \] is zero. On \(U\), its first arrow is an isomorphism, contradicting the chosen nonzero restriction. This proves the support assertion. Let \(j:V\hookrightarrow Y\) and \(h:U\hookrightarrow X\). Usual pushforward restriction and two adjunctions give, naturally in \(L\) and \(N\), \[ \begin{aligned} \operatorname{Hom}_X(L,a_fRj_*N) &=\operatorname{Hom}_Y(Rf_*L,Rj_*N)\\ &=\operatorname{Hom}_V(R(f_V)_*h^*L,N)\\ &=\operatorname{Hom}_X(L,Rh_*a_{f_V}N). \end{aligned} \] Hence \(a_fRj_*\simeq Rh_*a_{f_V}\). Apply \(a_f\) to the localization triangle \(Q\to K\to Rj_*j^*K\), where \(Q\) is bounded below and supported outside \(V\). Restrict to \(U\). Its first term vanishes by the support assertion, and \(h^*Rh_*\) is the identity. The remaining arrow is exactly (6), proving the theorem. \(\square\) \textbf{Lemma 2.4 (proper flat finite presentation).} If \(f:X\to Y\) is proper, flat and of finite presentation, with \(Y\) qcqs, then \(Rf_*P\) is perfect for every perfect \(P\) on \(X\). In particular it is pseudo-coherent, and Theorem 2.3 proves (6) for this case as well, for every quasi-compact open \(V\) and every bounded-below input. \textbf{Proof: descending the finite data.} Work on an affine open \(\operatorname{Spec}A\) of \(Y\), and write \(A=\operatorname{colim}A_i\) as its finite-type \(\mathbb Z\)-subalgebras. \href{semicontinuity-and-grauerts-theorem.md}{Semicontinuity and Grauert's theorem}, Lemma 0.1, proves that a finite-presentation model of this proper flat \(X/A\) can be made proper and flat over a Noetherian \(A_i\): apply its scheme, properness and flatness descent with the sheaf \(\mathcal O_X\). We still have to descend the perfect complex, rather than treating it as one globally free complex on \(X\). Choose a finite affine cover \(U_1,\ldots,U_m\) on which \(P\) is represented by bounded complexes \(P_a\) of finite projective modules, in a common interval \([s,t]\). Such models follow from Lemma 1.B's finite-model argument. The intersections \(U_I\) are affine, since \(X\) is separated over an affine base. They, their inclusions and their rings descend to a common stage by the finite-open and finite-presentation descent in the earlier limits lesson and the separated-diagonal descent used in Lemma 0.1. Each projective term is the image of a finite idempotent matrix. Its differential is a finite matrix between those images. Thus these terms and differentials are specified by finitely many entries and the equations \(e^2=e\), \(ed=d=de\) and \(d^2=0\). The gluing also has finite chain data, including the necessary homotopies. On each affine intersection, choose chain maps between the restricted local models which represent their identification with \(P\), and homotopy inverses. A derived map between bounded projective complexes is represented by a chain map; equality is represented by a chain homotopy. Record the inverse identities up to homotopy. On triple intersections record the homotopy between the two compositions. Continue with higher intersections: for an increasing tuple of length \(q+1\), record the degree \(1-q\) map making the boundary of the previous homotopies vanish. These are precisely the finitely many equations which say that the sum of the local differential, the restriction differential and the higher homotopy maps in the ordered Čech total complex squares to zero. Here is why these coherent choices exist, and not just the pairwise homotopies. Use a common resolution representing the original global \(P\) on each intersection. A quasi-isomorphism to that resolution induces a quasi-isomorphism on the Hom complex out of each bounded projective local model. Inductively, the boundary of the next desired homotopy is a cycle. Its image in the common resolution is a boundary: there all comparisons are the restrictions of the same global object, and the preceding chosen comparisons and homotopies have already been recorded. Injectivity on the cohomology of the Hom complex makes the cycle a boundary in the local model. Choose its preimage, adjusting by a cycle when necessary