TITLE: Relating the sheaf associated to a cyclic module over an affine scheme to the structure sheaf
QUESTION [4 upvotes]: Let $(X,\mathcal O_X)$ be the affine scheme of a commutative Noetherian ring $R$. Let $J$ be an ideal of $R$. From the sheaf $\mathcal O_X$ and the closed subset $Z:=V(J)$ of $X$ , how do we recover the sheaf of $\mathcal O_X$-modules on $X$ associated to the $R$-module $R/J$ ? Is it  $\mathcal O_X|_Z:=i^{-1}(\mathcal O_X)$ or $i^*(\mathcal O_X)$ (where $i:Z \to X$ is the inclusion) or something else ?
(In case it is unclear, by the sheaf associated to an $R$-module $M$, I mean the sheaf which takes every basic open set of the form $D(f)$ of $X$ to the localization $M_f$ )
NOTE: I guess strictly speaking, neither $\mathcal O_X|_Z$ nor $i^*(\mathcal O_X)$ is a Sheaf on $X$ but rather on $Z$. By the comment of user Ruben du Burck below it seems like the sheaf on $X$ I'm looking for is $i^*(\mathcal O_X)$ extended to $X$ by zero outside of $Z$ i.e. just $i_*(i^*(\mathcal O_X))$. But I'm still on the lookout for a definite answer. Thanks.

REPLY [3 votes]: You cannot recover $R/J$ from just the sheaf $\mathcal{O}_X$ and the set $Z$; if you could, that would mean any two ideals $J,J'$ with $V(J)=V(J')$ are the same, but this is not true.
If you know $Z$ not just as a closed subset of $X$ but as the closed subscheme assoociated to the ideal $J$, then the sheaf on $X$ you seek is just $i_*(\mathcal{O}_Z)\cong i_*(i^*(\mathcal{O}_X))$.  This follows immediately from the fact that if $\varphi:A\to B$ is a homomorphism of rings with corresponding morphism of schemes $f:\operatorname{Spec} B\to \operatorname{Spec} A$, then the operation $f_*$ on quasicoherent sheaves corresponds to taking a $B$-module and considering it as an $A$-module via $\varphi$, and the operation $f^*$ corresponds to $B\otimes_A -$.  In this case, $A=R$ and $B=R/J$, so to get $R/J$ as an $R$-module, you just take $i_*(\mathcal{O}_Z)$, and $\mathcal{O}_Z$ is also the same as $i^*(\mathcal{O}_X)$ since $R/J\otimes_R R\cong R/J$.


TITLE: Calculating a surface integral over some ellipsoid
QUESTION [0 upvotes]: a) Let $S$ the surface $4x^2+9y^2+36z^2=36$, $z \ge 0$. Let $\vec{F}=y\vec{i}+x^2\vec{j}+(x^2+y^4)^{3/2}\sin(e^{xyz})\vec{k}.$  Calculate the integral $\iint (\text{curl }\vec{F})\cdot\vec{n}\,dS$.
b) Calculate the same integral when $S$ is the whole surface of the ellipsoid.
My solution: $\vec{F}=(F_1,F_2,F_3),\ F_1(x,y,z)=y,\ F_2(x,y,z)=x^2$
$\vec{σ}:[0,2\pi]\rightarrow \mathbb{R^2},\ \vec{σ}=(x(t),y(t)),\ x(t)=3cost,\ y(t)=2sint$
$\iint (\text{curl }\vec{F})\cdot\vec{n}\,dS=\int_{θS}\vec{F}\cdot d\vec{s}=\int_{C^+}F_1dx+F_2dy=\int_0^{2\pi}(F_1\frac{dx}{dt}+F_2\frac{dy}{dt})dt=\int_0^{2\pi} y\frac{dx}{dt}+x^2\frac{dy}{dt}=\int_0^{2\pi}[2sint(-3sint)+(3cost)^22cost]dt=-6\pi$
Is there any other solution?

REPLY [1 votes]: $$\large\iint\limits_{S}{\operatorname{curl}\,F\,.n\,ds=\iint\limits_{{{S}^{'}}}{\operatorname{curl}\,F\,.n\,ds'}}$$
where
$${S}'=\left\{ (x,y,0)\left| \,\,\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}\le 1 \right. \right\}$$
we have
$$\iint\limits_{S}{\operatorname{curl}\,F\,.n\,ds}=\oint\limits_{C}{F.dr} =\iint\limits_{S'}\left( \frac{\partial {{F}_{2}}}{\partial x}-\frac{\partial {{F}_{1}}}{\partial y} \right) ds'=\iint\limits_{S'}(2x-1)dA$$
let $x=3r\cos\theta\,$ and $y=2r\sin\theta\,$, thus
$$\iint\limits_{S}{\operatorname{curl}\,F\,.n\,ds}=\int_{0}^{2\pi }{\int_{0}^{1}{(6r\cos \theta -1)6r\,drd\theta }} =-6\pi$$


TITLE: How to convert an object into a sphere?
QUESTION [0 upvotes]: I'm not sure if I understand it enough, but doesn't the Poincare conjecture prove any shape can be turned into a sphere? How would I go about transforming such an object? Like let's say I have a rectangular prism with length, width, and height 3,4,5. How would I find the characteristics of the sphere after I create it. This is topology correct? What would I study or look up to understand?
Thanks for any help!
Update:
From the previous comment "A prism is the boundary of a convex subset of R3
R3 so just pick a point on the interior and project along a line from that point onto a sphere which is large enough to enclose the prism. This will define a homeomorphism from the prism to the sphere."
Can someone explain this in more detail?

REPLY [1 votes]: $\newcommand{\Reals}{\mathbf{R}}$The stereogram shows a rectangular prism being mapped to a larger sphere by radial projection from the center. (The action on the vertices is indicated by dashed lines.)

Analytically, center the prism $P$ at the origin, let $R$ denote the radius of the sphere $S$, and let
$$
T(x, y, z) = \frac{R}{\sqrt{x^{2} + y^{2} + z^{2}}}(x, y, z)
$$
denote radial projection from $\Reals^{3} \setminus\{(0, 0, 0)\}$, the complement of the origin, to the sphere $S$. The mapping $T$ is continuous on its domain, the prism $P$ is a closed, bounded (i.e., compact) subset of $\Reals^{3}$ that does not contain the origin and that intersects each ray from the origin exactly once. It follows that $T$ defines a continuous bijection from $P$ to $S$. Since $P$ is compact, this bijection is a homeomorphism.


text:
\begin{document}

\maketitle

\begin{abstract}
    We extend Orlov's result that certain functors between derived categories of smooth projective varieties are Fourier--Mukai transforms to the case when those varieties are smooth and proper.
\end{abstract}

\section{Introduction}
In the landmark paper \cite{orlov1997equivalences}, Orlov proved that for smooth projective varieties $X, Y$ over a field $k$, a fully faithful, $k$-linear, exact functor $D^b_{coh}(X) \to D^b_{coh}(Y)$ having a right adjoint is isomorphic to a Fourier--Mukai transform. In this paper, we will show how to use modifications of Orlov's argument introduced by Ballard, de Jong, and this author to prove Orlov's Theorem in the smooth proper case:

\begin{thm}
Let $X, Y$ be smooth proper varieties over a field $k$. Let
$F : D^b_{coh} (X) \to D^b_{coh} (Y)$ be a fully faithful functor which is exact and $k$-linear. Then there is an object $E$  of $D^b_{coh} (X \times Y)$ such that $F$ is isomorphic to the functor
\begin{equation*}
    \Phi_E (K) = \mathbf{R} pr_{2 *} (\mathbf{L}pr_1^*K \otimes ^{\mathbf{L}}_{\mathcal{O}_{X \times Y}} E).
\end{equation*}
\end{thm}

We find the modifications to be pleasantly natural, so that what shines through is the power and elegance of Orlov's original proof. We also have access to the results of Bondal--Van den Bergh on strong generators and existence of adjoints (\cite{bondal2003}), which help simplify our statements and our proofs. The paper is organized as follows:

In Section 2, we show how to produce an object $E \in D^b_{coh} (X \times Y)$. Orlov's construction proceeds by embedding $X$ in projective space and then integrating a complex obtained from the Beilinson resolution of the diagonal. In \cite{2009arXiv0905.3148B}, Ballard shows how to replace the Beilinson resolution of the diagonal of $\mathbf{P}^N \times \mathbf{P}^N$ with any resolution of the diagonal of $X \times X$ by box products of vector bundles, thus immediately producing an object of $D^b_{coh} (X \times Y).$ There is no difficulty extending Ballard's ideas from the quasi-projective case to the smooth proper case since a smooth proper variety has enough vector bundles.

