Coherent Duality on Projective Cohen-Macaulay Schemes

On rings the adjunction is explicit: a map $u:M\to {\mathrm{Hom}}_A(B,I)$ corresponds to $m\mapsto u(m)(1)$, with inverse sending $v:M\to I$ to $m\mapsto (b\mapsto v(bm))$. Restriction of scalars is exact, so its right adjoint sends injectives to injectives: applying ${\mathrm{Hom}}_B(-,{\mathrm{Hom}}_A(B,I))={\mathrm{Hom}}_A(-,I)$ to an exact sequence proves this directly. Resolve $G$ by a bounded below complex $I^{\bullet }$ of injectives. Then ${\mathrm{Hom}}_A(B,I^{\bullet })$ is bounded below and injective over $B$; the displayed degreewise Hom identity, including the usual total-complex signs, computes the derived adjunction. Bounded below injective complexes compute these derived Homs because maps from acyclic complexes are null-homotopic, constructed successively from the lowest nonzero target degree using injectivity. The adjunction is natural in both arguments.

For a closed immersion, $i_{\ast }$ is exact on all module sheaves: at a point of $X$ its stalk is the original stalk, and outside $X$ the stalk vanishes. Its right adjoint is $i^{b}I=i^{-1}\mathcal{H}o{m}_Y(i_{\ast }{\mathcal{O}}_X,I)$, by the same evaluation formula, and the ideal defining $X$ annihilates this Hom. Since $i_{\ast }$ is exact and $i^b$ is its right adjoint, $i^b$ sends injectives to injectives: for injective $I$ the functor ${\mathrm{Hom}}_{{\mathcal{O}}_X}(-,i^{b}I)\cong {\mathrm{Hom}}_{{\mathcal{O}}_Y}(i_{\ast }-,I)$ is a composition of the exact functor $i_{\ast }$ with the exact functor ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(-,I)$. Restrictions of an injective module sheaf to an open subset remain injective, since extension by zero is exact and left adjoint to restriction. Applying the ringed-space adjunction on every open subset gives the internal Hom identity; applying it on all of $Y$ gives the global Hom identity. Injective resolutions now give the stated derived identities, and their counit is evaluation at $1$.

Extension by zero along a closed subset preserves flasque sheaves and is exact. Computing sheaf cohomology by flasque resolutions therefore identifies $R\mathrm{\Gamma }(Y,i_{\ast }K)$ with $R\mathrm{\Gamma }(X,K)$, first for sheaves and then for bounded complexes by totalizing their resolutions; this is the closed-immersion cohomology comparison of [F1], and the coherence of $i_{\ast }M$ for coherent $M$ is the same comparison read on an affine chart. AC enters through the resolution data; the adjunction formulas themselves involve no selections.

Every finite $A$-module has a bounded resolution by finite projective modules: finite generation of each kernel follows from Noetherianity, and the $n$th syzygy is projective by the global-dimension bound. Bounded complexes with finite cohomology are perfect as well, by truncation triangles and cones of such resolutions. For a bounded finite-projective complex $P$, the ordinary termwise map $P\to {\mathrm{Hom}}_A({\mathrm{Hom}}_A(P,L[s]),L[s])$ is an isomorphism of complexes with the signed evaluation convention: the two shifts and the two factors of $L$ cancel. Thus $R{\mathrm{Hom}}_A(-,L[s])$ has coherent biduality.

By [F1], $L$ has a bounded injective resolution $I^{\bullet }$. Applying ${\mathrm{Hom}}_A(B,-)$ gives a bounded complex of injective $B$-modules by [F2], representing $D_B$. Its cohomology is finite over $A$, since $B$ has a finite projective resolution, and the $B$-action makes it finite over $B$. For bounded finite $B$-cohomology, the adjunction identifies the underlying $A$-complex of ${\mathbb{D}}_B(M)$ with $R{\mathrm{Hom}}_A(M,L[s])$, so it too has bounded finite cohomology.

Apply that adjunction twice. The underlying $A$-complex of ${\mathbb{D}}_B{\mathbb{D}}_B(M)$ becomes the ambient double dual in step 1.1. Under these adjunctions the canonical $B$-evaluation becomes the ambient evaluation: both send an element $m$ to the functional $\phi \mapsto \phi (m)$, followed by evaluation at $1\in B$; resolving gives the same signed identity for complexes. It is therefore a quasi-isomorphism after forgetting the $B$-action, hence a quasi-isomorphism over $B$. At $M=B$, the identification ${\mathbb{D}}_B(B)=D_B$ shows that $B\to R{\mathrm{Hom}}_B(D_B,D_B)$ is precisely the homothety isomorphism. Together with step 2.1 this proves all conditions in [F3] and biduality. Finite projective resolutions show that derived Hom and evaluation commute with localization, so the construction agrees on intersections of affine charts. AC is used in the ambient global-dimension and resolution suppliers.