using surjectivity on that same cohomology. This supplies the next homotopy and the compatibility with the common resolution. Start with the chosen local quasi-isomorphisms and repeat on the increasing tuples. There are no tuples of length greater than \(m\); the finite interval of the models also kills sufficiently negative Hom degrees. Thus only finitely many finite matrices and finite equations are involved. This argument also records the augmentation which identifies the resulting total complex with the original \(P\). Represent all these entries in a common stage. Every stated matrix equality holds there after moving to a sufficiently late stage, since a zero element in the filtered colimit is zero eventually. Descend the equations for the inverse homotopies as well. The finite cover remains a cover at a late stage. Its descended coherent chain data glue to an object \(P_i\) on \(X_i\): explicitly use the finite ordered Čech totalization, with the higher homotopies as its extra differential components. Restriction to each \(U_{a,i}\) is equivalent to the local bounded projective model. The contraction which inserts the index \(a\) cancels the Čech columns on that open; the recorded homotopy equations make this contraction compatible with the extra components. Equivalently, eliminate the pairs of columns containing and not containing \(a\), starting from the longest tuples; their diagonal map is the recorded homotopy equivalence and the higher components are exactly its successive corrections. There are finitely many columns, so elimination terminates and leaves \(P_{a,i}\). Therefore \(P_i\) is perfect, and its derived pullback to \(X\) is the given \(P\). This describes the descent of the actual local complexes, equivalences, homotopies and their finite compatibility equations; it does not assume a resolution property for \(X\). \textbf{Proof: the finite projective image and arbitrary tensor.} Apply sections to this finite Čech model. It gives a bounded complex \(C_i\) of \(A_i\)-flat modules which computes \(R\Gamma(X_i,P_i)\). Each term is a finite sum of finite projective modules over an affine-intersection ring; that ring is \(A_i\)-flat because \(X_i\to\operatorname{Spec}A_i\) is flat. Direct summands of its finite free modules are consequently \(A_i\)-flat. The acyclic-affine Čech calculation applies to the total complex with its recorded homotopies, or to the finite filtration by its columns: on each column it is the earlier affine comparison, and finite induction proves the total assertion. The cohomology modules of \(C_i\) are finite. Indeed \(P_i\) is bounded coherent over the Noetherian \(X_i\), and the finite cohomology-sheaf filtration together with \href{proper-morphisms-and-coherent-direct-images.md}{Proper morphisms and coherent direct images}, Theorem 4.1, makes its derived sections bounded with finite cohomology. \href{base-change-and-the-grothendieck-complex.md}{Base change and the Grothendieck complex}, Lemma 3.1, therefore replaces \(C_i\), after shifting its finite interval, by a bounded complex \(K_i\) of finite projective \(A_i\)-modules, with a quasi-isomorphism which stays one after tensoring with every \(A_i\)-module. For any \(A_i\)-algebra \(B\), the cover, the idempotent matrices and every differential and homotopy in \(C_i\) base change termwise. The resulting total complex is \(C_i\otimes_{A_i}B\), and computes the derived pullback of \(P_i\) on \(X_{i,B}\). Its local terms are base-flat, so ordinary tensor here computes derived tensor. The same finite Čech calculation gives \[ K_i\otimes_{A_i}B \simeq R\Gamma(X_{i,B},LP_i|_{X_{i,B}}). \tag{2.B} \] Take \(B=A\). It follows that \(R\Gamma(X,P)\) is represented by the bounded finite projective complex \(K_i\otimes_{A_i}A\). Affine comparison makes \(Rf_*P\) perfect on this affine open of \(Y\). These opens cover \(Y\), so it is perfect globally. Lemma 2.2 and Theorem 2.3 now give the claimed proper restriction. All support-generation steps use the quasi-compact \(V\) specified in (6); any further assertion of locality is checked on quasi-compact affine opens. \(\square\) \subsection{3. Finite maps and closed immersions}\label{finite-maps-and-closed-immersions} Let \(f:X\to Y\) be finite between Noetherian schemes, and put \(B=f_*\mathcal O_X\), a coherent sheaf of \(\mathcal O_Y\)-algebras. Quasi-coherent sheaves on \(X\) correspond to quasi-coherent \(B\)-modules on \(Y\). In the derived category this correspondence is given by exact affine pushforward; it preserves and detects quasi-isomorphisms. \textbf{Theorem 3.1.