Having produced a kernel $E$, we conclude Section 2 by asking for which coherent $\mathcal{O}_X$-modules $\mathcal{F}$ is $F(\mathcal{F}) \cong \Phi_E(\mathcal{F})$. Orlov uses vanishing of cohomology to prove this for all powers of a very ample invertible sheaf $\mathcal{O}_X(1),$ and actually produces natural isomorphisms in this case. Not having ample line bundles at our disposal, we instead follow an idea of de Jong and show that $F$ and $\Phi_E$ agree on the spanning class of skyscraper sheaves of points.

In sections 3 and 4, we show that in fact $F$ and $\Phi_E$ differ only by tensor product with a line bundle on $X$. Since tensoring with a line bundle is a Fourier--Mukai functor and compositions of Fourier--Mukai functors are Fourier--Mukai, this completes the proof. The task is split into two separate parts: First we show in Section 3 that this is true when we restrict to the category of coherent sheaves. Then in Section 4 we extend this to the entire derived category. Section 4 should be compared with the proof of Proposition 2.16 in \cite{orlov1997equivalences} from which our section is taken almost verbatim. The key difference is that while Orlov uses an ample sequence $\mathcal{O}_X(-n)$, we use the the \emph{sequences} $\mathcal{O}_X (-nD_i)$ where $D_i$ are the complements of finitely many affine opens $U_i$ which cover $X$. We are able to make this work by exploiting the vanishing of
\begin{equation*}
    \mathrm{colim}_{n \geq 0}H^j (X, \mathcal{F}(nD_i))
\end{equation*}
for $j>0$, $\mathcal{F}$ coherent as a replacement for actual Serre vanishing.

The author would like to warmly thank Johan de Jong for asking him this question and for being so enthusiastic and available for mathematical discussions, especially during these strange times. This work would not have been possible without his idea to use points instead of negative line bundles.

\section{Producing a Kernel}

We produce here a candidate for a kernel of our functor. This section is basically an exposition of the construction in \cite[\href{https://stacks.math.columbia.edu/tag/0G07}{Tag 0G07}]{stacks-project}, so some parts are sketched. The outcome is summarized in Proposition 1, which is all that is needed for the sequel.

Let $X, Y$ be smooth proper varieties over a field $k$, and let $F : D^b_{coh} (X) \to D^b_{coh} (Y)$ be a fully faithful, exact, $k$-linear functor. The following Lemma will play a similar role in our construction to the role Lemma 2.4 of \cite{orlov1997equivalences} plays in Orlov's.
\begin{lemma}
$F$ is bounded: There is $m>0$ such that for every coherent sheaf $\mathcal{F}$ on $X$, $F(\mathcal{F})$ has nonzero cohomology sheaves only in degrees $[-m, m].$
\end{lemma}
\begin{proof}
This is proved in \cite[\href{https://stacks.math.columbia.edu/tag/0FZ8}{Tag 0FZ8}]{stacks-project}. It is a consequence of Bondal--Van den Bergh's theorem that $D^b_{coh} (X)$ has a strong generator. See \cite{bondal2003}.
\end{proof}
Since $X$ has enough vector bundles (see Lemma A in Section 4), a very general argument shows that every coherent sheaf on $X \times X$ is the quotient of a vector bundle $\mathcal{E} \boxtimes \mathcal{F}$ where $\mathcal{E}$, $\mathcal{F}$ are vector bundles on $X$ (\cite[\href{https://stacks.math.columbia.edu/tag/0FZ0}{Tag 0FZ0}]{stacks-project}). Therefore, we may construct a resolution of the structure sheaf of the diagonal $\Delta \subset X \times X$ by such vector bundles:
\begin{equation}
    \cdots \to \mathcal{E}_n \boxtimes \mathcal{F}_n \to \cdots \to \mathcal{E}_0 \boxtimes \mathcal{F}_0 \to \mathcal{O}_\Delta \to 0
\end{equation}
Then by applying $F$ to the second box tensor factors, we obtain a complex
\begin{equation}
    \cdots \to \mathcal{E}_n \boxtimes F(\mathcal{F}_n) \to \cdots \to \mathcal{E}_0 \boxtimes F(\mathcal{F}_0)
\end{equation}
in $D^b_{coh} (X \times Y)$ (the pullbacks and tensor products defining the box products in (2) are derived). By the K\"unneth formula,
\begin{align*}
    \mathrm{Ext}^k _{\mathcal{O}_{X\times Y}}(\mathcal{E}_i \boxtimes F(\mathcal{F}_i), \mathcal{E}_j \boxtimes F(\mathcal{F}_j))
    &= \bigoplus _{m + n = k} \mathrm{Ext}^m_{\mathcal{O}_X} (\mathcal{E}_i , \mathcal{E}_j) \otimes _k \mathrm{Ext}^n_{\mathcal{O}_Y} (F(\mathcal{F}_i), F(\mathcal{F}_j)) \\
    &= \bigoplus _{m + n = k} \mathrm{Ext}^m_{\mathcal{O}_X} (\mathcal{E}_i , \mathcal{E}_j) \otimes _k \mathrm{Ext}^n_{\mathcal{O}_X} (\mathcal{F}_i, \mathcal{F}_j),
\end{align*}
where we have used fully faithfulness for the second equality. This vanishes if $k<0,$ so there is no obstruction to integrating the complex (2) to a right Postnikov system:

\begin{equation}
    \begin{tikzcd}[column sep=tiny]
         \cdots & &E_2 \ar[ll, "+1"'] \ar[dr] & &E_1 \ar[ll, "+1"'] \ar[dr] \ar[dr] & &E_0 \ar[ll, "+1"'] \ar[dr, "\cong"] &\\
         & \cdots \ar[rr] & & \mathcal{E}_2 \boxtimes F(\mathcal{F}_2) \ar[rr] \ar[ur] & & \mathcal{E}_1 \boxtimes F(\mathcal{F}_1) \ar[rr]\ar[ur] & & \mathcal{E}_0 \boxtimes F(\mathcal{F}_0)
    \end{tikzcd}
\end{equation}
The triangles with an arrow of degree $+1$ are distinguished and the other triangles commute. The objects $E_n$ are unique up to non-unique isomorphism. See \cite[\href{https://stacks.math.columbia.edu/tag/0D7Y}{Tag 0D7Y}]{stacks-project}. Note that we use right Postnikov systems instead of left systems as in \cite{orlov1997equivalences}. This is because our complexes are unbounded to the left.