The quotient's depth over $R$ equals its depth over $B$, since multiplication by a sequence of elements and regularity are unchanged after passing to their images in $B$. Thus [F1] gives ${\mathrm{pd}}_{R}B=N-d=c$, proving vanishing for $q>c$. By [F3], $\mathrm{ht}I=c$: the dimension of $R\mathrm{/}I$ is the maximum of the dimensions $R\mathrm{/}\mathfrak{p}$ for minimal primes $\mathfrak{p}$ over $I$.

Inductively maintain a regular tuple $f_1,\dots ,f_j\in I$ with $Q=R\mathrm{/}(f_1,\dots ,f_j)$ CM of dimension $N-j$; the case $j=0$ is [F1]. For $j<c$, every associated prime of $Q$, viewed in $R$, has quotient dimension $N-j$, hence height $j$ by [F3]. None contains $I$, whose height is $c>j$. Prime avoidance [F2] gives $f_{j+1}\in I$ regular on $Q$. It avoids every minimal prime of $Q$ by [F5], so every prime containing $(f_1,\dots ,f_{j+1})$ strictly contains a height-$j$ minimal prime and has height at least $j+1$. By [F3], $e=\dim (Q\mathrm{/}{f}_{j+1}Q)\le N-j-1$. A parameter tuple of length $e$ on this nonzero quotient, supplied by [F5], lifts together with $f_{j+1}$ to a tuple with finite-length quotient on $Q$. The minimal-length assertion of [F5] gives $N-j\le e+1$, hence $e=N-j-1$ and this lifted tuple is a system of parameters for $Q$. Thus $f_{j+1}$ is a regular parameter element, and [F2] makes the next quotient CM, completing the induction. All quotients are nonzero since the ideals lie in the maximal ideal. This constructs the length-$c$ regular sequence, including the empty tuple if $c=0$.

If $f$ is regular in a ring $Q$ and annihilates a $Q$-module $M$, the two-term resolution $[Q\stackrel{f}{\to }Q]$ of $Q\mathrm{/}fQ$ shows $R{\mathrm{Hom}}_Q(Q\mathrm{/}fQ,Q)\cong (Q\mathrm{/}fQ)[-1]$. Derived adjunction [F4] gives ${\mathrm{Ext}}_Q^q(M,Q)={\mathrm{Ext}}_{Q\mathrm{/}fQ}^{q-1}(M,Q\mathrm{/}fQ)$. Apply this successively to the sequence in step 2.1, all of which annihilate $B$. The result is ${\mathrm{Ext}}_R^q(B,R)={\mathrm{Ext}}_{R\mathrm{/}(f_1,\dots ,f_c)}^{q-c}(B,R\mathrm{/}(f_1,\dots ,f_c))=0$ for $q<c$, since negative Ext between modules vanishes. Along with step 1.1 this proves concentration. An invertible module over a local ring is free of rank one, so the same assertion holds for $L$. AC is inherited from the cited suppliers, including the derived adjunction [F4]; no complete-intersection assumption on $B$ was introduced.

Choose the embedding in [F1]. Resolve ${\omega }_P[N]$ by injective module sheaves and apply $i^b$ from [F3]. This constructs $D_i$ with its actual ${\mathcal{O}}_X$-action, rather than treating an ambient locally free resolution as a complex of ${\mathcal{O}}_X$-modules. On a standard chart $U=\mathrm{Spec}A\subset P$, $X\cap U=\mathrm{Spec}(A\mathrm{/}I)$, and the derived complex corresponds to $R{\mathrm{Hom}}_A(A\mathrm{/}I,{\omega }_P(U)[N])$; finite resolutions in [F2] verify this correspondence and its compatibility with localization.

The rings $A$ are regular of dimension $N$, and the restriction of ${\omega }_P$ is invertible. Hence [F3] proves finite injective dimension, finite cohomology and homothety on $X\cap U$. These affine opens cover $X$. The finite ambient resolutions [F2] bound the cohomology degrees uniformly, and their local Hom cohomology sheaves are coherent, so $D_i\in D_{\mathrm{C}\mathrm{o}\mathrm{h}}^b(X)$ and is dualizing in the sense of Dualizing complexes and the normalized dualizing sheaf on a projective CM scheme. For any bounded coherent $K$, the same local argument gives bounded coherent duals and bidual evaluation is an isomorphism on this cover, hence globally. This proves existence and biduality before any use of duality on singular schemes. AC is used by the listed regularity and resolution suppliers.

