Let F be a quasi-coherent sheaf on a smooth projective variety X over an algebraically closed field and $D_X(?)=RHom(?,\omega _X[n])$ the dualizing functor. Is it the case that $D_X(D_X(F))$ is a sheaf? (i.e. $\mathcal H^n(D_X(D_X(F)))=0$ for $n\ne 0$). If so is there a convenient reference?
[Math] Is the derived double dual of a quasi-coherent sheaf a sheaf
ag.algebraic-geometry
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The answer to your question is positive and follows from Theorem 6.4.32 in Qing Liu's book Algebraic geometry and arithmetic curves.
Note that Liu uses Corollary 6.4.13 in the statement of his Theorem. Moreover, the base scheme is a locally Noetherian scheme, e.g., the spectrum of a field.
Let $\mathcal M$ be a left $\mathcal D$-module over a smooth curve $X$. Then the Koszul duality functor assigns to $\mathcal M$ the DG-module $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ over the de Rham complex $\Omega_X$. Viewed as a complex of sheaves, $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ is a two-term complex whose terms are quasi-coherent $\mathcal O_X$-modules, but the differential is not $\mathcal O_X$-linear. For this reason, even when $X$ is affine, one has to distinguish between the acyclicity of the complex of global sections of $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ and the acyclicity of this complex of sheaves itself.
Explicitly, let $x$ be a global coordinate on $X$ (assuming that one exists). Then the differential in the complex $\Omega_X\otimes_{\mathcal O_X}\mathcal M = (\mathcal M\to \Omega_X^1\otimes_{\mathcal O_X}\mathcal M)$ has the form $m\mapsto dx\otimes \partial/\partial x(m)$. The question about acyclicity of this complex of sheaves is, therefore, the question about injectivity and surjectivity of the operator $\partial/\partial x$ acting in the sections of $\mathcal M$.
In the case at hand, we have $\mathcal M=\mathcal O_X e^x$. So global sections of $\mathcal M$ over $\operatorname{Spec} \mathbb C[x]$ are expressions of the form $p(x)e^x$, where $p(x)$ is a polynomial in $x$. Hence one can easily see that the complex of global sections of $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ over $\operatorname{Spec} \mathbb C[x]$ is acyclic.
The complexes of sections of $\operatorname{Spec} \mathbb C[x]$ over Zariski open subsets of $\operatorname{Spec} \mathbb C[x]$ are not acyclic, however. It suffices to consider sections over $\operatorname{Spec} \mathbb C[x,x^{-1}]$. As is well known, the function $x^{-1}e^x$ does not lie in the image of $\partial/\partial x$ acting in the space of Laureant polynomials in $x$ multiplied with $e^x$. Therefore, the two-term complex of sheaves $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ is not acyclic. Its differential is an injective, but not surjective morphism of sheaves of $\mathbb C$-vector spaces over $\operatorname{Spec} \mathbb C[x]$.
On the other hand, one can consider the complex of sheaves of analytic forms $\Omega_X^{an}\otimes_{\mathcal O_X}\mathcal M$ in the analytic topology of the set of closed points of $\operatorname{Spec} \mathbb C[x]$. Then every function from $\mathcal O_X^{an}e^x$ will have a primitive analytic function locally in the analytic topology. So the differential in the two-term complex of sheaves $\Omega_X^{an}\otimes_{\mathcal O_X}\mathcal M$ is now surjective. However, it is no longer injective, as the constant functions are sections of $\mathcal O_X^{an}e^x$, the function $e^{-x}$ being analytic. This is what Kapranov is doing in his paper.
Having, as I hope, answered your question, let me now point out, as a side note, that what I would consider a superior alternative of the $\mathcal D{-}\Omega$ duality theory of the Beilinson--Drinfeld preprint can be found in Appendix B to my AMS Memoir "Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence", http://arxiv.org/abs/0905.2621 .
The point is that what you call "$\mathcal D$-quasi-isomorphisms" are defined in my paper in terms intrinsic to DG-modules over $\Omega$ (without any reference to differential operators). The corresponding localization is called the coderived category of quasi-coherent DG-modules over $\Omega_X$, and subsequently it is proven to be equivalent to the derived category of quasi-coherent $\mathcal D_X$-modules (for a smooth variety $X$ over any field, $\mathcal D_X$ denoting the crystalline differential operators in the case when the characteristic is finite).
