text:
\section{Introduction}

A function $\phi$ in the local ring ${\mathcal O}_0$ in one
complex variable belongs to the ideal $J$ generated by $z^m$
if and only if
$$
{\mathcal L}_\ell\phi(0)=0,\  \ell=0,\ldots, m-1,
$$
where  ${\mathcal L}_\ell=\partial^\ell/ \partial z^\ell$.
These conditions  can be elegantly expressed  by the single
equation $\phi\bar\partial(1/z^m)=0$,
where $1/z^m$ is the usual principal value distribution. Moreover,
the current $\mu=\bar\partial(1/z^m)$ is canonical
up to a non-vanishing holomorphic factor. There is a well-known
multivariable generalization.
Let $f=(f^1,\ldots,f^p)$  be a tuple of
holomorphic functions in a neighborhood  of the origin in ${\mathbb C}^n$ that
defines  a complete   intersection, i.e., the codimension
of $Z^f=\{f=0\}$ is equal to $p$. Then the Coleff-Herrera product
$$
\mu^f=\bar\partial\frac{1}{f^1}{\wedge}\ldots{\wedge}\bar\partial\frac{1}{f^p},
$$
introduced in \cite{CH}, is a  $\bar\partial$-closed $(0,p)$-current  with support on  $Z^f$,
and it is independent (up to a nonvanishing holomorphic factor) of the
choice of generators of the ideal ${\mathcal J}$ generated by $f$.
It was proved in \cite{DS} and
\cite{P1}  that the sheaf ${\mathcal J}$ coincides with the ideal sheaf
${\rm ann\,}\mu^f$ of holomorphic functions $\phi$ such that  the current
$\mu^f\phi$ vanishes. This is often referred to as the duality principle.

\smallskip

The Coleff-Herrera product  is the model for a general
Coleff-Herrera current:  Given a variety $Z$ of pure codimension
$p$ we say that a (possibly vector-valued) $(0,p)$-current $\mu$ (with support on $Z$)
is a Coleff-Herrera current on $Z$, $\mu\in {\mathcal{CH}}_Z$, if it is $\bar\partial$-closed,
annihilated by $\bar {\mathcal I}_Z$ (i.e., $\bar \xi\mu=0$ for each holomorphic
$\xi$ that vanishes on $Z$),
and has the standard extension property SEP. This means, roughly speaking, that
 $\mu$ has no ``mass'' concentrated on any subvariety of higher codimension;
in particular that $\mu$ is determined by its values on $Z_{reg}$, see, e.g.,
\cite{JEB4} or \cite{A12}. The SEP  implies that
${\rm ann\,} \mu$ has pure dimension, see, e.g., Proposition~5.3 in \cite{AW2}.
The condition $\bar{\mathcal I}_Z\mu=0$ means that $\mu$ only involves holomorphic
derivatives;  following Bj\"ork, see  \cite{JEB4} or  Section~\ref{noe} below,
one can quite easily   find  a finite number of
holomorphic differential operators ${\mathcal L}_\ell$ such that
$\phi\mu=0$ if and only if
${\mathcal L}_1\phi=\cdots ={\mathcal L}_\nu\phi=0$ on $Z$; i.e.,
a (complete) set of Noetherian  operators for
${\rm ann\,}\mu$.

In this paper we use the residue theory developed in \cite{AW1} and \cite{AW2}
to extend the  duality for a complete intersection to a general
pure-dimensional ideal (or submodule) sheaf.
In particular we can express  such an ideal
as the annihilator of a  finite set of Coleff-Herrera currents
(Theorem~\ref{thm1} and its corollaries).
Jan-Erik Bj\"ork has pointed out
to us that one can obtain the same duality result from a fundamental theorem of
Jan-Erik Roos, \cite{Roos},  about purity for a module in terms
of the bidualizing sheaves,  combined with some  other known facts that
will be described below. However our approach gives a representation
of the duality and the Coleff-Herrera currents in terms of
one basic residue current, that we first describe.

\smallskip

To begin with, let ${\mathcal J}$ be any  coherent subsheaf of a locally free sheaf ${\mathcal O}(E_0)$
over a complex manifold $X$, and assume that
\begin{equation}\label{acomplex}
0\to {\mathcal O}(E_N)\stackrel{f_N}{\longrightarrow}\ldots
\stackrel{f_3}{\longrightarrow} {\mathcal O}(E_2)\stackrel{f_2}{\longrightarrow}
{\mathcal O}(E_1)\stackrel{f_1}{\longrightarrow}{\mathcal O}(E_0)
\end{equation}
is  a locally free resolution of ${\mathcal F}={\mathcal O}(E_0)/{\mathcal J}$.
Here ${\mathcal O}(E_k)$ is the locally free sheaf
associated to the vector bundle $E_k$ over $X$.
If $X$ is Stein,
then one can find such a resolution in a neighborhood of any given compact
subset.
We will  assume that ${\mathcal F}$ has codimension $p>0$;   cf., Remark~\ref{pnoll}.
Then $f_1$ is (can be assumed to be) generically surjective,
and the analytic set $Z$ where it
is not surjective has codimension  $p$ and coincides with the
zero set of the ideal sheaf ${\rm ann\,}{\mathcal F}$.
In \cite{AW1} we defined, given Hermitian metrics
on $E_k$,  a residue current $R=R_p+R_{p+1}+\cdots $  with support on $Z$,
where $R_k$ is a $(0,k)$-current that takes values in
${\rm Hom\, }(E_0,E_k)$, such that a holomorphic section
$\phi\in{\mathcal O}(E_0)$ is in ${\mathcal J}$ if and only if  $R\phi=0$.
Recall that ${\mathcal F}$ has {\it pure} codimension $p$ if
the associated prime ideals (of each stalk) all have codimension $p$.

The starting point in this paper is the following
result that follows from  \cite{AW2}
(see Section~\ref{k>p} below);
as  we will see later on
it is in a way  equivalent to Roos' characterization of purity.

\begin{thm}\label{oxy}
The sheaf ${\mathcal F}={\mathcal O}(E_0)/{\mathcal J}$ has pure codimension $p$ if and only if
${\mathcal J}$ is equal to the annihilator of $R_p$, i.e.,
$$
{\mathcal J}=\{\phi\in{\mathcal O}(E_0);\  R_p\phi=0\}.
$$
\end{thm}

If ${\mathcal F}$ is Cohen-Macaulay we can choose a resolution
\eqref{acomplex} with  $N=p$, and then
$R=R_p$ is a matrix of ${\mathcal{CH}}_Z$-currents
which thus  solves our problem.   However, in general
$R_p$ is not $\bar\partial$-closed even if ${\mathcal F}$ has pure codimension.
Let
\begin{equation}\label{dualcomplex}
0\to {\mathcal O}(E_0^*)\stackrel{f_1^*}{\longrightarrow}{\mathcal O}(E_1^*)
\stackrel{f_2^*}{\longrightarrow} \ldots
\stackrel{f^*_{p-1}}{\longrightarrow} {\mathcal O}(E_{p-1}^*)\stackrel{f^*_p}{\longrightarrow}
{\mathcal O}(E_p^*)\stackrel{f^*_{p+1}}{\longrightarrow}
\end{equation}
be the dual complex of \eqref{acomplex} and let
\begin{equation}\label{extrep}
{\mathcal H}^k({\mathcal O}(E^*_\bullet))=\frac{{\rm Ker\,  }_{f^*_{k+1}}{\mathcal O}(E_k^*)}{f^*_k{\mathcal O}(E_{k-1}^*)}
\end{equation}
be the associated  cohomology sheaves.
It turns out that for each choice of $\xi\in{\mathcal O}(E_p^*)$ such that $f_{p+1}^*\xi=0$,
the current $\xi R_p$ is in ${\mathcal{CH}}_Z(E_0^*)$, and  we have in fact   a
bilinear (over ${\mathcal O}$)  pairing
\begin{equation}\label{pelle}
{\mathcal H}^p({\mathcal O}(E_\bullet^*))\times{\mathcal F}\to {\mathcal{CH}}_Z,   \quad  (\xi,\phi)\mapsto \xi R_p\phi.
\end{equation}
Moreover, \eqref{pelle} is independent of the choice of Hermitian metrics
on $E_k$.
It is well-known  that the sheaves in \eqref{extrep}
represent the intrinsic  sheaves  ${\mathcal Ext}^k_{\mathcal O}({\mathcal F},{\mathcal O})$.
(If $Z$ does not have pure codimension $p$  then we define
${\mathcal{CH}}_Z$ as  ${\mathcal{CH}}_{Z'}$, where $Z'$ is the union of irreducible components of
codimension $p$; this  is reasonable, in view of the SEP.)