We also have a trivial Postnikov system
\begin{equation}
    \begin{tikzcd}[column sep=tiny]
         \cdots & &K_2 \ar[ll, "+1"'] \ar[dr] & &K_1 \ar[ll, "+1"'] \ar[dr] \ar[dr] & &K_0 \ar[ll, "+1"'] \ar[dr, "\cong"] &\\
         & \cdots \ar[rr] & & \mathcal{E}_2 \boxtimes \mathcal{F}_2 \ar[rr] \ar[ur] & & \mathcal{E}_1 \boxtimes \mathcal{F}_1 \ar[rr]\ar[ur] & & \mathcal{E}_0 \boxtimes \mathcal{F}_0,
    \end{tikzcd}
\end{equation}
with $K_n$ the $n^{th}$ naive truncation of the complex (1). Thus $K_n$ has cohomology sheaves only in degrees 0 and $-n$. Since $X\times Y$ has finite homological dimension, for $n\gg 0$ we have $K_n \cong \mathcal{O}_\Delta \oplus \mathcal{H}^{-n} (K_n)[n],$ and $\mathcal{H}^{-n}(K_n)$ is a locally free $\mathcal{O}_{X \times Y}$-module. Our task in the remainder of the section is to show that $E_n$ has a similar decomposition $E \oplus C_n$ where $E$ is the desired kernel and $C_n$ slides off to $- \infty$.

Let us compute $\Phi_{E_n} (\mathcal{F})$ for $\mathcal{F}$ a coherent $\mathcal{O}_X$-module whose support has dimension 0: Apply the functor
\begin{equation*}
    K \mapsto \mathbf{R}pr_{2 *} (\mathbf{L}pr_1^*(\mathcal{F}) \otimes _{\mathcal{O}_{X \times Y}}^{\mathbf{L}} K)
\end{equation*}
to the entire diagram (3) to see that $\Phi_{E_n} (\mathcal{F})$ is an $n^{th}$ convolution of the complex
\begin{equation}
\Gamma (X, \mathcal{F} \otimes \mathcal{E}_{\bullet}) \otimes _k F(\mathcal{F}_\bullet)
\end{equation}
(the functor $\Gamma$ is underived since $\mathcal{F}$ has $0$-dimensional support).

On the other hand, we can apply the functor
\begin{equation*}
    K \mapsto F(\mathbf{R} pr_{2 *} (\mathbf{L}pr_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times X}}^{\mathbf{L}} K))
\end{equation*}
to the entire diagram (4). This transforms the second row of (4) into the complex (5). Thus, for $n \gg 0$, another $n^{th}$ convolution of the complex (5) is
\begin{align*}
    F(\mathbf{R} pr_{2 *} (\mathbf{L}pr_1^*\mathcal{F} &\otimes _{\mathcal{O}_{X \times X}}^{\mathbf{L}} K_n)) \\ & = F(\mathbf{R} pr_{2 *} (\mathbf{L}pr_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times X}}^{\mathbf{L}} \mathcal{O}_{\Delta})) \oplus F(\mathbf{R} pr_{2 *} (\mathbf{L}pr_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times X}}^{\mathbf{L}} \mathcal{H}^{-n}(K_n)))[n] \\ & = F(\mathcal{F}) \oplus F(\mathcal{K}_n)[n],
\end{align*}
where $\mathcal{K}_n$ is a coherent sheaf on $X$.

Therefore, by uniqueness of convolutions (note that negative Ext's vanish between the terms of the complex (5) since $F$ is fully faithful), we obtain isomorphisms
\begin{equation*}
    \Phi_{E_n} (\mathcal{F}) \cong F(\mathcal{F}) \oplus F(\mathcal{K}_n)[n].
\end{equation*}
By Lemma 1, we deduce that there is $m>0$ so that for every $\mathcal{F}$ coherent with 0-dimensional support and $n \gg 0$, $\Phi_{E_n} (\mathcal{F})$ has cohomology sheaves only in degrees $[-n-m, -n+m] \cup [-m, m].$ But knowing this for every such $\mathcal{F}$ implies that $E_n$ has cohomology sheaves only in degrees $[-n-m, -n+m] \cup [-m, m]$ (see \cite[\href{https://stacks.math.columbia.edu/tag/0FZ9}{Tag 0FZ9}]{stacks-project} for details). Thus if we take $n$ to be very large, since $X \times Y$ has finite homological dimension, we will have a decomposition $E_n = E \oplus C_n,$ where $E$ has cohomology sheaves in degrees $[-m,m]$ and $C_n$ has cohomology sheaves in degrees $[-n-m, -n+m].$ Then it follows that $\Phi_E(\mathcal{F}) \cong F(\mathcal{F})$ and $\Phi_{C_n} (\mathcal{F}) \cong F(\mathcal{K}_n)[n].$ Thus we have proved:

\begin{prop}
Let $X, Y$ be smooth proper varieties over a field $k$. Let $F : D^b_{coh} (X) \to D^b_{coh} (Y)$ be a fully faithful functor which is exact and $k$-linear. Then there is an object $E$  of $D^b_{coh} (X \times Y)$ such that $F(\mathcal{F}) \cong \Phi_E(\mathcal{F})$ for every coherent sheaf $\mathcal{F}$ on $X$ whose support has dimension 0.
\end{prop}

\section{Fully Faithful Functors Agreeing on Points}

The following lemma should be compared with Lemma 2.15 in \cite{orlov1997equivalences}.

\begin{lemma}
Let $X, Y$ be smooth proper varieties over a field $k$ and
$F: D^b_{coh}(X) \to D^b_{coh}(Y)$ an exact, $k$-linear functor. Suppose
\begin{equation*}
F: \mathrm{Ext}^*_{\mathcal{O}_X} (k(x_1), k(x_2)) \to \mathrm{Ext}^*_{\mathcal{O}_Y} (F(k(x_1)), F(k(x_2)))
\end{equation*}
is an isomorphism for every pair of closed points $x_1, x_2 \in X$. Then $F$ is fully faithful.
\end{lemma}

\begin{proof}
By \cite{bondal2003}, $F$ has exact $k$-linear left and right adjoints $R$ and $L$. By our assumption,
\begin{align*}
    \mathrm{Ext}^*_{\mathcal{O}_X} (k(x_1), k(x_2)) &=  \mathrm{Ext}^*_{\mathcal{O}_X} (F(k(x_1)), F(k(x_2)))
    \\ &= \mathrm{Ext}^* _{\mathcal{O}_X} (k(x_1), RF(k(x_2)))
\end{align*}
for every pair $x_1, x_2$. Since the sheaves $k(x_1)$ form a spanning class, we conclude that the unit $\mathbf{1}_{D^b_{coh}(X)} \to RF$ is an isomorphism on the objects $k(x)$. Now let $K$ be an object of $D^b_{coh}(X)$ and $x \in X$ a closed point. Then
\begin{align*}
    \mathrm{Ext}^*_{\mathcal{O}_X} (LF(K), k(x)) &= \mathrm{Ext}^*_{\mathcal{O}_Y} (F(K), F(k(x))) \\ &= \mathrm{Ext}^*_{\mathcal{O}_X} (K, RF(k(x))) \\
    &=\mathrm{Ext}^*_{\mathcal{O}_X}(K , k(x)),
\end{align*}
induced by the co-unit $LF \to \mathbf{1}_{D^b_{coh} (X)}.$ Again, since the objects $k(x)$ form a spanning class, we conclude that $LF(K) \to K$ is an isomorphism for every $K$, so $F$ is fully faithful.
\end{proof}

\begin{lemma}
Let $X, Y$ be smooth proper varieties over a field $k$. Let $F, G : D^b_{coh} (X) \to D^b_{coh}(Y)$ be exact, $k$-linear functors. Suppose $F$ is fully faithful and $F(k(x)) \cong G(k(x))$ for every closed point $x \in X$. Then the essential image of $G$ is contained in the essential image of $F$.
\end{lemma}