} For \(K\in D^+_{\mathrm{QCoh}}(Y)\), the object \(a_fK\), expressed as a \(B\)-complex on \(Y\), is \[ R\mathcal H om_{\mathcal O_Y}(B,K), \qquad (b\varphi)(b')=\varphi(bb'). \tag{7} \] For a closed immersion \(i:Z\hookrightarrow Y\), this is the derived functor of the annihilator sheaf \[ i^bK=\mathcal H om_Y(i_*\mathcal O_Z,K), \] regarded as a sheaf on \(Z\). Its derived value is often denoted \(Ri^bK\). \textbf{Proof.} The sheaf-level coinduction identity is \[ \operatorname{Hom}_B(M,\mathcal H om_Y(B,J)) \simeq\operatorname{Hom}_{\mathcal O_Y}(M,J). \tag{8} \] Evaluation at \(1\) gives the forward map; the inverse sends \(v\) to \(m\mapsto[b\mapsto v(bm)]\). This works on every open subset and glues. Restriction of scalars is exact, so its right adjoint sends injectives to injectives. Resolve \(K\) by a bounded-below injective complex \(J\). Formula (8), applied to complexes, identifies the derived adjunction with coinduction into \(\mathcal H om_Y(B,J)\). This complex has quasi-coherent cohomology. Locally on the Noetherian base, resolve the finite module \(B\) by finite free modules in nonpositive degrees. The bounded-below hypothesis on \(K\) makes each fixed total degree involve only finitely many terms. Hom therefore commutes with localization and computes quasi-coherent Ext sheaves. The equivalence for affine pushforward now transports the complex back to \(X\). It represents exactly the Hom functor (1), proving (7). For \(B=\mathcal O_Y/\mathcal I\), its terms are annihilated by \(\mathcal I\), so they are the asserted sheaves on \(Z\). \(\square\) The \(B\)-action in (7) is essential. An ambient internal Hom complex without its algebra action has not yet specified an object on \(X\). The trace in this model is evaluation at \(1\), not an unspecified ring trace. \textbf{Example 3.2 (an effective Cartier divisor).} If \(Z=D\) is an effective Cartier divisor in \(Y\), resolve \(\mathcal O_D\) by \[ 0\longrightarrow\mathcal O_Y(-D)\longrightarrow\mathcal O_Y \longrightarrow\mathcal O_D\longrightarrow0. \] Applying Hom into \(\mathcal O_Y\) gives \(\mathcal O_Y\to\mathcal O_Y(D)\) in degrees 0 and 1. The map is injective, and its cokernel is the normal line \(N_{D/Y}=\mathcal O_D(D)\). Thus \[ a_i(\mathcal O_Y)=N_{D/Y}[-1]. \tag{9} \] The minus sign in the shift places the normal line in cohomological degree 1. \subsection{4. Projective space and the classical trace}\label{projective-space-and-the-classical-trace} \textbf{Theorem 4.1.} For \(\pi:P=\mathbf P^n_k\to\operatorname{Spec}k\), \[ a_\pi(k)\simeq\mathcal O_P(-n-1)[n]=\omega_P[n]. \tag{10} \] The counit is the trace whose ordered Čech representative extracts the coefficient of \(T_0^{-1}\cdots T_n^{-1}\). \textbf{Proof.} The projective-space calculation gives \(R\Gamma(P,\omega_P[n])\simeq k\), with the specified trace. Transpose that map to obtain \(c:\omega_P[n]\to a_\pi(k)\). For a coherent sheaf \(F\), testing \(c\) against every shift of \(F\) gives the maps \[ \operatorname{Ext}^{n+r}_P(F,\omega_P) \longrightarrow\operatorname{Hom}_{D(k)}(R\Gamma(P,F),k[r]) \simeq H^{-r}(P,F)^\vee. \] These are precisely the Yoneda trace pairings proved in \href{ext-sheaves-and-serre-duality-on-projective-space.md}{Serre duality on projective space}, Theorem 5.1. They are isomorphisms for all \(r\), including the zero groups outside the cohomological range. Finite truncation triangles extend this assertion from coherent sheaves to every bounded complex with coherent cohomology. In particular it holds for a perfect generator \(G\) of \(D_{\mathrm{QCoh}}(P)\), which is bounded coherent because \(P\) is Noetherian and quasi-compact. If \(C\) is the cone of \(c\), then \(\operatorname{Hom}(G,C[r])=0\) for every \(r\). The generator property forces \(C=0\). Thus the map is an isomorphism in the full unbounded category, not merely on coherent test objects. When \(n=0\), this is the identity adjunction on the field; every twist on the point is the same line. \(\square\) \textbf{Theorem 4.2 (the relative projective bundle).