Serre duality on schemes

 Lecture topic

This post goes through the statement and proof of Serre duality for arbitrary projective schemes, as presented in Chapter III.7 of Hartshorne. Only the necessary tools and definitions to prove the statement are introduced.

Recall a scheme is a topological space $X$ and a sheaf of rings $\mathcal O_X$ such that for every open set $U\subset X$, $\mathcal O_X(U)\cong \Spec(R)$ for some ring $R$. Its dimension is its dimension as a topological space. A projective scheme is a scheme where $X\subset \P^n$. A sheaf (or scheme) over a scheme $X$ is a sheaf (or scheme) $Y$ and a morphism $Y\to X$. Recall also the sheafification $\widetilde {\mathcal F}$ of a presheaf $\mathcal F$.

Definition 1: Let $\mathcal F$ be a sheaf over a projective scheme $X$. Then $\mathcal F$ is

• proper if it is the image of a proper morphism (separated, finite type, universally closed),
• quasi-coherent if there exists a cover $\{U_i=\Spec(A_i)\}$ of $X$ such that $\mathcal F|_{U_i} = \widetilde{M_i}$ for some $A_i$-module $M_i$,
• coherent if it is quasi-coherent and each $M_i$ is finitely-generated as an $A_i$-module,
• locally free if for every $x\in X$, there exists $U\owns x$ open such that $\mathcal F|_U = \bigoplus_{i\in I} \mathcal O_X|_U$,
• very ample if there is an immersion $i:X\to \P^n$ for some $n$ such that $i^*\mathcal O(1) \cong \mathcal F$.

Often we say $\mathcal F$ is very ample if it has "enough sections," as $\P^n$ has many sections.

Remark 1: Recall some basic definitions of the $\Ext$ functor. Let $\mathcal F,\mathcal G$ be sheaves of $\mathcal O_X$-modules, and $\mathcal L$ a locally free sheaf of finite rank. Then:

1. $\Ext^i(\mathcal O_X,\mathcal F) \cong H^i(X,\mathcal F)$ for all $i\>0$ (Proposition III.6.3)
2. $\Ext^i(\mathcal F\otimes \mathcal L,\mathcal G) \cong \Ext^i(F,\mathcal L^\vee\otimes \mathcal G)$ (Proposition III.6.7)
3. $\Ext^i_{\mathcal O_X}(\mathcal F_x,\mathcal G_x) \cong \mathcal Ext (\mathcal F,\mathcal G)_x$ (Proposition III.6.8)
4. $\Ext^i(\mathcal F,\mathcal G(q)) \cong \Gamma(X,\mathcal Ext^i(\mathcal F,\mathcal G(q))$ (Proposition III.6.9)
5. $\mathcal Ext^i(\mathcal F\otimes \mathcal L,\mathcal G) \cong \mathcal Ext^i(F,\mathcal L^\vee\otimes \mathcal G) \cong \mathcal Ext^i(\mathcal F,\mathcal G)\otimes \mathcal L^\vee$
6. $\mathcal Ext^0(\mathcal O_X,\mathcal F) \cong \mathcal F$
7. $\mathcal Ext^i(\mathcal O_X,\mathcal F) \cong 0$ for all $i>0$

Recall that a local ring of a scheme $X$ is $\mathcal O_{X,x}$ for $x\in X$. It is equivalently a ring with a unique maximal left or right ideal. A regular local ring is a local ring $R$ whose maximal ideal is generated by $\dim(R)$ elements.

Preliminary definitions and lemmas

Let $A,B$ be abelian categories (recall this means kernels and cokernels exist).

Definition 2: A $\delta$-functor between $A$ and $B$ is a collection of functors $T^i:A\to B$ that generalize derived functors, in the sense that $R^i\mathcal F = T^i$. A $\delta$-functor is universal if for any other $\delta$-functor $U$, there is a natural transformation $f:T^0\to U^0$ that induces a unique collection of morphisms $f^{i\geqslant 0}:T^i\to U^i$ that extend $f$.

See Weibel for a more thorough definition (and Grothendieck for the original setting). These functors may be covariant or contravariant, homological or cohomological. Note that $\delta$-functors are unique up to isomorphism.