\begin{proof}
By \cite{bondal2003}, $G$ has an exact, $k$-linear right adjoint $R$. Applying the same theorem to $F$ shows that the essential image $\mathcal{A}$ of $F$ is an admissible subcategory of $D^b_{coh} (Y)$. Thus we only have to show that $G(K) \in \prescript{\perp}{}{(\mathcal{A}^\perp)}$ for every $K$ in $D^b_{coh}(X).$ So, suppose $L$ is in $\mathcal{A}^\perp$. Then
\begin{equation}
    \mathrm{Hom} (G(K), L) = \mathrm{Hom} (K, R(L)),
\end{equation}
so it suffices to show $R(L) = 0.$ Applying (6) to the objects $K = k(x)[i]$ gives
\begin{equation*}
    \mathrm{Ext}^*_{\mathcal{O}_X} (k(x), R(L)) = \mathrm{Ext}^*_{\mathcal{O}_Y} (G(k(x)), L) = \mathrm{Ext}^*_{\mathcal{O}_Y} (F(k(x)), L) = 0
\end{equation*}
since $L \in \mathcal{A}^\perp$. But the objects $k(x)$ form a spanning class so it follows that $R(L) = 0$ as needed.
\end{proof}

\begin{prop}
Let $X$ be a smooth proper variety over a field $k$. Let $F: D^b_{coh}(X) \to D^b_{coh}(X)$ be an exact, $k$-linear functor. Suppose $F(\mathcal{F}) \cong \mathcal{F}$ for every coherent sheaf on $X$ with 0-dimensional support. Then $F$ is an equivalence of categories. Furthermore, there exists a line bundle $\mathcal{L}$ on $X$ and a $k$-linear automorphism $f : X \to X$ such that, if we denote by $G$ the functor
\begin{equation*}
    G(K) = \mathbf{L}f^* K \otimes _{\mathcal{O}_X}^\mathbf{L} \mathcal{L},
\end{equation*}
then $F|_{Coh(\mathcal{O}_X)} \cong G|_{Coh(\mathcal{O}_X)}.$
\end{prop}

\noindent \textbf{Remark.} In fact, since $F(k(x)) \cong k(x)$ for every closed point $x \in X$, the automorphism $f$ must be the identity on underlying topological spaces. This implies that $f$ is the identity if $X$ has dimension at least 1 \cite[\href{https://stacks.math.columbia.edu/tag/0G05}{Tag 0G05}]{stacks-project}. This is irrelevant to us though: We care only that $G$ is a Fourier--Mukai auto-equivalence.

\begin{proof}
\underline{Step 1:} $F$ is an equivalence of categories.

Applying Lemma 3 to the functors $\mathbf{1}_{D^b_{coh} (X)}$ and $F$, we see that $F$ is essentially surjective. To prove that $F$ is fully faithful, it suffices by Lemma 2 to prove that
\begin{equation*}
    F: \mathrm{Ext}^*_{\mathcal{O}_X} (k(x), k(y)) \to \mathrm{Ext}^*_{\mathrm{O}_X} (F(k(x)), F(k(y)))
\end{equation*}
is an isomorphism for every pair of closed points $x, y$. It follows from our assumptions that both sides are trivial if $x \neq y$, so there is nothing to prove. If $x = y,$ both sides are isomorphic to
\begin{equation*}
    \bigwedge_{k(x)} \nolimits^* \mathrm{Ext}^1_{\mathcal{O}_X} (k(x), k(x)),
\end{equation*}
and moreover, $F$ is a $k$-algebra homomorphism. It will therefore suffice to show $F$ is surjective in degrees 0 and 1 (or equivalently injective or bijective). In degree 0, every non-zero homomorphism $k(x) \to k(x)$ is an isomorphism, and functors preserve isomorphisms, so indeed $F$ is injective in degree 0. Suppose given a non-zero extension class $\xi$ of $k(x)$ by $k(x),$ represented by a non-split short exact sequence
\begin{equation*}
    0 \to k(x) \to \mathcal{E} \to k(x) \to 0
\end{equation*}
of coherent sheaves. Then $F(\mathcal{E})$ is a coherent sheaf, and $F(\xi)$ is represented by the short exact sequence
\begin{equation*}
    0 \to F(k(x)) \to F(\mathcal{E}) \to F(k(x)) \to 0
\end{equation*}
of coherent sheaves. Furthermore, we have abstract isomorphisms $F(k(x)) \cong k(x)$ and $F(\mathcal{E}) \cong \mathcal{E}$ by assumption. But now we simply note that an arbitrary extension $\mathcal{E}'$ of $k(x)$ by $k(x)$ is trivial if and only if $\mathcal{E}' \cong k(x) \oplus k(x).$ Thus $F( \mathcal{E}) \cong \mathcal{E} \not \cong k(x) \oplus k(x),$ so $F(\xi) \neq 0$.

\underline{Step 2:} If $\mathcal{E}$ is a locally free $\mathcal{O}_X$-module of rank $r$, then $F(\mathcal{E}[0])$ is also locally free rank $r$ sitting in degree zero.

Because locally free sheaves of rank $r$ are those objects $K$ of $D^b_{coh}(X)$ such that
\begin{equation*}
    \mathrm{Ext}^*_{\mathcal{O}_X} (K, k(x)) \cong k(x)^{\oplus r} [0]
\end{equation*}
for every closed point $x \in X$.

\underline{Step 3:} If $D$ is an effective divisor on $X$, then $F(\mathcal{O}_D) \in Coh(\mathcal{O}_X)$.

We have an exact triangle
\begin{equation*}
    F(\mathcal{O}_X(-D)) \to F(\mathcal{O}_X) \to F(\mathcal{O}_D) \to F(\mathcal{O}_X(-D))[1],
\end{equation*}
thus we see that the cohomology sheaves of $F(\mathcal{O}_D)$ are zero except possibly in degrees $0$ and $-1$, and there is a long exact sequence
\begin{equation*}
    0 \to \mathcal{H}^{-1} (F(\mathcal{O}_D)) \to F(\mathcal{O}_X(-D)) \to F(\mathcal{O}_X) \to \mathcal{H}^{0} (F( \mathcal{O}_D)) \to 0,
\end{equation*}
where by Step 1, $F(\mathcal{O}_X(-D))$ and $F(\mathcal{O}_X)$ are line bundles on $X$. We need to show that $F(\mathcal{O}_X(-D)) \to F(\mathcal{O}_X)$ is injective, but here that's equivalent to
\begin{equation*}
    \mathrm{Supp} \hspace{.5em} \mathcal{H}^{0} (F( \mathcal{O}_D)) \subsetneq X.
\end{equation*}
This follows from
\begin{align*}
    \mathrm{Hom} _{\mathcal{O}_X} (\mathcal{H}^0 (F(\mathcal{O}_D)), k(x)) &= \mathrm{Hom} _{\mathcal{O}_X} (F(\mathcal{O}_D), k(x)) \\
    &= \mathrm{Hom} _{\mathcal{O}_X} (F(\mathcal{O}_D), F(k(x)))\\
    &= \mathrm{Hom}_{\mathcal{O}_X} (\mathcal{O}_D, k(x))
\end{align*}
for $x \in X$ closed. In fact, this shows $\mathcal{H}^0(F(\mathcal{O}_D))$ and $\mathcal{O}_D$ have the same support.

\underline{Step 4.} Set $\mathcal{L} = F(\mathcal{O}_X).$ Then $F(\mathcal{O}_D) \cong \mathcal{L}|_D$.