\begin{thm}\label{thm1} Assume that  ${\mathcal F}$ has  codimension $p>0$.
The pairing  \eqref{pelle}  induces  an intrinsic  pairing
\begin{equation}\label{ext}
{\mathcal Ext}_{\mathcal O}^p({\mathcal F},{\mathcal O})\times {\mathcal F}\to {\mathcal{CH}}_Z.
\end{equation}
If ${\mathcal F}$ has pure codimension, then the pairing is non-degenerate.
\end{thm}

Notice that ${\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z)$   is the subsheaf of
${\mathcal Hom\, }({\mathcal O}(E_0),{\mathcal{CH}}_Z)={\mathcal{CH}}_Z(E_0^*)$ consisting of all Coleff-Herrera currents
$\mu$ with values in $E_0^*$ such that
$\mu\phi=0$ for all $\phi\in {\mathcal J}$.
It follows that we have the equality
\begin{equation}\label{dual3}
{\mathcal J}=\{\phi\in{\mathcal O}(E_0);\   \mu \phi=0\ \text{for\ all}\ \mu\in{\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z)\}
\end{equation}
if ${\mathcal F}$ is pure. The
sheaf   ${\mathcal H}^p({\mathcal O}(E_\bullet^*))$ is coherent and thus locally finitely generated.
Therefore we have now a solution to our problem:

\begin{cor}\label{labbe} Assume that ${\mathcal F}$ has pure codimension.
If $\xi_1,\ldots,\xi_\nu\in{\mathcal O}(E_p^*)$ generate ${\mathcal H}^p({\mathcal O}(E^*_\bullet))$,
then  $\mu_j=\xi_j R_p$ are in ${\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z)$  and
\begin{equation}\label{decomp}
{\mathcal J}=\cap_{j=1}^\nu{\rm ann\,} \mu_j.
\end{equation}
\end{cor}

\begin{remark}
If ${\mathcal J}$ is not pure,  one obtains a decomposition  \eqref{decomp}
after a preliminary decomposition ${\mathcal J}=\cap{\mathcal J}_\nu$,
where each ${\mathcal J}_\nu$ has pure codimension.
\end{remark}

In case of a complete intersection, ${\mathcal Ext}^p({\mathcal F},{\mathcal O})$ is isomorphic to ${\mathcal F}$ itself.
If  ${\mathcal F}={\mathcal O}(E_0)/{\mathcal J}$ is   a sheaf of Cohen-Macaulay modules there is
also a certain symmetry:
If  \eqref{acomplex} is  a resolution with $N=p$, then
it is well-known, cf., also  Example~\ref{cmcase} below,
that  the dual complex  \eqref{dualcomplex} is a resolution
of  ${\mathcal O}(E_p^*)/{\mathcal J}^*$, where  ${\mathcal J}^*=f_p^*{\mathcal O}(E_{p-1}^*)\subset{\mathcal O}(E_p^*)$,
and we have

\begin{cor}\label{chcor}
If ${\mathcal O}(E_0)/{\mathcal J}$ is Cohen-Macaulay,  then ${\mathcal O}(E_p^*)/{\mathcal J}^*$ is Cohen-Macaulay
as well and we have a non-degenerate  pairing
$$
{\mathcal O}(E_0)/{\mathcal J}\times {\mathcal O}(E_p^*)/{\mathcal J}^*\to {\mathcal{CH}}_Z, \quad  (\xi,\phi)\mapsto \xi R_p\phi.
$$
\end{cor}

\begin{remark}\label{pnoll}
Assume that ${\mathcal F}$ has codimension $p=0$, or equivalently, ${\rm ann\,}{\mathcal F}=0$.
If it is pure, i.e., $(0)$ is the only associated prime ideal, then
there is a homomorphism $f_0\colon{\mathcal O}(E_0)\to{\mathcal O}(E_{-1})$
such that ${\mathcal J}={\rm Ker\,  } f_0$. It is natural to  consider $f_0$ as a Coleff-Herrera
current $\mu$ associated with the zero-codimensional ``variety'' $X$.
Then
 ${\mathcal J}={\rm ann\,} \mu$ and thus analogues of
Theorem~\ref{oxy}  and Corollary~\ref{labbe} still hold.
\end{remark}

The duality discussed here  leads to a generalization of the
Dickenstein-Sessa decomposition that we now  will describe.
It was proved by Malgrange,  see, e.g.,  \cite{JEB4},
that the analytic sheaf of distributions ${\mathcal C}$ is stalkwise injective.
Thus the double complex
\begin{equation}\label{dubb}
{\mathcal Hom\, }_{\mathcal O}({\mathcal O}(E_\ell),{\mathcal C}^{0,k})={\mathcal C}^{0,k}(E_\ell^*),
\end{equation}
with differentials $f^*$ and $\bar\partial$,  is exact except at $k=0$ and  $\ell=0$,
where we have the cohomology sheaves ${\mathcal O}(E_\ell^*)$ and
${\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,\bullet})$, respectively.
By standard homological algebra,
we therefore have natural isomorphisms
\begin{equation}\label{nathom}
{\mathcal H}^k({\mathcal O}(E^*_\bullet),{\mathcal O})
\simeq
{\mathcal H}^k({\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,\bullet})).
\end{equation}
The residue calculus also gives

\begin{thm}\label{galopp}
Assume that ${\rm codim\,}{\mathcal F}=p>0$. Both mappings
\begin{equation}\label{natmap}
{\mathcal H}^p({\mathcal O}(E_{\bullet}^*))
\stackrel{\Psi}{\simeq}
{\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z)\simeq
{\mathcal H}^p({\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,\bullet}))
\end{equation}
are isomorphisms, and the composed mapping coincides with  the  isomorphism \eqref{nathom}.
\end{thm}

These isomorphisms seem to be known as ``folklore'' since long ago,
cf., Section~\ref{bjork} below. Our  contribution should be the
proof by residue calculus, and  in particular, the
realization of the mapping $\Psi$ as $\xi\mapsto \xi R_p$.

\begin{ex} If $\mu\in{\mathcal{CH}}_Z$ is annihilated by ${\mathcal J}$ it follows that we have the
factorization $\mu=\xi R_p$. There are analogous isomorphisms
where ${\mathcal O}$ is replaced by $\Omega^r$, the sheaf of holomorphic $(r,0)$-forms,
and Coleff-Herrera currents of bidegree $(r,p)$,
${\mathcal{CH}}^r_Z={\mathcal{CH}}_Z(\Omega^r)$ (the definition is the obvious one).
For instance it follows that there is  a factorization
$$
[Z]=\xi R_p,
$$
where $[Z]$ is the Lelong current, and $\xi$ is in $\Omega^p(E^*_p)$
with  $f_{p+1}^*\xi=0$.
\end{ex}

\begin{ex}
We can rephrase the second isomorphism in \eqref{natmap} as  the decomposition
\begin{equation}\label{kraka}
{{\rm Ker\,  }}_{\bar\partial} {\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,p})={\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z)\oplus \bar\partial{\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,p-1}).
\end{equation}
For a given $\bar\partial$-closed $(0,p)$-current $\mu$
(with values in $E_0^*$ and    annihilated by ${\mathcal J}$),
its canonical projection in  ${\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z)$ is given by $\xi R_p$,
where $\xi$ is obtained from $\mu$ via the isomorphism \eqref{nathom}.
\end{ex}

\begin{ex}
Assume that $Z$ has pure codimension $p$ and let
${\mathcal C}_Z^{0,k}$ denote the sheaf of $(0,k)$-currents with support on $Z$.
If  ${\mathcal F}$ has support on $Z$, then  ${\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,k})={\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,k}_Z)$.
Since any current with support on $Z$ must be annihilated
by some power of ${\mathcal I}_Z$, \eqref{kraka} implies  the
  decomposition
\begin{equation}\label{ds}
{\rm Ker\,  }_{\bar\partial}{\mathcal C}^{0,p}_Z={\mathcal{CH}}_Z\oplus\bar\partial{\mathcal C}_Z^{0,p-1}
\end{equation}
first proved in \cite{DS} by Dickenstein and Sessa
(in the case of a complete intersection;   see \cite{JEB4}
for the general case).
\end{ex}

The main results are  proved in Sections~\ref{sec3} and \ref{proofs}.
In Section~\ref{bjork} we sketch  a purely algebraic proof
of Theorem~\ref{thm1} (except for the explicit residue representation)
based on Roos' theorem.
In Section~\ref{absolut} we consider in some more detail the
absolute case, i.e., $p=n$, and in Section~\ref{coho}
we briefly discuss a cohomological variant of the duality.
In Section~\ref{k>p} we consider a partial generalization of \eqref{natmap}
to $k>p$; again we can trace residue manifestations of Roos' theorem.