Definition 3: Let $F:A\to B$ be a functor. $F$ is effaceable if for every $X\in A$ there exists a monomorphism $u\in \Hom_A(X,Y)$ such that $F(u)=0$. Similarly, $F$ is coeffaceable if for every $X\in A$ there exists an epimorphism $v\in \Hom_A(Y,X)$ such that $F(v)=0$.

Lemma 1: If a covariant (or contravariant) cohomological $\delta$-functor is effaceable for every $i>0$, then it is universal. Similarly, if a covariant (or contravariant) homological $\delta$-functor is coeffaceable for every $i>0$, then it is universal.

This appears as Proposition II.2.2.1 in Grothendieck and Exercise 2.4.5 in Weibel. Now let $\mathcal F$ be a sheaf over a projective scheme $X$.

Lemma 2: (Theorem III.5.2 in Hartshorne) If $\mathcal F$ is coherent, there is $q\gg 0$ such that $H^i(X,\mathcal F(q))=0$ all $i>0$.

Definition 4: The dualizing sheaf of $X$ is a coherent sheaf $\omega_X^\circ$ and a trace map $t:H^n(X,\omega_X^\circ) \to k$ such that the isomorphism $\Hom(\mathcal F,\omega_X^\circ)\to H^n(X,\mathcal F)^\vee$ is induced by the natural pairing
$$\Hom(\mathcal F,\omega_X^\circ)\times H^n(X,\mathcal F) \to H^n(X,\omega_X^\circ)$$
composed with $t$.

Lemma 3: (Corollary II.5.18 in Hartshorne) If $\mathcal F$ is coherent, then it is a quotient of $\bigoplus_{i=1}^N \mathcal O_X(-q)$ for $q\gg 0$.

Next we recall some ring theory. Let $A$ be a ring and $M$ an $A$-module.

Definition 5: A sequence $a_1,\dots,a_n\in M$ is $M$-regular if $a_i$ is not a zero divisor of $M/(a_1,\dots,a_{i-1})M$ and $M\neq (a_1,\dots,a_i)M$ for all $i$. The depth of $M$ is the maximal length of an $M$-regular sequence of elements in some maximal ideal $\mathfrak m\leqslant M$. A local Noetherian ring is Cohen-Macaulay if $\text{depth}(A)=\dim(A)$, where dimension is Krull dimension (maximal length of prime ideal chains). A scheme $X$ is Cohen-Macaulay if every point $x\in X$ has a neighborhood $U$ such that the local ring $\mathcal O_X(U)$ is Cohen--Macaulay.

Lemma 4: Let $A$ be a regular local ring of dimension $n$ and $M,N$ be $A$-modules. Then:

1. $\text{pd}(M)\leqslant n$ iff $\Ext^i(M,N)=0$ for all $i>n$ (Proposition III.6.10)
2. $\text{pd}(M)+\text{depth}(M)=n$ if $M$ is f.g. (Proposition III.6.12A)

Main theorem and proof

First we state the duality theorem for $X=\P^n$, without proof. Let $\omega_X$ be the canonical sheaf of $X$.

Theorem 1: (Theorem III.7.1 in Hartshorne) For $\mathcal F$ coherent over $\P^n$, for $i\geqslant 0$ there are natural isomorphisms
$$\Hom(\mathcal F,\omega_X) \cong H^n(X,\mathcal F)^\vee, \hspace{2cm} \Ext^i(\mathcal F,\omega) \cong H^{n-i}(X,\mathcal F)^\vee.$$

Now we give the duality theorem for an arbitrary projective scheme, going through the proof as in Hartshorne.

Theorem 2: (Theorem III.7.6 in Hartshorne) Let $X$ be a projective scheme of dimension $n$ such that $\mathcal O(1)$ is very ample. For $\mathcal F$ coherent,
$$\begin{align*} \Ext^i(\mathcal F, \omega_X^\circ) \cong H^{n-i}(X,\mathcal F)^\vee &\ \iff\ H^i(X,\mathcal F(-q))=0 \text{ for all $\mathcal F$ locally free, }i<n,q\gg0,\\ &\ \iff\ X\text{ is CM and equidimensional.} \end{align*}$$