We have a short exact sequence of coherent sheaves
\begin{equation*}
    0 \to F( \mathcal{O}_X(-D)) \otimes _{\mathcal{O}_X} \mathcal{L}^{-1} \to \mathcal{O}_X \to F( \mathcal{O}_D) \otimes_{\mathcal{O}_X} \mathcal{L}^{-1} \to 0,
\end{equation*}
and we need to show
\begin{equation*}
    F( \mathcal{O}_D) \otimes_{\mathcal{O}_X} \mathcal{L}^{-1} \cong \mathcal{O}_D.
\end{equation*}
We know that the left hand side $= \mathcal{O}_{D'}$ for some effective divisor $D',$ and from the proof of Step 3, it follows that $D$ and $D'$ have the same underlying closed subset. Thus $D = \sum n_i D_i$ and $D' = \sum m_i D_i$ with $D_i$ prime divisors and $n_i, m_i$ positive integers. To show that $n_i = m_i,$ let $x$ be a general point of $D_i,$ (so that $D_i$ is regular at $x$ and $x \not \in D_j$ for $j \neq i$). Choose a regular system of parameters $f_1, \dots , f_d$ for $X$ at $x$ such that $D_i = V(f_1)$ in a neighborhood of $x$. Let $\mathcal{F}_N$ be the structure sheaf of the scheme $\mathrm{Spec} (\mathcal{O}_{X,x} / (f_1^{N}, f_2, \dots , f_d)).$ Then we can recover $n_i$ as
\begin{equation*}
    \dfrac{1}{[k(x): k]}\mathrm{dim}_{k} \mathrm{Hom} _{\mathcal{O}_X}(\mathcal{O}_D,\mathcal{F}_N)
\end{equation*}
for $N \gg 0$. By applying $F$ to $\mathcal{O}_D$ and $\mathcal{F}_N$ we see $n_i = m_i$ so $D = D'$.

\underline{Step 5:} Define a natural isomorphism on line bundles $\mathcal{O}_X(-D), D$ effective.

By Step 3 we have unique isomorphisms $F(\mathcal{O}_X(-D)) \to \mathcal{O}_X(-D) \otimes_{\mathcal{O}_X} \mathcal{L}$ making the diagram
\begin{equation*}
\begin{tikzcd}
     F(\mathcal{O}_X(-D)) \ar[rr] \ar[dr] & &\mathcal{L} \\
      & \mathcal{O}_X(-D) \otimes_{\mathcal{O}_X} \mathcal{L} \ar[ur] &
\end{tikzcd}
\end{equation*}
commute, where the horizontal arrow is $F$ of the morphism $\mathcal{O}_X(-D) \to \mathcal{O}_X.$ These isomorphisms are natural with respect to inclusions of divisors $D \subset E$.

\underline{Step 6:} $K \in D^b_{coh} (X)$ is in the subcategory $Coh (\mathcal{O}_X)$ if and only if $F(K)$ is.

Let $U$ be an affine open of $X$ with complement $D$ (reduced induced subscheme structure). Applying fully-faithfulness of $F$ and the isomorphisms in Step 5 gives
\begin{align*}
    \Gamma (U, \mathcal{H}^i(F(K))) &=  \mathrm{colim}_{n}\mathrm{Ext}^i_{\mathcal{O}_X} (\mathcal{O}_X(nD), F(K)) \\ &= \mathrm{colim}_{n}\mathrm{Ext}^i_{\mathcal{O}_X}(\mathcal{O}_X (nD), K \otimes _{\mathcal{O}_X}^{\mathbf{L}} \mathcal{L}) \\
    &= \Gamma (U, \mathcal{H}^i (K \otimes _{\mathcal{O}_X}^{\mathbf{L}} \mathcal{L})),
\end{align*}
so $K$ and $F(K)$ have non-zero cohomology sheaves in the same degrees.

\underline{Step 7:} Conclude.

By Step 6, $F$ restricts to a $k$-linear auto-equivalence of the subcategory $Coh(\mathcal{O}_X).$ But these are classified by Gabriel's Theorem (see \cite{gabriel} or \cite[\href{https://stacks.math.columbia.edu/tag/0FZR}{Tag 0FZR}]{stacks-project}): They are all isomorphic to
\begin{equation*}
    \mathcal{F} \mapsto f^*\mathcal{F} \otimes _{\mathcal{O}_X} \mathcal{L}
\end{equation*}
for some $k$-linear automorphism $f$ of $X$ and line bundle $\mathcal{L}$ on $X$, as needed.
\end{proof}

\section{Fully Faithful Functors Agreeing on Coherent Sheaves}

For the following three lemmas, let $X$ be a smooth proper variety of dimension at least 1 over a field $k$. Choose a finite affine open covering $X = \bigcup_i U_i$ such that $\emptyset \subsetneq U_i \subsetneq X$ for every $i$. Let $D_i = X \setminus U_i$ (reduced induced subscheme structure). We will consider the systems of sheaves
\begin{equation*}
   \cdots \subset \mathcal{O}_X(-nD_i) \subset \mathcal{O}_X(-(n-1)D_i) \subset \cdots \subset \mathcal{O}_X(-D_i).
\end{equation*}
Compare Lemmas A, B, C below to the properties (a), (b), (c) of an ample sequence in \cite{orlov1997equivalences}.

\begin{lemmaa}
Given a coherent sheaf $\mathcal{F}$ on $X$, there exists a surjection from a sheaf of the form
\begin{equation}
\bigoplus _i \mathcal{O}_X(-n_i D_i)^{\oplus r_i}
\end{equation}
to $\mathcal{F}$. Furthermore, we can choose this surjection so that for $m_i \geq n_i$, the induced map \\ $\bigoplus _i \mathcal{O}_X(-m_i D_i)^{\oplus r_i} \to \mathcal{F}$ remains surjective.
\end{lemmaa}

\begin{proof}
Choose a surjection $ \varphi_i : \mathcal{O}_{U_i} ^{\oplus r_i} \to \mathcal{F}|_{U_i},$ then using the formula
\begin{equation*}
    \mathrm{Hom}_{\mathcal{O}_{U_i}} (\mathcal{O}_{U_i} ^{\oplus r_i}, \mathcal{F}|_{U_i}) = \mathrm{colim}_{n_i} \mathrm{Hom}_{\mathcal{O}_X} (\mathcal{O}_X(-n_iD_i)^{\oplus r_i}, \mathcal{F}),
\end{equation*}
we find an $n_i$ and a map $\mathcal{O}_X(-n_iD_i)^{\oplus r_i} \to \mathcal{F}$ which agrees with $\varphi_i$ when restricted to $U_i$. Note that if we increase $n_i,$ the induced map still agrees with $\varphi_i$ on $U_i$. Thus putting these maps together for every $i$ we are done.
\end{proof}

\begin{lemmab}
Assume $K \in D^b_{coh}(X), \mathcal{H}^i(K) = 0$ for $i \geq 0$. Then given a map $\bigoplus _i \mathcal{O}_X(-n_iD_i)^{\oplus r_i}[0] \to K$, there exists $N$ such that the induced map $\bigoplus_i \mathcal{O}_X(-m_iD_i)^{\oplus r_i}[0] \to K$ is zero for $m_i \geq N.$
\end{lemmab}
\begin{proof}
Since
\begin{equation*}
    \mathrm{colim}_N \mathrm{Hom} _{\mathcal{O}_X}(\mathcal{O}_X(-ND_i), K) = \mathrm{Hom}_{ \mathcal{O}_{U_i}}(\mathcal{O}_{U_i}, K|_{U_i}) = 0.
\end{equation*}
\end{proof}