For the reader's convenience we sketch in Section~\ref{noe}
Bj\"ork's  construction of Noetherian operators for the ideal ${\rm ann\,} \mu$.

All  results  above have  natural analogues in the algebraic case,
i.e., for polynomial ideals and modules;
this is briefly discussed in Section~\ref{algcase}.

To further exemplify the utility of the residue
calculus,  we conclude this paper by a  proof of Malgrange's theorem by means of
residues and integral formulas in Section~\ref{malgrange}.

\smallskip
{\bf Acknowledgement}  I am grateful to Jan-Erik Bj\"ork for
invaluable discussion on these matters, and  for
communicating  the arguments in Section~\ref{bjork}.

\section{Residue currents associated with Hermitian  complexes}\label{sec3}

We first have to recall the construction in \cite{AW1}. Let
\begin{equation}\label{hcomplex}
0\to E_N\stackrel{f_N}{\longrightarrow}\ldots
\stackrel{f_3}{\longrightarrow} E_2\stackrel{f_2}{\longrightarrow}
E_1\stackrel{f_1}{\longrightarrow}E_0\to\cdots
\stackrel{f_{-M+1}}{\longrightarrow} E_{-M}\to 0
\end{equation}
be a generically exact complex of Hermitian vector bundles over
a complex manifold $X$, and let
\begin{equation}\label{complex}
0\to {\mathcal O}(E_N)\stackrel{f_N}{\longrightarrow}\ldots
\stackrel{f_1}{\longrightarrow}{\mathcal O}(E_0)\longrightarrow
\cdots \stackrel{f_{-M+1}}{\longrightarrow} {\mathcal O}(E_{-M})\to 0
\end{equation}
be the corresponding complex of locally free sheaves. Assume that
\eqref{hcomplex} is pointwise exact outside the variety $Z$. Furthermore,
over $X\setminus Z$ let $\sigma_k\colon E_{k-1}\to E_k$ be the minimal
inverses of $f_k$. Then
$
f\sigma+\sigma f=I,
$
where $I$ is the identity on $E=\oplus E_k$, $f=\oplus f_k$ and $\sigma=\oplus\sigma_k$.
The bundle $E$ has a natural superbundle structure
$E=E^+\oplus E^-$, where $E^+=\oplus E_{2k}$ and $E^-=\oplus E_{2k+1}$,
and $f$ and $\sigma$ are odd mappings with respect to this structure,
see, e.g., \cite{AW1} for more details.
The operator   $\nabla=f-\bar\partial$ acts on ${\mathcal C}^{0,\bullet}(X, E)$ and extends to
a mapping $\nabla_{\rm End}$ on ${\mathcal C}^{0,\bullet}(X, {\rm End} E)$
and  $\nabla_{{\rm End}}^2=0$.
If
$$
u=\frac{\sigma}{\nabla_{{\rm End}}\sigma}=\sigma+\sigma(\bar\partial\sigma)+\sigma(\bar\partial\sigma)^2+\cdots
$$
it turns out that $\nabla_{\rm End} u=I$ in $X\setminus Z$.
One can define a canonical current extension $U$ of $u$
across $Z$ as the analytic continuation  to $\lambda=0$ of
$|F|^{2\lambda}u$, where $F$ is any holomorphic function that
vanishes on $Z$. In the same way we  can define
the current
$R=\bar\partial|F|^{2\lambda}{\wedge} u|_{\lambda=0}$
with support on $Z$, and then
\begin{equation}\label{res}
\nabla_{\rm End} U=I-R.
\end{equation}
More precisely
$$
R=\sum_\ell  R^\ell=\sum_{\ell k} R^\ell_k,
$$
where $R^\ell_k$ is a $(0,k-\ell)$-current that takes values in ${\rm Hom\, }(E_\ell,E_k)$, i.e.,
$$
R^\ell_k\in{\mathcal C}^{0,k-\ell}(X,{\rm Hom\, }(E_\ell,E_k)).
$$
Moreover we have (Proposition~2.2 in \cite{AW1})
\begin{prop} \label{keps}
If $\phi\in{\mathcal O}(E_\ell)$ and
$f_\ell\phi=R^\ell\phi=0$, then $\phi= f_{\ell+1}\psi$ has local
solutions $\psi\in{\mathcal O}(E_{\ell+1})$.
If $R^{\ell+1}=0$, then $\phi=f_{\ell+1}\psi$ has local holomorphic solutions
$\psi$  if and only if
$f_\ell\phi=R^\ell\phi=0$.
\end{prop}

Since \eqref{hcomplex} is generically exact, so is its dual complex
\begin{equation}\label{dualcomplex2}
0\to E_{-M}^*\stackrel{f^*_{-M+1}}{\longrightarrow}
\cdots \stackrel{f^*_N}{\longrightarrow}E^*_N\to 0
\end{equation}
of Hermitian vector bundles,
and we have the corresponding dual complex of locally free sheaves
\begin{equation}\label{dualcomplex3}
0\to {\mathcal O}(E_{-M}^*)\stackrel{f^*_{-M+1}}{\longrightarrow}
\cdots \stackrel{f^*_N}{\longrightarrow}{\mathcal O}(E^*_N)\to 0.
\end{equation}
Using the induced metrics, we get a residue current
$$
R^*=\sum_k (R^*)^k=\sum_{k,\ell} (R^*)^k_{\ell},
$$
where $(R^*)^k_{\ell}$ takes values in ${\rm Hom\, }(E_k^*,E^*_\ell)$.
\begin{prop}\label{res2}
Using the natural isomorphisms ${\rm Hom\, }(E_k^*,E^*_\ell)={\rm Hom\, }(E_\ell,E_k)$
we have that
$
(R^*)^k_\ell=R^\ell_k.
$
\end{prop}

\begin{proof} It is readily verified that the adjoint $\sigma^*\colon E^*\to E^*$ of
$\sigma\colon E\to E$ over $X\setminus Z$ is the minimal inverse of $f^*$. Therefore,
$$
u^*=(\sigma+\sigma(\bar\partial\sigma)+\sigma(\bar\partial\sigma)^2+\cdots)^*=
\sigma^*+\sigma^*(\bar\partial\sigma^*)+\sigma^*(\bar\partial\sigma^*)^2+\cdots,
$$
since, see \cite{AW1}, $\sigma^*\bar\partial\sigma^*=(\bar\partial\sigma^*)\sigma^*$. Now the proposition
follows.
\end{proof}

If $\xi\in{\mathcal O}(E_k^*)$ and $\phi\in{\mathcal O}(E_\ell)$ we write
$$
\xi R^\ell_k\phi=\phi(R^*)^k_\ell\xi.
$$

In \cite{AW2} we introduced the sheaf of {\it pseudomeromorphic}  currents
${\mathcal{PM}}$ and pointed out that the currents  $U$ and $R$ are pseudomeromorphic.
The sheaf ${\mathcal{PM}}$ is closed under $\bar\partial$ and multiplication with smooth forms.
From \cite{AW2} (Corollary~2.4 and Proposition~2.5) we have

\begin{prop}\label{hyp}
If $\mu\in{\mathcal{PM}}$ with  bidegree $(r,p)$ has  support on
a variety $V$ of codimension $k>p$ then $\mu=0$.

Furthermore,
if $V$ has  codimension $p$, then the sheaf of Coleff-Herrera currents
${\mathcal{CH}}_V^r={\mathcal{CH}}_V(\Omega^r)$ coincides with the subsheaf of  ${\mathcal{PM}}$ of $\bar\partial$-closed
$(r,p)$-currents with support on $V$.
\end{prop}

\section{Proofs of the main results}\label{proofs}

Assume that ${\mathcal F}$ is a coherent sheaf of positive codimension $p$, and let
\eqref{acomplex}
be a (locally) free resolution of ${\mathcal F}={\mathcal O}(E_0)/{\mathcal J}$.
Moreover, assume that  $f_1$ is generically
surjective so that  the corresponding vector bundle complex
\begin{equation}\label{nupcomplex}
0\to E_N\stackrel{f_N}{\longrightarrow}\ldots
\stackrel{f_3}{\longrightarrow} E_2 \stackrel{f_2}{\longrightarrow}
E_1 \stackrel{f_1}{\longrightarrow} E_0\to 0
\end{equation}
is generically exact.
It follows from  Proposition~\ref{hyp}  that
$$
R^0=R^0_p+R^0_{p+1}+\cdots .
$$
By  Theorem~3.1 in \cite{AW1},  $R^\ell_k=0$ for each $\ell\ge 1$,
i.e., $R=R^0$,   and
combining with  Proposition~\ref{keps} above we find that
a $\phi\in{\mathcal O}(E_0)$ is in ${\mathcal J}$ if and only if $R\phi=0$.
It is proved in Section~5 of  \cite{AW2} that
${\mathcal F}$ has pure codimension $p$ if and only if
${\rm ann\,} R={\rm ann\,} R_p$, i.e.,    Theorem~\ref{oxy} holds.