Proof: Natural maps $\Ext^i(\mathcal F,\omega^\circ_X)\to H^{n-i}(X,\mathcal F)^\vee$ exist, as $\Ext^i(-,\omega_X^\circ):\text{Coh}(X)\to \text{Mod}$ is a coeffaceable $\delta$-functor for every $i>0$, hence universal by Lemma 1. Indeed, by Lemma 3, we have a surjection

$$\underbrace{\bigoplus_{j=1}^N \mathcal O_X(-q)}_{\mathcal E}\tov u \mathcal F\to 0,$$
for which
$$\Ext^i(\mathcal E,\omega_X^\circ) = \bigoplus_{j=1}^N \Ext^i(\mathcal O_X(-q),\omega_X^\circ)= \bigoplus_{j=1}^N \Ext^i(\mathcal O_X,\omega_X^\circ (q)) = 0$$
for $i>0$. The first equality was distributing $\Ext^i$ over the sum and the second was by applying Remark 1.2. Hence $\Ext^i(-,\omega_X^\circ)(u)=0$ for $i>0$, so the functor is coeffeaceable for $i>0$, and so universal. By Definition 2 there exist maps generalizing the map $\Ext^0$ from Definition 4.

First iff $\Leftarrow$: Since universal $\delta$-functors are unique (up to isomorphism), we show $H^{n-i}(X,-)^\vee:\text{Coh}(X)\to \text{Mod}$ is also universal contravariant, which follows as it is coeffaceable for $i>0$. Using the same sequence and sheaf as in the equation above, we have that
$$H^{n-i}(X,\mathcal E) = \bigoplus_{j=1}^N H^{n-i}(X,\mathcal O_X(-q)) = 0$$
whenever $n-i<n$ by hypothesis, or equivalently, when $i>0$. The dual module is then also zero for $i>0$, so we are done.

First iff $\Rightarrow$: Assume the hypothesis with index $n-i$ and a locally free sheaf $\mathcal F(-q)$ for $q\gg 0$, for which
$$\begin{align*} H^i(X,\mathcal F(-q))^\vee & \cong \Ext^{n-i}(\mathcal F(-q),\omega_X^\circ) & (\text{hypothesis}) \\ & \cong \Ext^{n-i}(\mathcal O_X,\mathcal F^\vee \otimes \mathcal O_X(q) \otimes\omega_X^\circ) & (\text{Remark 1.2}) \\ & \cong H^{n-i}(X,(\mathcal F^\vee \otimes\omega_X^\circ)\otimes \mathcal O_X(q)). & (\text{Remark 1.1}) \end{align*}$$
Tensoring with $\mathcal O_X(q)$ is twisting by $q$, and Lemma 2 says that $H^{n-i}(X,\mathcal G(q))=0$ for $\mathcal G$ coherent, for all $n-i>0$, for $q$ large enough. So for $i<n$ and $q$ large enough $H^i(X,\mathcal F(-q))^\vee=0$, and so its dual, the original cohomology group, is also trivial.

Second iff $\Leftarrow$: Embed $X\hookrightarrow \P^N$. As $X$ is Cohen--Macaulay and equidimensional of dimension $n$, for $\mathcal F$ locally free on $X$, a stalk $\mathcal F_x$ of a closed point $x\in X$ has depth $n$. Also, $\mathcal F_x\subset \mathcal O_{\P^N,x}$, and $\mathcal O_{\P^n,x}$ is regular as $\P^N$ is smooth over $k$. By Lemma 4.2, we have that
$$\text{pd}(\mathcal F_x) +n \leqslant \text{pd}(\mathcal O_{\P^N,x})+n=N,$$
so Lemma 4.1 and Remark 1.3 gives us that, for $i>N-n$,
$$\Ext^i(\mathcal F_x,-)=0 \ \ \implies\ \ \mathcal Ext^i(\mathcal F_x,-)=0 \ \ \implies\ \ \mathcal Ext^i(\mathcal F,-)=0.$$
Applying Theorem 1, Remark 1.4, and letting the functor $\mathcal Ext^i(\mathcal F,-)$ act on $\omega_{\P^N}(q)$, we have
$$H^i(X,\mathcal F(-q))^\vee \cong \Ext_{\P^n}^{N-i}(\mathcal F,\omega_{\P^N}(q)) \cong \Gamma(\P^N,\mathcal Ext_{\P^N}^{N-i}(\mathcal F, \omega_{\P^n}(q))) \cong \Gamma(\P^N, 0) = 0$$
for $q\gg0$ and $N-i>N-n$, or $i<n$. Since the dual is trivial, the cohomology group $H^i(X,\mathcal F(-q))$ is also trivial.