\begin{lemmac}
Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then there exists $N$ such that if $n_i \geq N,$
\begin{equation*}
    \mathrm{Hom}_{\mathcal{O}_X} (\mathcal{F}, \bigoplus _i \mathcal{O}_X(-n_i D_i)^{\oplus r_i}) = 0.
\end{equation*}
\end{lemmac}
\begin{proof}
It suffices to show that if $D$ is a non-empty effective divisor on $X$, then
\begin{equation*}
    \mathrm{Hom}_{\mathcal{O}_X} (\mathcal{F}, \mathcal{O}_X(-nD)) = 0
\end{equation*}
for $n \gg 0$. There is by Lemma A a surjection $\bigoplus _i \mathcal{O}_X(-n_iD_i)^{\oplus r_i} \to \mathcal{F}$. Thus it suffices to prove the lemma for $\mathcal{F} = \mathcal{O}_X(-E)$ for some effective divisor $E$. That is, we must show $H^0(X, \mathcal{O}_X(E-nD)) = 0$ for $n\gg 0$. These form a descending sequence of finite dimensional $k$-subspaces of $k(X)$, so it suffices to show the intersection is zero. Let $f$ be in the intersection. Let $Z$ be a prime divisor occurring with positive multiplicity in the divisor $D$. Then $f \in k(X)$ has infinite order of vanishing along $Z$, so $f = 0$.
\end{proof}

For a complex $K$, we will write
\begin{equation*}
    w(K) = \mathrm{sup} \{b-a | \mathcal{H}^b(K) \neq 0 \neq \mathcal{H}^a(K)\}
\end{equation*}
for the \emph{width} of $K$.

\begin{prop} Let $X, Y$ be smooth proper varieties over a field $k$, and let $F, G : D^b_{coh} (X) \to D^b_{coh} (Y)$ be exact, $k$-linear functors. If $F$ is fully faithful and
\begin{equation*}
    F |_{Coh(\mathcal{O}_X)} \cong G |_{Coh(\mathcal{O}_X)},
\end{equation*}
then $F \cong G$.
\end{prop}

\begin{proof}
Lemmas 2 and 3 imply that $G$ is fully faithful with essential image contained in the essential image of $F$. Therefore, $F^{-1} \circ G$ (makes sense and) is fully faithful. Our assumption implies that
\begin{equation*}
   F^{-1} \circ G |_{Coh (\mathcal{O}_X)}
\end{equation*}
is isomorphic to the inclusion $Coh(\mathcal{O}_X) \to D^b_{coh} (X).$ If we can show $F^{-1} \circ G \cong \mathbf{1}_{D^b_{coh} (X)},$ this will imply $F \cong G$ and we will be done. Thus we are reduced to proving the Proposition in the special case $X = Y, G = \mathbf{1}_{D^b_{coh} (X)}.$

Next, if $X$ has dimension 0, i.e., $X = \mathrm{Spec} (k')$ for $k'$ a finite separable extension of $k$, then the result follows from the fact that $D^b_{coh} (X)$ is equivalent to the category of graded $k'$-vector spaces. Thus we may assume $\mathrm{dim} (X) \geq 1$, and we may use Lemmas A, B, C as well as their notation.

For this, we construct by induction on $w = w(K)$ isomorphisms $\alpha_K: K \to F(K)$ for all complexes of width $\leq w$ compatible with all morphisms between such $K$.

The base case is $w(K) = 0$, i.e., $K = \mathcal{F}[n]$ for $\mathcal{F}$ coherent on $X$ and $n \in \mathbf{Z}.$ We are given an isomorphism $\alpha_{\mathcal{F}} : \mathcal{F} \to F(\mathcal{F})$, and we set $\alpha_K = \alpha_{\mathcal{F}} [n]$. One must still check compatibility with all morphisms $K \to K'$ with $w(K) = w(K') = 0$: This is known if $K$ and $K'$ are concentrated in the same degree, but it is not obvious otherwise. However, Orlov's proof of compatibility (see 2.16.4 of \cite{orlov1997equivalences}) carries through unchanged in our context. See also \cite[\href{https://stacks.math.columbia.edu/tag/0FZW}{Tag 0FZW}]{stacks-project}.

\underline{Inductive Step:} We assume given isomorphisms $K \to F(K)$ for all $K$ with $w(K) \leq w-1$ compatible with all morphisms between such $K$.

\underline{Construction:} Let $K$ be a complex of width $w = b-a$ with cohomology sheaves living in $[a,b]$. There is a surjection from a sheaf $\mathcal{P}$ of the form (7) to $\mathcal{H}^b(K)$. There is an obstruction to lifting this surjection along $K[b] \to \mathcal{H}^b(K)$ which lives in $\mathrm{Ext}^b (\mathcal{P}, \tau_{< b} K).$ Thus by Lemma B, after possibly increasing the integers $n_i$, we can assume (a) the surjection factors through $K[b] \to \mathcal{H}^b(K).$ By Lemma C, after possibly increasing the integers $n_i$ again, we can assume (b) that $\mathrm{Hom}_{\mathcal{O}_X} (\mathcal{H}^b(K), \mathcal{P}) = 0.$ From this it follows that $\mathrm{Hom}(K, \mathcal{P}[-b]) = 0.$ Then there are distinguished triangles as follows:
\begin{equation*}
    \begin{tikzcd}
    \mathcal{P}[-b] \ar[r]\ar[d] &K \ar[r] \ar[d, dashed] \ar[r] &L \ar[r, "+1"] \ar[d] & { } \\
    F(\mathcal{P}[-b]) \ar[r] & F(K) \ar[r] &F(L) \ar[r, "+1"] & { }
    \end{tikzcd}
\end{equation*}
The solid vertical arrows are from the inductive hypothesis (note that $L$ has cohomology sheaves in $[a,b-1]$ by construction), and the dashed arrow exists by the axioms of a triangulated category. It is an isomorphism since the other two vertical arrows are. Since $\mathrm{Hom}(K, \mathcal{P}[-b]) = \mathrm{Hom}(K, F(\mathcal{P}[-b]))  = 0$, the dashed arrow $K \to F(K)$ can be characterized as the unique arrow making the right hand square commute.

\underline{Independence of Surjection:} If $\mathcal{P}'\to \mathcal{H}^b(K)$ is another surjection where $\mathcal{P}'$ has the form (7) and (a) and (b) are satisfied, we want to show that we get the same definition of the arrow $K \to F(K).$ Note that $\mathcal{P} \oplus \mathcal{P}' \to \mathcal{H}^b(K)$ also satisfies (a) and (b) so we may assume the map $\mathcal{P}'[-b] \to K$ factors through $\mathcal{P}[-b] \to K.$ Choose a distinguished triangle $\mathcal{P}'[-b'] \to K \to L' \to \mathcal{P}'[-b+1],$ and write $\alpha, \alpha'$ for the arrow constructed using $\mathcal{P}, \mathcal{P}'$, respectively. Then $K \to L$ factors through $K \to L'$ since by the axioms of a triangulated category there is a fill in:
\begin{equation*}
    \begin{tikzcd}
         \mathcal{P}'[-b] \ar[r] \ar[d] &K \ar[r] \ar[d, "="] & L' \ar[r, "+1"] \ar[d, dashed] & {} \\
         \mathcal{P}[-b] \ar[r] & K \ar[r] &L \ar[r, "+1"] &{}
    \end{tikzcd}
\end{equation*}
But then by the definition of $\alpha'$ and the inductive hypothesis, the diagram
\begin{equation*}
    \begin{tikzcd}
         K \ar[r] \ar[d, "\alpha'"] &L' \ar[r] \ar[d] &L \ar[d] \\
         F(K) \ar[r] &F(L') \ar[r] & F(L)
    \end{tikzcd}
\end{equation*}
commutes. Thus by uniqueness of $\alpha$, $\alpha = \alpha'$.