\begin{proof}[Proof of Theorem~\ref{thm1}]
It follows from \eqref{res} that
\begin{equation}\label{opera}
\bar\partial R_k=f_{k+1}R_{k+1}
\end{equation}
for each $k$.
If $\xi\in{\mathcal O}(E_k^*)$ and $f^*_{k+1}\xi=0$  we therefore have
$$
\bar\partial(\xi R_k)=\pm \xi\bar\partial R_k=\pm \xi f_{k+1} R_{k+1}=\pm (f_{k+1}^*\xi)R_{k+1}=0.
$$
Thus $\xi R_p$ is $\bar\partial$-closed
and since it is also  pseudomeromorphic, cf., Proposition~\ref{hyp},   it
is in ${\mathcal{CH}}_Z$. Moreover, if
$\xi=f_p^*\eta$, then
$$
\xi R_p=(f^*_p\eta)R_p=\eta f_pR_p=\eta\bar\partial R_{p-1}=0
$$
since $R_k=0$ for $k<p$. Thus $\xi R_p$ only depends on the cohomology class
of $\xi $ in $H^p({\mathcal O}(E_\bullet^*))$.
We now choose another Hermitian metric on $E$ and let $\tilde R$ denote the current
associated with  the new metric.
It is showed  in \cite{AW1} (see the proof of Theorem 4.4) that then
$$
R_p-\tilde R_p=f_{p+1}M_{p+1}^0
$$
for a certain residue current $M$. Thus $\xi R_p=\xi\tilde R_p$.
It follows that the mapping \eqref{pelle}  is
well-defined and independent of the Hermitian metric on $E$.

It is enough to prove the invariance at a fixed point $x$, so we consider
stalks of the  sheaves at $x$.  It is well-known that then  our resolution
${\mathcal O}_x(E_\bullet), f_\bullet$ can be written
$$
{\mathcal O}_x(E'_\bullet\oplus E''_\bullet)\simeq{\mathcal O}_x(E'_\bullet)
\oplus {\mathcal O}_x(E''_\bullet),\    f_\bullet=f'_\bullet\oplus f''_\bullet,
$$
where ${\mathcal O}_x(E'_\bullet)$ is a resolution of ${\mathcal F}_x$ and (since we assume
that $E_0$ has minimal rank)  ${\mathcal O}_x(E''_k), k\ge 1$,  is a resolution of
${\mathcal O}_x(E_0'')=0$.
It follows that the natural mapping
$H^p({\mathcal O}_x((E'_\bullet)^*)\to H^p({\mathcal O}_x((E_\bullet)^*)), \  \xi'\mapsto (\xi',0)$,
is an isomorphism. Moreover,
if we choose a metric on $E_k=E'_k\oplus E''_k$ that respects the direct
sum, then the resulting current $R$ is $R'\oplus 0$, where $R'$ is the
current associated with ${\mathcal O}_x(E'_\bullet)$. Since all minimal
resolutions are isomorphic, the  mapping \eqref{ext} is therefore well-defined.

\smallskip

It remains to check that  \eqref{ext} is non-degenerate.
If $\xi\in{\mathcal O}(E_p^*)$ with $f^*_{p+1}\xi=0$ and $\xi R_p\phi=0$ for all
$\phi\in{\mathcal O}(E_0)$, then clearly $\xi R_p=0$.
Since $R=R^0_p$, by Proposition~\ref{res2} therefore
$(R^*)^p_\ell\xi=0$ for all $\ell$,  and now it follows from
Proposition~\ref{keps} that $\xi=f_p^*\eta$ for some $\eta$.
Thus (the class of)  $\xi$ is zero in ${\mathcal H}^p({\mathcal O}(E^*_\bullet))$.

Now, assume that  $\xi R_p\phi=0$ for all $\xi$ such that $f_{p+1}^*\xi=0$.
If  ${\mathcal F}$ is Cohen-Macaulay and  $N=p$,  then
$f^*_{p+1}=0$ so the assumption  implies that $R_p\phi=0$, and thus
$\phi\in{\mathcal J}$.  However,
generically on $Z$, ${\mathcal F}$ is Cohen-Macaulay,
and hence for an arbitrary resolution
we must have that $R_p\phi=0$ outside a variety of codimension $\ge p+1$.
Since $R_p\phi$ is pseudomeromorphic with bidegree $(0,p)$
it follows  from Proposition~\ref{hyp} that $R_p\phi$ vanishes identically.
If we in addition assume that ${\mathcal F}$ has pure codimension it follows
from  Theorem~\ref{oxy} that  $\phi\in{\mathcal J}$. Thus the pairing is
non-degenerate.
\end{proof}

\begin{ex}[The Cohen-Macaulay case]\label{cmcase}
It is well-known, see, e.g., \cite{Eis1},
that ${\mathcal F}$ is Cohen-Macaulay if and only if
it admits resolutions of length $p={\rm codim\,} Z$.
If \eqref{acomplex} is a resolution with $N=p$, then
$R=R^0_p$, and hence $R^*=(R^*)^0_p$. It follows from
Proposition~\ref{keps},  applied to $R^*$, that the dual complex
\eqref{dualcomplex} is a resolution of ${\mathcal O}(E_0^*)/{\mathcal J}^*$ and,  in particular,
that  ${\mathcal O}(E_0^*)/{\mathcal J}^*$ is Cohen-Macaulay.
\end{ex}

\begin{proof}[Proof of  Theorem~\ref{galopp}]
Let
$$
{\mathcal L}^\nu=\sum_{\ell+k=\nu} {\mathcal C}^{0,k}(E_\ell^*)
$$
be the  total complex with differential $\nabla^*=f^*-\bar\partial$,
associated with the double complex \eqref{dubb}.
We thus have   natural isomorphisms
\begin{equation}\label{isos}
{\mathcal H}^k({\mathcal O}(E_\bullet^*))\simeq {\mathcal H}^k({\mathcal L})=_{def}
\frac{{\rm Ker\,  }_{\nabla^*}{\mathcal L}^k}{\nabla^*{\mathcal L}^{k-1}}\simeq
{\mathcal H}^k({\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,\bullet})).
\end{equation}
The naturality means that the ismorphisms  are induced by the natural mappings
${\mathcal O}(E_k^*)\to{\mathcal L}^k$ and ${\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,\ell})\to {\mathcal L}^k$, respectively, and that
$\xi\in{\mathcal O}(E^*_k)$ such that $f^*_{k+1}\xi=0$
defines the same class as $\mu\in{\rm Hom\, }({\mathcal F},{\mathcal C}^{0,k})$ with $\bar\partial\mu=0$ if and
only if there is $W\in{\mathcal L}^{k-1}$ such that
$
\nabla^*W=\xi-\mu.
$

If now $\xi\in{\mathcal O}(E_k^*)$ and $f^*_{k+1}\xi=0$, then $\nabla^*\xi=0$, and hence
\begin{equation}\label{astrid}
\nabla^* (U^*)^k\xi=\xi- (R^*)^k\xi=\xi-\xi R_k,
\end{equation}
cf., \eqref{res} and Proposition~\ref{res2} above.
Therefore  the composed mapping
in \eqref{natmap} coincides with the isomorphisms  in \eqref{nathom}.
It is readily verified that the second mapping in \eqref{natmap}  is injective,
see, e.g., Lemma~3.3  in \cite{A12}, and hence
both mappings  must be isomorphisms. Thus
Theorem~\ref{galopp} is  proved.
\end{proof}

We think it may be  enlightening with  a proof of the first isomorphism
in \eqref{natmap}  that does not rely on Malgrange's theorem.
We already know from Theorem~\ref{thm1} that this mapping  is injective,
so we have to prove the surjectivity.
The proof
is based on the following lemma.

\begin{lma}\label{chlma}
If there is a current $W\in{\mathcal L}^{p-1}$ such that
$\nabla^*W=\mu\in{\mathcal{CH}}_Z(E_0^*)$, then $\mu=0$.
\end{lma}

\begin{proof}
Let $u$ be a  smooth form $u$ such that $\nabla^*_{{\rm End}}u=I_{E^*}$ in $X\setminus Z$.
For a given neighborhood $\omega$ of $Z$, take a cutoff function $\chi$
with support in $\omega$ and equal to $1$ in some neighborhood of $Z$.
Then $g=\chi I_{E^*}-\bar\partial\chi{\wedge} u$ is smooth with compact
support in $\omega$, equal to $I_{E^*}$ in
a neighborhood of $Z$, and moreover $\nabla^* g=0$.
Therefore, $\nabla^* (g W)=g \mu=\mu$ and hence, for degree reasons,
we have a solution  $\bar\partial w=\mu$ with support in $\omega$.
Since $\omega\supset Z$ is arbitrary it follows, cf., Lemma~3.3 in \cite{A12},
that $\mu=0$.
\end{proof}

Since \eqref{dubb} is  exact in $k$ except at $k=0$,
the  first equivalence in \eqref{isos} holds.
Take  $\mu\in{\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z)$. Then  $\nabla^*\mu=(f_1^*-\bar\partial)\mu=0$
so by \eqref{isos} (with $k=p$)  there is $\xi\in{\mathcal O}(E_p^*)$ such that
$
\nabla^* W=\xi-\mu
$
has a current solution $W\in{\mathcal L}^{p-1}$. In view of
\eqref{astrid} it now follows from  Lemma~\ref{chlma} that
$\mu=\xi R^0_p$.