By the cohomology calculation [F1], $R\mathrm{\Gamma }(P,{\omega }_P[N])$ has $k$ as its sole cohomology, in degree zero; the Laurent trace identifies it with $k$. Every coherent complex on $P$ is perfect: [F2] gives this for a sheaf, and finite truncation triangles give it for bounded coherent cohomology. Thus derived internal Hom and evaluation are computed locally by bounded finite locally free complexes. Compose the derived cup map and evaluation with the trace to obtain $R\mathrm{\Gamma }(\mathcal{R}\text{\hspace{0.17em}⁣}Hom(K,{\omega }_P[N])){\otimes }_k^{\mathbf{L}}R\mathrm{\Gamma }(K)\to k$. The tensor-Hom adjoint is the map in the statement. It is natural and respects triangles because it is induced by chain evaluation and the signed total-complex differential.

For $K={\mathcal{O}}_P(m)$, the map on degree $a$ cohomology is precisely $H^{N+a}(P,\mathcal{O}(-m-N-1))\to H^{-a}(P,\mathcal{O}(m){)}^{\vee }$, which is an isomorphism in all degrees by [F1], including the zero groups outside their ranges. Hence the map is a quasi-isomorphism for twists and their finite direct sums and shifts. Both functors in the assertion take triangles to triangles contravariantly; exact duality of vector spaces makes the right-hand cohomology equal to the dual of the opposite-degree cohomology. The long exact sequences of [F3] and the five lemma [F3] extend the isomorphism across a cone. Apply this finitely many times to a resolution in [F2], then to truncation triangles of a bounded coherent complex. This proves the assertion for every $K$ and supplies compatibility with connecting maps. AC enters through [F2] and the derived-category and product suppliers.

The adjunction [F2] identifies $R{\mathrm{Hom}}_X(K,D_i)$ with $R{\mathrm{Hom}}_P(i_{\ast }K,{\omega }_P[N])$. The pushforward $i_{\ast }K$ is bounded coherent, so [F3] identifies the latter with $R{\mathrm{Hom}}_k(R\mathrm{\Gamma }(P,i_{\ast }K),k)$, which equals $R{\mathrm{Hom}}_k(R\mathrm{\Gamma }(X,K),k)$ by [F2]. Cohomology in degree $r$ gives the displayed formula. The comparison is the evaluation pairing: for maps $\alpha :K\to D_i[r]$ and $\beta :{\mathcal{O}}_X\to K[-r]$, adjunction takes $\alpha$ to the ambient map followed by the counit. It takes $\beta$ followed by $i_{\ast }{\mathcal{O}}_X$ to the same ambient composition; pulling back the unit ${\mathcal{O}}_P\to i_{\ast }{\mathcal{O}}_X$ makes the equality immediate by evaluation at $1$. Thus the paired scalar is $t_i(\alpha [-r]\circ \beta )$, not merely an unspecified vector-space isomorphism.

At $r=0$, each embedding gives a representation on $D_{\mathrm{C}\mathrm{o}\mathrm{h}}^b(X)$ of the same contravariant functor $K\mapsto {\mathrm{Hom}}_k(H^0(X,K),k)$. Both representing objects lie in that category by [F1]. The Yoneda evaluation bijection [F3] supplies a unique isomorphism: evaluate the natural comparison at the first representing object on its identity, and do the reverse at the second; naturality says that the two composites are identities. The trace itself is recovered by evaluating the represented functional for a map ${\mathcal{O}}_X\to D_i$ at $1\in H^0(X,{\mathcal{O}}_X)$, so this isomorphism preserves it. Naturality under shifts then preserves every degree and triangle comparison. These unique isomorphisms compose transitively, proving independence of the embedding with its normalization; an unnormalized dualizing complex alone is not claimed to be uniquely isomorphic. AC is retained from [F1]–[F3].

At a closed point $x\in X$, its residue field is finite algebraic over $k$ by Finite-type maps from Jacobson rings induce finite residue-field extensions at maximal ideals, so [F3] gives $\dim {\mathcal{O}}_{X,x}=d$ and $\dim {\mathcal{O}}_{P,x}=N$. The local ring of $P$ is a localization of a polynomial chart. CM gives depth $d$ for the quotient. Hence [F2], with the local trivialization of ${\omega }_P$, shows that ${\mathcal{H}}^a(D_X{)}_x=0$ unless $a=(N-d)-N=-d$. Every cohomology sheaf is coherent by [F1]. A nonzero coherent sheaf on a finite-type scheme over a field has a closed point in its support: on an affine chart its support is the vanishing locus of its annihilator, and a maximal ideal of this quotient has finite algebraic residue field; such points are closed in the finite-type scheme. Thus the asserted vanishing at closed points proves vanishing everywhere. Canonical truncation gives $D_X\cong {\omega }_X[d]$, and the construction gives the ambient sheaf Ext formula.