\underline{Compatibility:} We need to show the isomorphisms constructed are compatible with all morphisms $K \to K'$ between objects of width at most $w$. We do this by induction on $w(K)+w(K').$ Let $K$ have cohomology sheaves in degrees $[a,b]$ and $K'$ in degrees $[a',b'].$ Note that all cases with $w(K), w(K') < w$ are known by the first induction hypothesis.

\underline{Case 1:} $b'< b$.

Choose a triangle $\mathcal{P}[-b] \to K \to L \to \mathcal{P}[-b+1]$ as in the construction. Composing with $K \to K'$ we get a map $\mathcal{P}[-b] \to K'$. Then using Lemma B we may assume by increasing the $n_i$ that this map is zero. Then $K \to K'$ factors through $L.$ Since $L$ has cohomology sheaves in $[a,b-1],$ our  inductive hypotheses imply that the right square
\begin{equation*}
    \begin{tikzcd}
    K \ar[r] \ar[d] &L \ar[r] \ar[d] &K'\ar[d] \\
    F(K) \ar[r] &F(L) \ar[r] &F(K')
    \end{tikzcd}
\end{equation*}
is commutative, while the left square commutes by construction. So we're done in this case.

\underline{Case 2:} $b' \geq b$.

Choose a distinguished triangle $\mathcal{P}'[-b'] \to K' \to L' \to \mathcal{P}'[-b+1]$ as in the construction (for $K'$ instead of $K$) with the property that $\mathrm{Hom}(\mathcal{H}^{b'}(K), \mathcal{P}') = 0$. This is possible by Lemma C. Then by our choice of $\mathcal{P}',$
$\mathrm{Hom}(K, K') \to \mathrm{Hom} (K, L')$ is injective, and since $F(K) \cong F(K')$ and $F$ is fully faithful, $\mathrm{Hom}(K, F(K')) \to \mathrm{Hom} (K, F(L'))$ is injective as well. Consider the diagram:
\begin{equation*}
    \begin{tikzcd}
    K \ar[r] \ar[d] &K' \ar[r] \ar[d] &L' \ar[d] \\
    F(K) \ar[r] & F(K')  \ar[r] & F(L')
    \end{tikzcd}
\end{equation*}
The right square commutes by construction. Thus by the injectivity on Hom's, it suffices to show the outer square commutes. But this follows from our induction hypotheses since $L'$ has cohomology sheaves in $[a', b'-1].$
\end{proof}

\begin{proof}[Proof of Theorem]
By Proposition 1, there is $E \in D^b_{coh} (X \times Y)$ such that $F(\mathcal{F}) \cong \Phi_E(\mathcal{F})$ for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ with 0-dimensional support. By Lemma 3, the essential image of $\Phi_E$ is contained in the essential image of $F$, thus the functor $F^{-1} \circ \Phi_E$ makes sense. By combining Propositions 2 and 3, we obtain an isomorphism $F^{-1} \circ \Phi_E \cong G$ where $G$ is a Fourier--Mukai auto-equivalence of $D^b_{coh} (X)$. But then $F \cong \Phi_E \circ G^{-1}$, which is Fourier--Mukai.
\end{proof}

\newpage

\bibliographystyle{alpha}
\bibliography{references}

\end{document}

config: arxiv

file: 2006.15173/Orlov's Theorem in the Smooth Proper Case/main.tex


TITLE: Factorization of $(1+2i)z^2+6iz+2i-1$ to find the residues, but it doesn't work
QUESTION [4 upvotes]: I'm doing complex analysis and I have the following $$\int_{C}\frac{2dz}{(2i+1)z^2+6iz+2i-1}$$ where $C$ is the unit circle. I tried factorizing this polynomial and find the residues but I really can't solve this.
This is my work:
$$z_{1,2} = \frac{-3i\pm \sqrt{-9-(-4-1)} }{2i+1}=\frac{-3i\pm 2i}{1+2i} \cdot \frac{1-2i}{1-2i}= \frac{-3i-6\pm 2i\pm 4}{5}$$ which gives $z_1 = -\frac{2}{5}-\frac{i}{5}$ and $z_2 = -2-i$. Now clearly $z_2$ is outside the unit circle so I just have to consider the residue of $z_1$.
However I think I'm either making a factorisation mistake or a mistake in the roots. Indeed if we check the roots, I don't get the same polynomial! $$(z+2+i)\left(z+\frac{2}{5}+\frac{i}{5}\right) = z^2 +z\left(\frac{12}{5}+\frac{6i}{5}\right)+\frac{4i}{5}+\frac{3}{5}$$ which is clearly not the same polynomial I started with. Also if I try to calculate the residue using the limit, I don't get the correct one. Where's my mistake?
EDIT
As requested, I'll add some more calculations. To calculate the residue I did the following: $$\lim_{z\to z_1}(z+\frac{2}{5}+\frac{i}{5})\frac{2}{(z+2+i)\left(z+\frac{2}{5}+\frac{i}{5}\right)} = \frac{2}{\frac{8}{5}+\frac{4i}{5}} = \frac{5}{4+2i} = \frac{20-10i}{20}=1-\frac{i}{2}$$ which is clearly wrong as this integral is the mapping of a real integral of the first type (rational function in sine and cosine). Indeed by Cauchy's Residue Theorem this should give that the value of the initial real integral is $2\pi i (1-\frac{i}{2}) = 2\pi i+\pi$ which is a complex value, so this is wrong

REPLY [5 votes]: If you were only interested in the value of your integral $f$ you could let $z=\exp (i \theta )$ and get
$$fr=\int_0^{2 \pi } \frac{1}{\sin (\theta )+2 \cos (\theta )+3} \, d\theta$$
In order to calculate the integral we attempt to use the fundamental theorem of calculus, i.e. taking the difference of the anitiderivative at both ends of the integration interval. But this holds only for continuous antiderivatives. If the antiderivative contains jumps these must be taken into account.
An antiderivative of the integrand is
$$fa=-\tan ^{-1}\left(\frac{2}{\tan \left(\frac{\theta }{2}\right)+1}\right)$$
In the region of integration the denominator vanishes at $\theta =\frac{3 \pi }{2}$. Here the antiderivative makes a jump from
$$\lim_{\theta \to \left(\frac{3 \pi }{2}\right)^-} \, fa=\frac{\pi }{2}$$
to
$$\lim_{\theta \to \left(\frac{3 \pi }{2}\right)^+} \, fa=-\frac{\pi }{2}$$
i.e. it jumps by an amount of $\pi$.
Now the antiderivative at both ends of the integration interval is equal to $-\tan ^{-1}(2)$. Hence the difference is zero, and we are left with the jump which gives $f=fr=\pi$.
EDIT
I'm afraid I have made it more complicated than necessary.
Let us therefore apply a "standard" procedure and do this for the more general integral (displayed here in original form and reduced form)
$$\int_0^{2 \pi } \frac{1}{a+b \cos (\theta )+c \sin (\theta )} \, d\theta =\int_0^{2 \pi } \frac{1}{a+\sqrt{b^2+c^2} \cos (\phi )} \, d\phi =\frac{2 \pi }{\sqrt{a^2-b^2-c^2}}$$
which holds for $a^2-b^2-c^2>0$
Making the "standard" substitution $\theta \to 2 \tan ^{-1}(t)$, $\text{d$\theta $}=\frac{2 \text{dt}}{t^2+1}$ and splitting the integral into the two parts from $0$ to $\pi$ and form $\pi$ to $2 \pi$ gives the complete integration range in to from $-\infty$ to $\infty$.
We have
$$\cos (\theta )=\cos ^2\left(\frac{\theta }{2}\right)-\sin ^2\left(\frac{\theta }{2}\right)=\left(1-t^2\right) \cos ^2\left(\frac{\theta }{2}\right)$$
$$\sin (\theta )=2 \sin \left(\frac{\theta }{2}\right) \cos \left(\frac{\theta }{2}\right)=2 t \cos ^2\left(\frac{\theta }{2}\right)$$
$$\frac{1}{\cos ^2\left(\frac{\theta }{2}\right)}=t^2+1$$
The integral becomes
$$2 \int_{-\infty }^{\infty } \frac{1}{a \left(t^2+1\right)+b \left(1-t^2\right)+2 c t} \, dt$$
Supplementing the square, shifting the integration varibale (this makes no dfference in our infnte interval) and extracting factors gives finally
$$\frac{\int_{-\infty }^{\infty } \frac{2}{v^2+1} \, dv}{\sqrt{a^2-b^2-c^2}}=\frac{2 \pi }{\sqrt{a^2-b^2-c^2}}$$