\section{An algebraic approach}\label{bjork}

In this section we indicate  how Theorem~\ref{thm1} and its corollaries
(except for the concrete representation $\xi\mapsto \xi R_p$)
can be proved  algebraically. This  material has
been communicated to us by Jan-Erik Bj\"ork.
Whereas our residue proof above was based on   Theorem~\ref{oxy},
the algebraic proof instead relies on the following
fundamental result due to  J-E Roos, \cite{Roos}:

\begin{thm}\label{roosthm}
The sheaf ${\mathcal F}$  has pure codimension $p$ if and only if
the natural mapping
\begin{equation}\label{roos}
{\mathcal F}\to{\mathcal Ext}^p({\mathcal Ext}^p({\mathcal F},{\mathcal O}),{\mathcal O})
\end{equation}
is injective.
\end{thm}

Assume that
$$
0\to {\mathcal J}\to{\mathcal O}(E_0)\to{\mathcal F} \to 0
$$
is exact as before. Moreover,
let us  assume to begin with that we already know the isomorphisms \eqref{natmap}.
In particular we then have that
$$
{\mathcal Ext}^p({\mathcal F},{\mathcal O})\simeq{\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z).
$$
Thus  we can choose (locally) a finite number of generators $\mu_\alpha$, $\alpha\in A$,  for
${\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z)$ and get an  exact sequence
$$
0\to {\mathcal I}\to {\mathcal O}^A\to {\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z)\to 0,
$$
and therefore we  have, with ${\mathcal M}={\mathcal Hom\, }({\mathcal F},{\mathcal{CH}}_Z)\simeq{\mathcal Ext}^p({\mathcal F},{\mathcal O})$,  that
$$
{\mathcal Ext}^p({\mathcal M},{\mathcal O})\simeq{\mathcal Hom\, }({\mathcal M},{\mathcal{CH}}_Z).
$$
We claim that the canonical mapping \eqref{roos} is given by
$$
{\mathcal O}(E_0)\ni\phi\mapsto (\mu_\alpha\phi)_\alpha.
$$
In fact,  clearly it is a mapping from ${\mathcal F}={\mathcal O}(E_0)/{\mathcal J}$, since each $\mu_\alpha$ is.
Moreover, if $(\psi_\alpha)\in{\mathcal I}$, then by definition
$\sum_\alpha\psi_\alpha\mu_\alpha=0$, and hence $(\mu_\alpha\phi)_\alpha$ defines an element
in
$$
{\mathcal Hom\, }({\mathcal O}^A/{\mathcal I},{\mathcal{CH}}_Z)\simeq{\mathcal Ext}^p({\mathcal Ext}^p({\mathcal F},{\mathcal O}),{\mathcal O}).
$$
One can  verify that this mapping is independent of the choice of generators,
and  must be the canonical mapping \eqref{roos}.

It follows that \eqref{roos} is injective if and only if
the equality \eqref{dual3} holds. If ${\mathcal F}$ has pure
codimension $p$, by Theorem~\ref{roosthm} therefore
\eqref{dual3} holds and then
Corollary~\ref{labbe}  as well as Theorem~\ref{thm1} follow.

\begin{remark}
We actually get a residue  proof of Theorem~\ref{roosthm}:
If ${\mathcal F}$ has pure codimension $p$ we know that  \eqref{dual3} holds
by the residue theory, and thus  \eqref{roos} is injective.
On the other hand, it is not hard to see, e.g., it follows from \cite{AW2}, that
the annihilator of (a set of) currents in ${\mathcal{CH}}_Z$
must have pure codimension $p$. Thus  the injectivity  of
\eqref{roos} implies that  ${\mathcal F}$ has pure codimension.
\end{remark}

Let us conclude with a brief discussion of  \eqref{natmap}.
The  Dickenstein-Sessa decomposition \eqref{ds}
is well-known;  see \cite{DS} in  case of a complete
intersection and \cite{JEB4} for the general case.
Malgrange also proved that  ${\mathcal C}_Z$ is stalkwise injective as an analytic sheaf.
Using these two facts and considering  the spectral
sequence obtained from the double complex
$$
{\mathcal Hom\, }({\mathcal O}(E_\ell),{\mathcal C}_Z^{0,k}),
$$
one can conclude that the second mapping in \eqref{natmap} is indeed
an isomorphism, and hence both of them. However, we omit the details.

\section{The absolute  case and Bezoutians}\label{absolut}

Let $J\subset{\mathcal O}_0(E_0)$ be a submodule of the free module ${\mathcal O}_0(E_0)$
over the local ring ${\mathcal O}_0$  such that the zero variety of ${\rm ann\,}({\mathcal O}_0(E_0)/J)$
is  $Z =\{0\}$.
Moreover, let \eqref{acomplex} be a resolution of ${\mathcal O}_0(E_0)/J$ of length $n$.
From Corollary~\ref{chcor} we have the non-degenerate pairing
$$
{\mathcal O}_0(E_0)/J\times{\mathcal O}_0(E_n^*)/J^*\to{\mathcal{CH}}_{\{0\}}.
$$
Let $\alpha\in\Omega^n_0$ be a germ of a
nonvanishing holomorphic $(n,0)$-form at the origin,
and let
$$
{\mathcal{CH}}_{\{0\}}\to{\mathbb C},\quad \mu\to \mu.\alpha=\int\alpha{\wedge} \mu.
$$
Then we have
\begin{prop} The composed mapping
\begin{equation}
{\mathcal O}_0(E_0)/J\times{\mathcal O}_0(E_n^*)/J^*\to {\mathbb C}
\end{equation}
is a non-degenerate pairing.
\end{prop}

\begin{proof}
If $\phi\in{\mathcal O}_0(E_0)$  is not in $J$, then there is some
$\xi\in{\mathcal O}_0(E_n^*)$ such that
$\mu=\xi R_n\phi$ is not identically zero. Since $\mu$ is in
${\mathcal{CH}}_{\{0\}}$   there is some holomorphic
$\psi\in{\mathcal O}_0$ such that $\psi\mu\neq 0$.  Thus
$
\psi\xi R_n \phi.\alpha=\psi\mu.\alpha\neq 0.
$
Since we can interchange the roles of $J$ and $J^*$ the proposition follows.
\end{proof}

In particular, we obtain an (non-canonical) isomorphism
\begin{equation}\label{holm}
({\mathcal O}_0(E_0)/J)^*\simeq {\mathcal O}_0(E_n^*)/J^*.
\end{equation}

Notice that the form $u_n^0$ defines a ${\rm Hom\, }(E_0,E_n)$-valued
Dolbeault cohomology class $\omega$ in ${\mathcal U}\setminus\{0\}$.
Since  $\bar\partial U_n^0=R_n$ we  have
\begin{equation}\label{svarv}
(\xi,\phi)=\int_{|\zeta|=\epsilon}\alpha{\wedge} \xi \omega \phi
\end{equation}
From  (5.4) in \cite{AW1} we get the representation
$$
\phi(z)=f_1(z)\int H^1U\phi{\wedge} g +\int H^0_nR_n\phi{\wedge} g, \quad \phi\in{\mathcal O}(E_0);
$$
here $H^1$ is a holomorphic ${\rm Hom\, }(E,E_1)$-vlaued form,
$H_n^0$ is a  ${\rm Hom\, }(E_n,E_0)$-valued holomorphic $(n,0)$-form,
so that
$$
H_n^0=h_n^0\alpha,
$$
and $g$ is the  form (5.2) in \cite{AW1}.
It has compact support and
depends  holomorphically
in $z$. Moreover,
$g=\chi+\cdots,$
where the dots  denote smooth forms of positive bidegree, so
modulo $J$ we  have
$$
\phi(z)\equiv_{mod J}\int_{|\zeta|=\epsilon}H_n^0(\cdot,z)\omega\phi,
$$
and hence
$$
\phi(z)\equiv_{mod J} (h_n^0(\cdot, z),\phi).
$$
This  means that we  can consider $h_n^0$ as a (generalized)  Bezoutian, cf., \cite{CD}.
For  each analytic functional $\mu$ on ${\mathcal O}_0(E_0)$
that vanishes on  $J$  there is a unique
element $\xi$ in ${\mathcal O}_0(E_n^*)/J^*$ such that the action on ${\mathcal O}_0(E_0)$ modulo
$J$ coincides with
$\mu.\phi=(\xi,\phi)$ in view of \eqref{holm}.
More explicitly we have
$$
\xi(\zeta)=\mu_z(h_n^0(\zeta,\cdot)).
$$