Best Answer
For coherent sheaves, by using resolutions, one checks that the assertion in question is true: for any perfect complex $K$, one even has $D_X^2(K) \simeq K$. However, for larger quasicoherent sheaves $F$, the value $D_X^2(F)$ fails to even be a sheaf; I give an example below in Corollary 5.
Fix a countable set $I$, and a PID $A$ with at least two non-zero irreducible elements $p,q \in A$ such that $(p,q) = (1)$. We begin with the following basic observation that lies at the heart of the argument:
Lemma 1: One has $Hom_A(\prod_I A,A) \simeq \oplus_I A$ via the natural map in $D(A)$.
Proof: See Is it true that, as $\Bbb Z$-modules, the polynomial ring and the power series ring over integers are dual to each other?.
This Lemma shows that $\prod_I A$ is not projective. More precisely, we get:
Lemma 2: The complex $K := RHom_A(\prod_I A,A)$ has non-zero $H^1$.
Proof: If $k$ is a residue field of $A$ at a maximal ideal (ex: $k = A/(p)$), then $k$ is $A$-perfect, so $- \otimes_A k$ commutes with all (homotopy) limits and colimits. This gives $K \otimes_A k \simeq Hom_k(\prod_I k, k)$, which is a vector space of uncountable dimension. On the other hand, by Lemma 1, if we assume that $H^1(K) = 0$, then we get $K \otimes_A k \simeq \oplus_I k$, which is a vector space of countable dimension; so we conclude $H^1(K) \neq 0$.
Now we globalize. Set $X = \mathbf{P}^1_k$ over some field $k$. All complexes below are considered as living in the derived category $D(O_X)$ of all $O_X$-modules on $X$, and similarly for functors. (I prefer this to using $D_{qc}(O_X)$ as the description of products in the latter is more subtle.) The global analogue of Lemma 2 is:
Lemma 3: The complex $K := RHom_X(\prod_I O_X,O_X)$ has a non-zero $H^1$.
Proof: This does not formally follow from Lemma 2 as $\prod_I O_X$ is not quasi-coherent. However, we can argue as follows: (a) first show that $H^0(K) := Hom_X(\prod_I O_X,O_X) \simeq \oplus_I O_X$ via the natural map, and (b) show that $K \otimes_{O_X} k \simeq Hom_k(\prod_I k,k)$ is a vector space of uncountable dimension, where $k$ is the residue field at $0$ viewed as an $O_X$-complex. Then (a) and (b) immediately imply the claim by the argument of Lemma 2. Moreover, (b) is clear since $k$ is perfect over $O_X$, so it remains to show (a). Consider the natural map $\eta:\oplus_I O_X \to Hom(\prod_I O_X,O_X)$. Clearly $\eta$ is injective. For surjectivity, fix some $f:\prod_I O_X \to O_X$. We want to show that $f$ has finite support, i.e., that $f$ factors through a projection to finitely many components of the product on the left. For any affine open $U \subset X$, the map $f(U)$ has finite support by Lemma 1. A glueing argument then finishes the proof.
Writing a product as the dual of a sum then gives:
Lemma 4: The complex $RHom_X(RHom_X(\oplus_I O_X,O_X),O_X)$ has non-zero cohomology in degree $0$ and $1$.
Proof: This follows formally from Lemma 3 as $RHom_X(-,O_X)$ carries coproducts to products.
Here is the promised example:
Corollary 5: On $X = \mathbf{P}^1_k$, if $D_X = RHom_X(-,\omega_X[1])$, then the functor $D_X^2$ carries the sheaf $\oplus_I O_X$ to a complex with non-zero $H^0$ and $H^1$.
Proof: As $\omega_X$ is invertible, one has $D_X = RHom_X(-,O_X) \otimes \omega_X[1]$, and hence $D_X^2 = RHom_X(RHom_X(-,O_X),O_X)$, so the claim follows form Lemma 4.