} Let \(E\) be locally free of rank \(n+1\geq1\) on a qcqs scheme \(Y\), and use the quotient convention for \(\pi:\mathbf P(E)\to Y\). Then \[ a_\pi(\mathcal O_Y)\simeq \pi^*\det E\otimes\mathcal O_{\mathbf P(E)}(-n-1)[n]. \tag{11} \] \textbf{Proof of the trace.} On an affine free-frame open, \href{cohomology-of-projective-space.md}{Cohomology of projective space}, Theorem 2.2, gives just one nonzero cohomology module for \(\mathcal O(-n-1)\), in degree \(n\), generated by \([1/(T_0\cdots T_n)]\). Its ordered trace sends that generator to one. Under a change of frame by \(M\), the generator transforms by \(\det(M)^{-1}\). Here is a presentation-independent verification, valid over every ring: the Koszul resolution of the quotient of \(A[T_0,\ldots,T_n]\) by its variables has top exterior term \(\bigwedge^{n+1}A^{n+1}\). Its dual class maps, through the dual power-Koszul complexes, to the reciprocal-monomial class in top Čech degree. The direct-limit identification is the termwise localization calculation (1.C), now for the variables. Changing the variables multiplies the top exterior term by \(\det M\), hence its dual by the inverse determinant. The map into Čech cohomology is defined by the quotient and localization maps and respects a change of presentation. This proves the transformation law, including elementary changes with nondiagonal entries. Thus tensoring by \(\det E\) glues these local traces to a canonical isomorphism \[ R\pi_*D\simeq\mathcal O_Y, \qquad D=\pi^*\det E\otimes\mathcal O(-n-1)[n]. \tag{4.A} \] Transpose it to \(c:D\to a_\pi\mathcal O_Y\). \textbf{Proof of detection on projective space.} On \(\mathbf P_A^n\), the objects \(\mathcal O,\ldots,\mathcal O(-n)\) detect zero unbounded quasi-coherent complexes. The sheafified Koszul complex of the homogeneous coordinates is exact, since at every point one coordinate is a unit and contracts it. Twisting this complex successively shows that all \(\mathcal O(-m)\), \(m\geq0\), are finite extensions of shifts of the displayed twists. If their Hom groups to a complex \(C\), in all degrees, vanish, then \(R\Gamma(\mathbf P_A^n,C(m))=0\) for every \(m\geq0\). On a standard chart \(D_+(T_i)\), localization is the telescope of multiplication by \(T_i\) on these twists. This assertion follows on every trivializing affine from the exact module localization \(\operatorname{colim}(M\xrightarrow{T_i}M\to\cdots)=M_{T_i}\), and the telescope is that colimit by its injective \(1-\mathrm{shift}\) map. Derived sections commute with these telescopes by the earlier coproduct theorem. Consequently derived sections of \(C\) on every standard chart vanish. Affine comparison gives \(C=0\). This proves detection for unbounded complexes over arbitrary \(A\), without invoking coherent resolutions over \(A\). \textbf{Proof that the transpose is an isomorphism.} Take the perfect generator \(G\) on \(Y\) from Theorem 1.E. The objects \[ L_i=\pi^*G\otimes\mathcal O(i),\qquad -n\leq i\leq0, \] detect zero on \(\mathbf P(E)\). Indeed, vanishing of their shifted Hom groups implies \(R\pi_*(C\otimes\mathcal O(-i))=0\), by adjunction and detection by \(G\). Restrict usual pushforward to an affine free-frame open on the base and use the projective-space detector just proved. This makes \(C\) vanish on an open cover. For \(-n\leq i<0\), the earlier twist calculation gives both \(R\pi_*\mathcal O(i)=0\) and \(R\pi_*(D\otimes\mathcal O(-i))=0\). Perfect duality and the projection formula identify all Hom groups from \(L_i\) to both the source and target of \(c\) with zero groups. For \(i=0\), the groups