In the classical case of a complete intersection
$J=(f^1,\ldots,f^n)$, if we choose Hefer forms,
i.e., $(1,0)$-forms $h_k=\sum_j h_{jk} d\zeta_j$ such that
$\sum h_{jk}(\zeta_j-z_j)=f^k(z)-f^k(\zeta)$,
and $\alpha=d\zeta_1{\wedge}\ldots{\wedge} d\zeta_n$, then, cf., \cite{A7},
it turns out that $h_n^0=\det(h_{jk})$; this  is a well-known
formula, cf., \cite{CD}, for the   Bezoutian in this case.

\section{Cohomological  residues}\label{coho}

The Coleff-Herrera currents admit a nice intrinsic way
of expressing the action of holomorphic differential operators,
but the very definition relies on Hironaka's theorem
about the existence of resolutions of singularities.
In \cite{DS} and \cite{P1} there is also
a cohomological way of expressing the duality for a complete
intersection.
We have the following cohomological version  of
Corollary~\ref{labbe}.

\begin{thm}\label{gurka}
With the assumptions and notation as in  Corollary~\ref{labbe}
and with $w_j=\xi_j u^0_p$ outside $Z$ we have for $\phi\in{\mathcal O}(E_0)$
that
\begin{equation}\label{bulle}
w_j\phi.\psi=0,  \quad \psi\in{\mathcal D},\quad \bar\partial\psi=0 {\text \  close\ to\ } Z
\end{equation}
if and only if $\phi\in {\mathcal J}$.
\end{thm}

In fact, $w_j$ can be extended to $W_j=\xi_j U^0_p$ across
$Z$ and
$$
\bar\partial W_j =\bar\partial (\xi_j U^0_p)=\pm \xi_j R_p=\pm \mu_j.
$$
One can now verify that \eqref{bulle} implies that $\mu_j\phi=0$,
see, e.g.,  Theorem~6.1 in \cite{A12}.
Thus Theorem~\ref{gurka} follows from  Corollary~\ref{labbe}.
Moreover,  \eqref{bulle} is equivalent
to that $w_j\phi$ are $\bar\partial$-exact in $X\setminus Z$ for
each Stein neighborhood $X$.

\smallskip
In the complete intersection case, as well as the Cohen-Macaulay case,
see \cite{JL}, the proof is algebraic and only involves ``cohomological'' residues,
whereas the proof of Theorem~\ref{gurka} here is
obtained via the residue calculus.
It is reasonable to believe that one can
produce  a purely algebraic proof (thus avoiding Hironaka's theorem) based on
Roos' theorem,  cf., Section~\ref{bjork}.

\section{Higher ${\mathcal Ext}$ sheaves}\label{k>p}

Let ${\mathcal F}={\mathcal O}(E_0)/{\mathcal J}$ has codimension $p$ as before.
 In view of
Proposition~\ref{hyp} and \eqref{natmap} one could guess that
${\mathcal Ext}^k({\mathcal F},{\mathcal O})$ for $k>p$ could be represented by
cohomology of pseudomeromorphic currents. We have
the following partial result.

\begin{thm}\label{orm}
Assume that \eqref{acomplex} is a resolution of ${\mathcal F}$ and $R$ is the
associated residue current (with respect to some given metric).
For each $k$, ${\mathcal O}(E_k^*)\ni\xi\mapsto \xi R_k$
induces an invariant injectice mapping
\begin{equation}\label{kmap1}
{\mathcal Ext}^k({\mathcal F},{\mathcal O})\to {\mathcal H}^k({\mathcal Hom\, }({\mathcal F},{\mathcal{PM}}^{0,\bullet})).
\end{equation}
Moreover,  the composed mapping
\begin{equation}\label{kmap2}
{\mathcal H}^k({\mathcal O}(E_\bullet^*))\to
{\mathcal H}^k({\mathcal Hom\, }({\mathcal F},{\mathcal{PM}}^{0,\bullet}))\to
{\mathcal H}^k({\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,\bullet}))
\end{equation}
coincides with the natural isomorphism \eqref{nathom}.
\end{thm}

\begin{remark}\label{anka}
If we widen the definition of ${\mathcal{PM}}$ slightly so that it is
preserved under any surjective holomorphic mapping
rather than just modifications, then
the $\bar\partial$-complex ${\mathcal{PM}}_{0,\bullet}$ is exact and thus it is
a fine resolution of the sheaf ${\mathcal O}$. It is reasonable to believe
that  ${\mathcal{PM}}$ so defined is stalkwise injective. If this is true,
by considering the double complex ${\mathcal{PM}}_{0,k}(E^*_\ell)$, we could conclude
that the first mapping in \ref{kmap2} is an isomorphism for any $k$,
and hence that both mappings are.
\end{remark}

Let $Z_k$ be the analytic set where the mapping $f_k$ in \eqref{acomplex}
does not have optimal rank. It is well-known and not hard to see that
these sets are independent of the resolution and hence invariants
of the sheaf ${\mathcal F}$. Then
$$
\cdots Z_{k+1}\subset Z_k\subset\cdots \subset Z_{p+1}\subset Z_{sing}\subset Z_p=Z
$$
and the Buchsbaum-Eisenbud theorem,
see \cite{Eis1}, states that the codimension of $Z_k$ is at least $k$.
Moreover, by Corollary~20.14 in \cite{Eis1}, ${\mathcal F}$ has pure codimension $p$
if and only if
\begin{equation}\label{plast}
{\rm codim\,} Z_k\ge k+1, \quad k>p.
\end{equation}
Notice also that ${\mathcal F}$ is Cohen-Macaulay if and only if
$Z_k=\emptyset$ for all $k>p$.

\begin{proof}[Proof of Theorem~\ref{oxy}]
If ${\mathcal F}$ is pure,  using \eqref{plast},
it follows as in (the proof of) Lemma~5.2 in \cite{AW2},
that ${\rm ann\,} R_p={\rm ann\,} R$, and thus  ${\rm ann\,} R_p={\mathcal J}$. Conversely,
assume that ${\rm ann\,} R_p={\mathcal J}$.
It follows from Proposition~5.3 in \cite{AW2} that
${\rm ann\,} R_p$ must be an intersection of primary modules
of codimension $p$, and hence ${\mathcal J}={\rm ann\,} R_p$ is pure.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{orm}]
Since $R\phi=0$ for $\phi\in{\mathcal J}$, it follows from
\eqref{opera} that the first mapping in \eqref{kmap2} is
well-defined, and in view of \eqref{astrid}
the composed mapping coincides with the natural isomorphism.
It  follows that the first mapping must be injective.
(This is also easily seen by a direct argument
that avoids Malgrange's theorem: Assume that $\xi R_k=\bar\partial\gamma$
for some $\gamma$ in ${\mathcal Hom\, }({\mathcal F},{\mathcal{PM}}^{0,k-1})$. Then
$\xi R_k=\nabla^*\gamma$, so that
$
\nabla^*(U^*\xi-\gamma)=\xi
$
and hence $\xi=0$ in ${\mathcal H}^k({\mathcal O}(E_\bullet^*))$ in view of the first isomorphism
in \eqref{isos}.)