Second iff $\Rightarrow$: Omitted (techniques are similar to previous step, but use many others not used elsewhere). $\square$

Addendum

In certain cases, Serre duality holds for the canonical sheaf instead of the dualizing sheaf.

Proposition 1: For $X$ a smooth projective variety over $k=\overline k$, $\omega_X^\circ \cong \omega_X$.

References: Grothendieck (Tohoku paper), Hartshorne (Algebraic Geometry, Section III.7), Weibel (An introduction to homological algebra, Section 2.1), Matsumura (Commutative algebra, Chapter 6)

Generalizing planar detection to k-plane detection

In this post the planar detection algorithm in $\R^3$ of Bauer and Polthier in Detection of Planar Regions in Volume Data for Topology Optimization is generalized to detect $k$-planes with largest density in $\R^n$. Let $\Omega\subset \R^n$ be the compact support of a piecewise-constant probability density function $\rho:\R^n\to \R_{\geqslant 0}$.

Definition: Let $(G,\rho)$ be a grid, where $G \subset \lambda \Z^n + c \subset \R^n$ is a lattice in $\Omega$.  A cell $x$ of the grid is $B_\infty(x,\lambda/2) = \{y\in \R^n\ :\ ||x-y||_\infty\leqslant\lambda/2\}$, for $x\in G$. Every cell is assigned a value
$$\int_{B_\infty(x,\lambda/2)}\rho\ dx,$$
called the mass of the cell, which may be though of as a type of Radon transform of $\rho$.

Assuming that $k$ is a global variable, running $Recursive(G,w,k)$ will give the desired result. This algorithm is the naive generalization of Bauer and Polthier, and suffers from calculating mass along the same $k$-plane several times, whenever $k<n-1$ (as any $k$-plane does not lie in a unique $(k+1)$-plane).

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Algorithm: $k$PlaneFinder
$Recursive(G,w,k)$:
Input: A grid $(G,\rho)$, a width $w$ of fattened $k$-planes, the current plane dimension $k\leqslant k'<n$
Output: A $k$-planar connected component covering most mass in $G$

discretize the unit $(k'-1)$-hemisphere in an appropriate manner
order the vertices by a Hamiltonian path
for each vertex $\textbf{n}$:
    sort the grid cells in direction $\textbf{n}$
    discretize the range in direction $\textbf{n}$ equidistantly
    for each $k'$-plane $(\textbf{n},d)$:
        collect the cells closer than $w$ to the $k'$-plane into a graph $G'$
        if $k'\neq k$:
            run $Recursive(G',w,k'-1)$
        else:
            compute the connected component having the most mass in $G'$
return the connected $k$-component having most mass (and the corresponding $k$-plane)

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Measuring along connected components of a $k$-plane works the same way as in the original version, as the gird on $\R^n$ similarly induces a connectivity graph.

Remark: Bauer and Polthier cite Kantaforoush and Shahshahani in evenly sampling points on the unit 2-sphere, but it is not clear how their method (using the inscribed icosahedron) generalizes. Another method would be uniformly sampling random points on $S^{k-1}$ and take all on one hemisphere. A Hamiltonian path could then be taken from an arbitrary point and then using the greedy algorithm (with respect to Euclidean distance) to find consecutive vertices (to keep down the time of consecutive sorting operations).

Recall the Grassmannian $Gr(n,k)$ of all $k$-planes in $\R^n$ through the origin, a compact manifold of dimension $k(n-k)$. Note that any $k$-plane $P\subset \R^n$ is a translation of an element $Q\in Gr(n,k)$ by an element of $Q^\perp$ (we conflate notation for $Q$ and its natural embedding in $\R^n$).

Remark: $Gr(n,k)$ is parametrizable, so by choosing directions in the unit $(n-k)$-hemisphere, the process of choosing $k$-planes in the algorithm may be completely parametrized. The quick sorting of points that was available in Bauer and Polthier's $n=3,k=2$ case may be replaced by an iterated restriction of the original data set through a complete flag $P\subset \cdots \subset \R^n$.

References: Bauer and Polthier (Detection of Planar Regions in Volume Data for Topology Optimization), Katanforoush and Shahshahani (Distributing points on the Sphere 1)