on its source are \[ \operatorname{Hom}_Y(G,R\pi_*D[q]) =\operatorname{Hom}_Y(G,\mathcal O_Y[q]), \] while its target groups, by the adjunction, are \[ \operatorname{Hom}_Y(R\pi_*\pi^*G,\mathcal O_Y[q]) =\operatorname{Hom}_Y(G,\mathcal O_Y[q]). \] The map is the identity under these identifications because \(c\) was transposed from (4.A), and the projection comparison is defined by evaluation. Thus the cone of \(c\) is undetected by all the \(L_i\), and is zero. This proves (11). When \(n=0\), the only detector is \(\pi^*G\), \(\mathbf P(E)=Y\), and \(\mathcal O(1)=\pi^*E\); the two line factors cancel. No proper-locality theorem was used in this argument. \(\square\) \subsection{5. Proper duality over a field}\label{proper-duality-over-a-field} Let \(p:X\to\operatorname{Spec}k\) be proper, and define \[ \omega_X^\bullet=a_p(k). \] \textbf{Theorem 5.1.} For every \(K\in D_{\mathrm{QCoh}}(X)\) and every integer \(i\), the trace induces \[ H^i(X,K)^\vee\simeq \operatorname{Ext}^{-i}_X(K,\omega_X^\bullet). \tag{12} \] If \(X\) is projective, Cohen--Macaulay and equidimensional of dimension \(n\), then \[ \omega_X^\bullet\simeq\omega_X^\circ[n], \] where \(\omega_X^\circ\) is the dualizing sheaf of the preceding lesson. \textbf{Proof.} Formula (1) gives \[ \operatorname{Hom}_X(K,\omega_X^\bullet[-i]) \simeq\operatorname{Hom}_{D(k)}(R\Gamma(X,K),k[-i]). \] Over a field, every complex splits into its cohomology complex and a contractible complex: choose complements to boundaries inside cycles and to cycles inside each term. Its maps to \(k[-i]\) in the derived category are consequently exactly the linear functionals on \(H^i\). This remains valid for unbounded complexes and infinite-dimensional cohomology; no finite-dimensional biduality has been used. It proves (12). For bounded coherent \(K\), proper coherence gives the familiar finite-dimensional interpretation. For the last assertion embed \(i:X\hookrightarrow P=\mathbf P^N_k\). Composition and Theorems 3.1 and 4.1 identify \[ \omega_X^\bullet=Ri^b(\omega_P[N]). \] The preceding lesson proves that \(Ri^b\omega_P\) has only one cohomology sheaf, in degree \(c=N-n\), and that this sheaf is \(\omega_X^\circ\). Thus it is \(\omega_X^\circ[-c]\); shifting by \(N\) gives \(\omega_X^\circ[n]\). The classical representing trace was defined from the same ambient pairing, so this identification respects traces. \(\square\) \textbf{Theorem 5.2 (the complex on an arbitrary proper scheme).} For every proper \(k\)-scheme \(X\), the complex \(\omega_X^\bullet=a_p(k)\) is bounded coherent, is locally dualizing, and has cohomology only in \([-\dim X,0]\). \textbf{Proof.} Fix an affine open \(U\subset X\). Embed \(U\) as a closed subscheme of \(\mathbf A_k^N\) using finite algebra generators, and let \(Z\) be its scheme-theoretic closure in \(\mathbf P_k^N\). It is projective, contains \(U\) as an open, and has dimension \(\dim U\): taking the closure introduces no new irreducible components unrelated to the dense open. The scheme-theoretic closure exists by taking on each affine the kernel of restriction to the prescribed open and gluing those quasi-coherent ideals; on a Noetherian scheme they are coherent. Let \(W\) be the scheme-theoretic closure of the graph of the two inclusions of \(U\) in \(X\times_k Z\). Both projections \(h:W\to X\) and \(q:W\to Z\) are proper, because the product factors are proper and \(W\) is closed. Both inverse images of \(U\) are precisely this common \(U\). Indeed, above \(U\subset X\) the graph of \(U\to Z\) is already closed, since \(Z\) is separated; above \(U\subset Z\) the graph of \(U\to X\) is already closed, since \(X\) is separated. Formation of scheme-theoretic closure restricts to these opens, so there is no extra scheme structure there. Composition of the adjoints and the Noetherian proper restriction theorem give \[ \begin{aligned} (a_p k)|_U &\simeq(a_h a_p k)|_U\\ &\simeq a_{W\to\operatorname{Spec}k}(k)|_U\\ &\simeq(a_q a_{Z\to\operatorname{Spec}k}k)|_U\\ &\simeq a_{Z\to\operatorname{Spec}k}(k)|_U. \end{aligned} \tag{5.A} \] The