\smallskip

If $\tilde R$ denotes the current associated with another metric, then as before
there is a current $M$ with support on $Z$ such that
$$
\nabla_{{\rm End}} M=R-\tilde R.
$$
In fact, if
$$
u=\sigma/\nabla_{{\rm End}}\sigma=\sigma+(\bar\partial\sigma)\sigma +(\bar\partial\sigma)^2\sigma+\cdots
$$
and  $\tilde u$ is  the analogous form corresponding to the new  metric, then, cf., \cite{AW1},
we can take
\begin{equation}\label{homotopi}
M=\bar\partial|F|^{2\lambda}{\wedge} u\tilde u|_{\lambda=0}.
\end{equation}
Now,  if $\xi\in{\mathcal O}(E_k^*)$ and $f_{k+1}^*\xi=0$, then
$$
\xi(R_k-\tilde R_k)
\xi(\nabla_{{\rm End}} M)^0=
\xi(f_{k+1} M^0_{k+1}-\bar\partial M^0_k)=-\pm \bar\partial(\xi M^0_k).
$$
Thus we must  show that $\xi M^0_k\phi=0$ for $\phi\in{\mathcal J}$.
If $\phi\in{\mathcal J}$, then $\phi=f_1\psi=\nabla\psi$ for some
$\psi\in{\mathcal O}(E_1)$. Thus
\begin{multline*}
\xi M^0_k\phi=\xi M(\nabla\psi)=\xi(\nabla_{{\rm End}}M)\psi-\xi\nabla(M\psi)= \\
\xi(R^1_k-\tilde R^1_k)\psi-
\xi f_{k+1}M^1_{k+1}\psi+\xi\bar\partial M_k^1\psi=\xi\bar\partial M^1_k\psi
\end{multline*}
since $R^1=\tilde R^1=0$,
so it is enough to check that  $M^1_k=0$, which we prove by induction
over $k$:
First notice that $M^1_k$ must vanish for $k\le p+1$ since it has bidegree
$(0,k-2)$ and  has support on $Z$ that has  codimension $p$.
Now suppose that we have proved that $M^1_k=0$.
Outside $Z_{k+1}$  the mapping $\sigma_{k+1}$ is smooth so we have, cf., \eqref{homotopi}
and the definition of $R$,
$$
M^1_{k+1}=\sigma_{k+1}\tilde R^1_{k+1}+(\bar\partial\sigma_{k+1})M^1_k=(\bar\partial\sigma_{k+1})M^1_k
=0$$
there since $\tilde R^1=0$. Thus $M^1_{k+1}$ has support on
$Z_{k+1}$ and for degree reasons  it must vanish identically.
\end{proof}

Outside the set $Z_k$, there is a resolution \eqref{acomplex}
of ${\mathcal F}$  with $N<k$, and it follows that then
${\mathcal Ext}^k({\mathcal F},{\mathcal O})\simeq{\mathcal H}^k({\mathcal O}(E_\bullet^*))$ vanishes there; i.e.,
${\mathcal Ext}^k({\mathcal F},{\mathcal O})$ has its support on $Z_k$.
On the other hand, if ${\mathcal Ext}^\ell({\mathcal F},{\mathcal O})=0$ for all $\ell\ge k$, then
$$
{\mathcal O}(E^*_{k-1})\stackrel{f^*_k}{\longrightarrow}
{\mathcal O}(E^*_k)
\stackrel{f^*_{k+1}}{\longrightarrow}\ldots
\stackrel{f^*_N}{\longrightarrow}{\mathcal O}(E_N^*)\to 0
$$
is exact, and hence all the mappings must have constant rank, so we must
be outside $Z_k$. It follows that
\begin{equation}\label{puma}
\text{supp}\,  {\mathcal Ext}^k({\mathcal F},{\mathcal O})\subset Z_k\subset \bigcup_{\ell\ge k}
\text{supp}\, {\mathcal Ext}^\ell({\mathcal F},{\mathcal O}).
\end{equation}
If ${\mathcal F}$ has pure codimension $p$, then it follows from
\eqref{puma} and Eisenbud's theorem mentioned above that
the support of $ {\mathcal Ext}^k({\mathcal F},{\mathcal O})$ has at least codimension
$k+1$; a fact that was already
established  by Roos in \cite{Roos}.

On the other hand, if we have not pure codimension somewhere, then for some $k>p$,
${\rm codim\,} Z_k$ has codimension $k$.
It follows from \eqref{puma} that then the support $V$ of
${\mathcal Ext}^k({\mathcal F},{\mathcal O})$ has codimension
$k$ here (since $\text{supp} {\mathcal Ext}^\ell({\mathcal F},{\mathcal O})$ for $\ell>k$ have higher codimension).
By the coherence there is $\xi\in{\mathcal O}(E_k^*)$ whose cohomology class
is (generically) nonvanishing on $V$. By Theorem~\ref{orm} then the current
$\xi R_k$ represents the corresponding  nonvanishing
class in ${\mathcal Hom\, }({\mathcal F},{\mathcal C}^{0,\bullet})$.

\section{Noetherian operators}\label{noe}

\begin{thm}[Bj\"ork \cite{JEB4}]
Let $V$ be a germ of an analytic variety of pure codimension $p$ at $0\in{\mathbb C}^n$.
There is a neighborhood $\Omega$ of $0$ such that for each
$\mu\in{\mathcal{CH}}_V(E_0^*)$ in $\Omega$, there are  holomorphic
differential operators ${\mathcal L}_1,\ldots,{\mathcal L}_\nu$ in $\Omega$ such that
for any $\phi\in{\mathcal O}(E_0)$,
$\mu\phi=0$ if and only if
\begin{equation}\label{bjarne}
{\mathcal L}_1\phi=\cdots={\mathcal L}_\nu\phi=0 \ \text{on}\ \ V.
\end{equation}
\end{thm}

\begin{proof}[Sketch of proof]
In a suitable pseudoconvex neighhborhood $\Omega$ of $0$
we can find  holomorphic functions $f_1,\ldots,f_p$,  forming
a complete intersection,  such that
$V\cap \Omega$, henceforth denoted just $V$,
is a union of irreducible components of $V_f=\{f=0\}$, and such that
$
df_1{\wedge}\ldots{\wedge} df_p\neq 0
$
on $V\setminus W$,  where $W$ is a hypersurface not containing any component
of $V_f$.
By a suitable choice of coordinates $(\zeta,\omega)\in{\mathbb C}^{n-p}\times{\mathbb C}^p$ we may
assume that  $W$ is the zero set of
$
h=\det (\partial f/\partial\omega).
$
Let
$
z=\zeta,  w=f(\zeta,\omega).
$
Since
$d(z,w)/d(\zeta,\omega)=h$,
locally in $\Omega\setminus W$,    $(z,w)$ is a holomorphic  coordinate system.
If we take the multiindex $M$ so large that $\mu$ is annihilated by
$f_j^{M_j+1}$, it  follows from
\cite{JEB4} (or Theorem~4.1 in \cite{A12}) that
there is a  holomorphic function $A\in\Omega$ such that
$$
\mu=A \bar\partial\frac{1}{f_1^{M_1+1}}{\wedge}\ldots {\wedge}\bar\partial\frac{1}{f_p^{M_p+1}}
$$
in $\Omega$. Thus locally in $\Omega\setminus W$,
$$
\mu.\xi=\int_{w=0}\sum_{\alpha\le M}c_\alpha\frac{\partial^{M-\alpha}A(z,0)}{\partial w^{M-\alpha}}
\gamma\neg \frac{\partial^\alpha\xi}{\partial w^\alpha},
$$
where $\gamma\neg$ is contraction with the vector field
$$
\gamma=\frac{\partial}{\partial w_p}{\wedge}\ldots{\wedge} \frac{\partial}{\partial w_1}.
$$
Now  $\mu\phi=0$ if and only if for all test forms $\xi$,
\begin{equation}\label{lotta}
0=\mu\phi.\xi=
\int_{w=0}\sum_{\ell\le M} (Q_\ell\phi)\gamma\neg \frac{\partial^\ell}{\partial w^\ell}\xi,
\end{equation}
where
$$
Q_\ell=\sum_{\ell\le\alpha\le M}
c_{\alpha,\ell}\frac{\partial^{M-\alpha}A}{\partial w^{M-\alpha}}
\frac{\partial^{\alpha-\ell}}{\partial w^{\alpha-\ell}}.
$$
Applying to $\xi=w^\ell\eta$
(induction over $\ell$ downwards) it follows that \eqref{lotta}  holds for all
$\xi$   if and only if
$Q_\ell\phi=0$ (locally) on $V\setminus W$ for all $\ell\le M$.

However,
$\partial\omega/\partial w=(\partial f/\partial\omega)^{-1}=\gamma/h
$
where $\gamma$ is a holomorphic matrix in $\Omega$.
It follows that ${\mathcal L}_\ell=h^NQ_\ell$ are well-defined and holomorphic in $\Omega$
if $N$ is large enough, and by the SEP  $\mu\phi=0$ in $\Omega$
 if and only if \eqref{bjarne} holds.
\end{proof}

One also get a global (in $\Omega$) representation of $\mu$ in this way:
Notice that for some $L$,  $\tilde\gamma=h^L\gamma$ is holomorphic
in $\Omega$. One can verify that (with ${\mathcal L}={\mathcal L}_0$), actually
$$
\mu.\xi=\int_Z\frac{1}{h^{M+L}}\tilde\gamma\neg {\mathcal L}\xi
$$
for all test forms, if the right hand side is interpreted as a principal
value.