At a closed point set $R={\mathcal{O}}_{P,x}$, $B={\mathcal{O}}_{X,x}$ and $c=N-d$. The proof of [F2] gives a finite free resolution of $B$ of length $c$. Dualizing it over $R$ gives an exact sequence $0\to F_0^{\vee }\to \cdots \to F_c^{\vee }\to E\to 0$, where $E={\mathrm{Ext}}_R^c(B,R)$; for $c=0$ this means $E=B^{\vee }$. Thus ${\mathrm{pd}}_{R}E\le c$. The module $E$ is nonzero: $R{\mathrm{Hom}}_R(B,R)$ is nonzero, since if it vanished then the biduality $B\cong R{\mathrm{Hom}}_R(R{\mathrm{Hom}}_R(B,R),R)$ of [F1] would give $B=0$, contrary to $B\mathrm{\ne }0$; and [F2] identifies this complex with $E[-c]$. It is annihilated by $I$, so its support has dimension at most $d$. By [F4], ${\mathrm{depth}}_{R}E=N-{\mathrm{pd}}_{R}E\ge d$; depth is at most support dimension by A finite local module has depth at most its dimension, so both equal $d$. Regularity over $R$ and over $B$ is measured by the same multiplication maps; hence $E$ is CM over $B$. Its localizations are CM by [F4], and every point specializes to a closed point, so ${\omega }_X$ is CM everywhere. Finally at any point $y$, if $({\omega }_X{)}_y$ vanished then $(D_X{)}_y$ would vanish, contradicting the local homothety for nonzero ${\mathcal{O}}_{X,y}$; therefore its support is all of $X$. AC is inherited through the dimension, resolution and CM suppliers.

Apply [F1] to $F$ with $r=-i$. Since $D_X={\omega }_X[d]$ by [F2], its left side becomes ${\mathrm{Hom}}_{D(X)}(F,{\omega }_X[d-i])={\mathrm{Ext}}_X^{d-i}(F,{\omega }_X)$; its right side is ${\mathrm{Hom}}_k(H^i(X,F),k)$. For $\eta \in H^i(X,F)$, [F3] represents it by $\beta :{\mathcal{O}}_X\to F[i]$. A class $\alpha :F\to {\omega }_X[d-i]$ pairs with it by $t_X(\alpha [i]\circ \beta )$. This is the Yoneda composition of [F3] and is exactly the evaluation comparison proved in [F1]. Thus it is bilinear, natural in $F$, and compatible with shifts and connecting maps.

Apply [F4] to the closed subscheme $X\hookrightarrow {\mathbb{P}}_k^N$ supplied by a projective embedding: its finite-dimensionality clause gives that each $H^i(X,F)$ is a finite-dimensional $k$-vector space. The isomorphism in step 1.1 proves that the adjoint map from Ext to the dual of cohomology is an isomorphism. For a finite-dimensional space, evaluation into its double dual is an isomorphism, so the other adjoint map is an isomorphism as well. Outside $0\le i\le d$, cohomology vanishes by [F4] and the negative-degree convention; step 1.1 then proves the corresponding Ext vanishing. For bounded coherent $K$, retain the actual global derived comparison used in [F1]: closed-immersion adjunction gives $R{\mathrm{Hom}}_X(K,D_X)\cong R{\mathrm{Hom}}_P(i_{\ast }K,{\omega }_P[N])$, and the projective-space evaluation/trace map identifies the latter with $R{\mathrm{Hom}}_k(R\mathrm{\Gamma }(P,i_{\ast }K),k)=R{\mathrm{Hom}}_k(R\mathrm{\Gamma }(X,K),k)$. This comparison is an isomorphism in $D(k)$ induced by the same counit and trace, so its naturality and signs are those of [F1]. The biduality $K\cong \mathcal{R}\text{\hspace{0.17em}⁣}Ho{m}_X(\mathcal{R}\text{\hspace{0.17em}⁣}Ho{m}_X(K,D_X),D_X)$ is the coherent biduality of the embedding construction [F1], valid after pushing into the ambient projective space, without imposing perfectness on $K$ over the singular scheme. AC enters through the cited resolution, derived-composition and cohomology suppliers.