local maps are isomorphisms because the restrictions of \(h\) and \(q\) over these opens are identity maps. This argument only constructs a common refinement of the two already existing proper schemes. It does not use a compactification theorem for arbitrary finite-type maps. By the closed-immersion and projective-space computations, the last object is the restriction of \[ C_Z=Ri^b\mathcal O_{\mathbf P^N}(-N-1)[N]. \] The preceding lesson, \href{dualizing-sheaves-and-serre-duality-for-projective-schemes.md}{Dualizing sheaves and Serre duality for projective schemes}, Proposition 8.1, proves its local dualizing property by derived coinduction and finite regular-local resolutions. It also proves bounded coherence. Its Lemma 2.1 kills the ambient Ext sheaves below \(N-\dim Z\); the regular-local resolution bound kills them above \(N\). Thus its shifted cohomology lies in \([-\dim Z,0]\). For the latter bound, the exact earlier provider is \href{../AG-CA/AG-CA-14.html}{Regular local rings}, Theorem 2.2 and Proposition 3.3: polynomial ambient local rings are regular of dimension at most \(N\), and have that finite global dimension. No regularity of \(Z\) or \(X\) is asserted. Equation (5.A) proves these properties on every affine open of \(X\). Bounded coherence and the dualizing condition are local; a finite affine cover supplies a common finite bound, and \(\dim U\leq\dim X\) supplies the stated uniform interval. This proves the theorem for proper schemes which are not projective and for schemes with nilpotents and embedded components. \(\square\) The embedded-point example \(X=V(T_0^2,T_0T_1)\subset\mathbf P^2\) from the preceding lesson has \(H^{-1}(\omega_X^\bullet)=\mathcal O_{\mathbf P^1}(-2)\) and \(H^0(\omega_X^\bullet)=k_p\). Dropping its second cohomology sheaf was exactly what made the single-sheaf duality formula fail. \subsection{6. Coinduction versus restriction}\label{coinduction-versus-restriction} For a finite \(k\)-algebra \(B\), Theorem 3.1 gives \(a_p(k)=\operatorname{Hom}_k(B,k)\) in degree zero. This is coinduction; \(k\) is injective as a vector space, so there is no higher cohomology. Now take \(A=k[t]\), \(B=k[x]\), with \(t=x^2\). As an \(A\)-module, \(B=A\oplus Ax\) is free. Let \(\lambda:B\to A\) extract the coefficient of \(x\). The pairing \((b,b')\mapsto\lambda(bb')\) has matrix \[ \begin{pmatrix}0&1\\1&0\end{pmatrix} \] in the basis \(1,x\), so \(B\to\operatorname{Hom}_A(B,A)\), \(b\mapsto b\lambda\), is a \(B\)-linear isomorphism in every characteristic. Hence \(a_f(\mathcal O_Y)\simeq\mathcal O_X\). Under this chosen trivialization the counit sends \(a+bx\) to \(b\). The algebraic trace of multiplication is a different functional: it sends \(1\) to \(2\), and in characteristic 2 it vanishes identically. For an open immersion \(j:U\hookrightarrow X\), the compactification functor \(j^!\) discussed below is restriction, but the right adjoint \(a_j\) of \(Rj_*\) generally differs. An explicit affine example is \[ j:\operatorname{Spec}A[t^{-1}]\hookrightarrow\operatorname{Spec}A, \qquad A=k[t]. \] Here \(a_j(A)=R\operatorname{Hom}_A(A[t^{-1}],A)\), with its \(A[t^{-1}]\)-action. In degree zero this is zero: the image of \(1\) under an \(A\)-linear map must lie in every \(t^mA\), whose intersection is zero; the same argument applies to each \(t^{-r}\). But \(j^!A=A[t^{-1}]\) has nonzero degree zero. Thus restriction cannot be the general right adjoint of pushforward. More precisely, the free telescope resolution has map \(e_r\mapsto e_r-te_{r+1}\) on \(\bigoplus_{r\geq0}A\). Its cokernel is \(A[t^{-1}]\), with \(e_r\) mapping to \(t^{-r}\); its injectivity follows by examining the last nonzero coefficient in a finite sum. Dualizing computes \(a_j(A)\) by \[ \prod_{r\geq0}A\longrightarrow\prod_{r\geq0}A, \qquad (b_r)\longmapsto(b_r-tb_{r+1}) \tag{13} \] in degrees 0 and 1. Its cokernel is \(k[[t]]/k[t]\). Indeed, sending \((c_r)\) to \(\sum t^rc_r\) modulo \(A\) is surjective. A sequence is in its kernel exactly when that sum is a polynomial \(s\); then \(b_r=(s-\sum_{j