\section{Polynomial ideals and Noetherian operators}\label{algcase}

Let $J$ be a submodule of ${\mathbb C}[z_1,\ldots,z_n]^r$, and assume that
$F={\mathbb C}[z_1,\ldots,z_n]^r/J$ has positive codimension $p$.
From a free resolution of the of the corresponding homogeneous
module over the graded ring $S={\mathbb C}[z_0,\ldots,z_n]$ we constructed
in \cite{AW1} a residue current on ${\mathbb P}^n$ whose restriction
$R$ to ${\mathbb C}^n$ has the property that $\Phi\in
{\mathbb C}[z_1,\ldots,z_n]^r$  is in $J$ if and only if $R\Phi=0$ in ${\mathbb C}^n$.
If $F$ has pure codimension, then precisely as in the
semi-global case above we have that
$\Phi$ is in $J$ if and only if $R_p\Phi=0$. By the same proofs we get
complete analogues of Theorem~\ref{thm1} and its corollaries.
In particular if $F$ is pure,
we get  a finite number of global Coleff-Herrera
currents $\mu_j=\xi R_p$ such that $\Phi\in J$ if and only if
$\mu_j\Phi=0$ for each $j$.  Moreover, since $R_p$  has a current
extension to ${\mathbb P}^n$, following  the proof in the
preceding section with $\Omega={\mathbb C}^n$,
one obtains for each $\mu_j$ a finite set of
differential operators ${\mathcal L}_{j\ell}$ with polynomial
coefficients such that
${\mathcal L}_{j\ell}\Phi$ vanishes on $Z$ for all $\ell$ if and only if
$\Phi$ is in the annihilator of $\mu_j$.
Starting with a primary decomposition of $J$ we obtain in this way
a complete proof of the existence of Noetherian
operators for an arbitrary polynomial ideal, a fact first proved
by Ehrenpreis and Palamodov as the corner stone  in the
celebrated fundamental principle, see
\cite{Ehr}, \cite{Pal}, \cite{Hor}, and  \cite{JEB1}.
For a discussion about effectivity, see  \cite{Ob}.

\section{The injectivity of the analytic sheaf   ${\mathcal C}$}\label{malgrange}

We conclude this paper with a  residue proof of
Malgrange's theorem.
Let ${\mathcal F}$ be any module over the local ring ${\mathcal O}_0$ and let
\eqref{acomplex} be a resolution of ${\mathcal F}$. We have to prove that then
the complex
\begin{equation}\label{mula}
0\to {\rm Hom\, }({\mathcal O}_0(E_0),{\mathcal C})\stackrel{f_1^*}{\longrightarrow}
{\rm Hom\, }({\mathcal O}_0(E_1),{\mathcal C}) \stackrel{f_2^*}{\longrightarrow}
\end{equation}
is exact except at $k=0$.
Fix a natural number $N$. Given a  smooth function $\phi$ in $X\subset{\mathbb C}^n$, let
$\tilde\phi$ be the function
$$
\tilde\phi(\zeta,\omega)=\sum_{|\alpha| <N}\partial^\alpha_{\bar\zeta}\phi(\zeta)(\omega-\bar\zeta)
^\alpha/\alpha!,
$$
in $\tilde X=\{(\zeta,\omega)\in{\mathbb C}^{2n};\ \zeta\in X\}$.
Then
$$
\tilde\phi(\zeta,\bar\zeta)=\phi(\zeta),\quad  \bar\partial\tilde\phi=O(|\omega-\bar\zeta|^N).
$$
Moreover, if $f$ is holomorphic then
$\widetilde{f\phi}=f\tilde\phi.$
Combining the formulas in \cite{A7} with the construction in \cite{A3},
we get
$$
\tilde\phi(z,\bar z)=
\int_{\zeta,\omega} (f_{k+1}(z)H^kU^k+H^kR^k+H^{k}U^{k-1}f_k)
{\wedge}(\tilde\phi+\bar\partial\tilde\phi{\wedge} v^z){\wedge} g,
$$
where $g$ is a suitable form in ${\mathbb C}^{2n}$ with compact support and
$v^z$ is the Bochner-Martinelli form in ${\mathbb C}^{2n}$ with pole at $(z,\bar z)$,
and $H^\ell$ are holomorphic forms.
Since $R^k=0$ for $k\ge 1$ when \eqref{acomplex} is a resolution,
we have the  homotopy formula
$$
\phi = f_{k+1}T_{k+1}\phi+T_k(f_k\phi),\quad  h\ge 1,
$$
where
$$
T_k\phi(z)=\int_{\zeta,\omega}H^{k}U(\tilde\phi+\bar\partial\tilde\phi{\wedge} v^z){\wedge} g^z.
$$
Moreover, as in \cite{A3} one can verify that
$T_k\phi$ is of class $C^M$ if $N$ is large enough.
If now $\mu$ has order at most $M$, then we have
$$
\mu=T^*_{k+1}f^*_{k+1}\mu+f_k^*T^*_k\mu,
$$
so if $f^*_{k+1}\mu=0$, then $\mu=f^*_k\gamma$  if $\gamma=T^*_k\mu$.
Thus \eqref{mula} is exact at $k$.

\def\listing#1#2#3{{\sc #1}:\ {\it #2},\ #3.}

redpajama_set_name: RedPajamaArXiv


Are We There Yet? — That Cost How Much?
With my summer vacation (finally!) on the horizon, I can't help but have resort wear on the brain. I think the things you wear on vacation should be your absolute favorites - items you can't wear in your normal day-to-day life, the things you long to wear all year round. They should transform and transport. For me, that always means an exotic print, a new silhouette, and as many tassels as possible.
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text:
Q: How to put a \n(new line) inside a list box? How to put the newline(\n) in list box .selected item? Here I have code it generates the all links but I want to have all those link in listbox, these code is working but the links are not coming in the new line  all comes in single line
and my code is:
var links = TextBox1.Text.Split(new string[] { "\n", "\r" },
StringSplitOptions.RemoveEmptyEntries);

  foreach (var link in links)
        {
            if (!IsLinkWorking(link))
            {
                //Here you can show the error. You don't specify how you want to show it.
                TextBox2.ForeColor = System.Drawing.Color.Green;
               TextBox2.Text += string.Format("{0}\nNot working\n\n ", link);
                //ListBox1.SelectedItem+= string.Format("{0}\nNot working\n\n ", link);
            }
            else
            {
               // ListBox1.SelectedValue += string.Format("{0}\nNot working\n\n ", link);
                TextBox2.Text += string.Format("{0}\n working\n\n", link);
            }

string[] values = TextBox2.Text.Split(',');

foreach (string value in values)
{
   if (value.Trim() == "")
       continue;
   ListBox1.Items.Add(value.Trim());
       }
    }
}

A: You can use Regex.Split to split up your string to multiple lines like this:
foreach (string s in Regex.Split(TheStringwithNewLines, "\n"))
ListBox.Items.Add(s);

A: Your code should work fine, I can only suspect that you are not splitting the string properly, are you sure that you have , delimiter in your string , because only then you will see all the items in one line, since split would produce only a single item
Just try the sample code and it will add the items in multiple lines.
string str = "somelink1,somelink2,somelink3";
string[] values = str.Split(',');

foreach (string value in values)
{
    if (value.Trim() == "")
        continue;
    ListBox1.Items.Add(value.Trim());
}

Now if for some reason you end up with string "somelink1 somelink2 somelink3", without the comma as delimiter , you will get a single string in your array and that will appear in a single line.
EDIT: Based on your edited question.
You are adding \n and then you are trying to split it on comma your split statement should be:
string[] values = TextBox2.Text.Split(new[] { "\n" }, StringSplitOptions.RemoveEmptyEntries);

A: Try to see if there is something like Listbox.Items.Row
or
ListBox.Items.Add(value.Trim() + "<BR />");

A: I guess you can use this code
&#13

redpajama_set_name: RedPajamaStackExchange


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The FA have not produced a hard copy Disciplinary Handbook again this season but they have produced an online interactive reference guide to aid Leagues, Clubs and Players through the forthcoming season.
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This year's guide also includes videos and resources that will go into greater depth to explain the disciplinary process.
Plus there is a new Leagues section to provide guidance for administrators regarding the Powers and Duties of the League.


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