text:
\begin{document}

\title{Co-ordinate Interleaved Distributed Space-Time Coding for Two-Antenna-Relays Networks}
\author{Harshan J~and
        B. Sundar Rajan,~\IEEEmembership{Senior Member,~IEEE}

\thanks{This work was supported through grants to B.S.~Rajan; partly by the IISc-DRDO program on Advanced Research in Mathematical Engineering, and partly by the Council of Scientific \& Industrial Research (CSIR, India) Research Grant (22(0365)/04/EMR-II). The material in this paper was presented in parts at the IEEE Global telecommunication conference (GLOBECOM 2007), Washington D.C., USA, Nov. 26-30, 2007. Harshan J and B. Sundar Rajan are with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore-560012, India. Email:\{harshan,bsrajan\}@ece.iisc.ernet.in.}
\thanks{Manuscript received August 08, 2007; revised November 03, 2007.}}

\markboth{IEEE Transactions on Wireless Communications ,~Vol.~xx, No.~xx, xxxx}{Harshan \MakeLowercase{and} Rajan: Co-ordinate Interleaved Distributed Space-Time Coding for Two-Antenna-Relays Networks}

\maketitle

\begin{abstract}

Distributed space time coding for wireless relay networks when the source, the destination and the relays have multiple antennas have been studied by Jing and Hassibi. In this set-up, the transmit and the receive signals at different antennas of the same relay are processed and designed independently, even though the antennas are colocated. In this paper, a wireless relay network with single antenna at the source and the destination and two antennas at each of the $R$ relays is considered. A new class of distributed space time block codes called Co-ordinate Interleaved Distributed Space-Time Codes (CIDSTC) are introduced where, in the first phase, the source transmits a $T$-length complex vector to all the relays and in the second phase, at each relay, the in-phase and quadrature component vectors of the received complex vectors at the two antennas are interleaved and processed before forwarding them to the destination. Compared to the scheme proposed by Jing-Hassibi, for $T \geq 4R$, while providing the same asymptotic diversity order of $2R$, CIDSTC scheme is shown to provide asymptotic coding gain with the cost of negligible increase in the processing complexity at the relays. However, for moderate and large values of $P$, CIDSTC scheme is shown to provide more diversity than that of the scheme proposed by Jing-Hassibi. CIDSTCs are shown to be fully diverse provided the information symbols take value from an appropriate multi-dimensional signal set.

\end{abstract}

\begin{keywords}
Cooperative communication, distributed space-time coding, co-ordinate interleaving, coding gain.
\end{keywords}

\section{Introduction and Preliminaries}
Co-operative diversity is proved to be an efficient means of achieving spatial diversity in wireless networks without the need of multiple antennas at the individual nodes. In comparison with single user colocated multiple antenna transmission, co-operative communication is based on the relay channel model where a set of distributed antennas belonging to multiple users in the network co-operate to encode the signal transmitted from the source and forward it to the destination so that the required diversity order is achieved \cite{SEA1,SEA2,LaW,NBK}.\\
\indent In \cite{JiH1}, the idea of space-time coding devised for point to point co-located multiple antenna systems is applied for a wireless relay network with single antenna nodes and PEP (Pairwise error probability) of such a scheme was derived. It is shown that in a relay network with a single source, a single destination with $R$ single antenna relays, distributed space time coding (DSTC) achieves the diversity of a colocated multiple antenna system with $R$ transmit antennas and one receive antenna, asymptotically. \\
\indent Subsequently, in \cite{JiH2}, the idea of \cite{JiH1} is extended to relay networks where the source, the destination and the relays have multiple antennas. But, co-operation between the multiple antennas of each relay is not used, i.e., the colocatedness of the antennas is not exploited. Hence, a total of $R$ relays each with a single antenna is assumed in the network instead of a total of $R$ antennas in a smaller number of relays. With this set up, for a network with $M$ antennas at the source, $N$ antennas at the destination and a total of $R$ antennas at $R$ relays, for large values of $P$, the PEP of the network, varies with $P$ as

\begin{eqnarray*}
 \left(\frac{1}{P}\right) ^{min(M,N)R} &\mbox{ if }& M \neq N \mbox{  and  }\\
\left(\frac{(\mbox{log}_{e}^{1/M}P)}{P}\right)^{MR} &\mbox{ if }& M = N.
\end{eqnarray*}

\noindent In particular, the PEP of the scheme in \cite{JiH2} for large $P$ when specialized to $M = N = 1$ with 2$R$ antennas at relays is upper-bounded by,

\begin{equation}
\label{JingPEP}
\left[\frac{32R}{T(\rho^\prime)^{2}}\right]^{2R}\left[\frac{(\mbox{log}_{e}(P))^{2R}}{P^{2R}}\right]
\end{equation}

\noindent where $(\rho^\prime)^{2}$ is the minimum singular value of $(\textbf{S} - \textbf{S}^\prime)^{H}(\textbf{S}- \textbf{S}^\prime)$ where $\textbf{S}$ and $\textbf{S}^\prime$ are the two distinct codewords of a distributed space time block code and $P$ is the total power per channel use used by all the relays for transmitting an information vector.\\
Following the work of \cite{JiH2}, constructions of distributed space time block codes for networks with multiple antenna nodes are presented in \cite{OgH1}, \cite{OgH2}.

We refer cooperative diversity schemes in which multiple antennas of a relay do not co-operate i.e when the transmitted vector from every antenna is function of only the received vector in that antenna, or when every relay has only one antenna as Regular Distributed Space-Time Coding (RDSTC).\\

The key idea in the proposed scheme is the notion of vector coordinate interleaving defined below:
\begin{definition}\label{def1} Given two complex vectors $\textbf{y}_{1}, \textbf{y}_{2} \in \mathbb{C}^{T},$ we define a Coordinate Interleaved Vector Pair of $\textbf{y}_{1}, \textbf{y}_{2},$ denoted as  $\mbox{CIVP}\left\{ \textbf{y}_1, \textbf{y}_2\right\}$ to be the pair of complex vectors $\left\{ {\textbf{y}_1^\prime, \textbf{y}_2^\prime}\right\},$ where  $\textbf{y}_1^\prime, \textbf{y}_2^\prime \in \mathbb{C}^{T},$ given by
\begin{equation*}
\textbf{y}_{1}^\prime = \mbox{Re} \,\textbf{y}_{1} + \textbf{j} \,\mbox{Im}\, \textbf{y}_{2}, ~ \textbf{y}_{2}^\prime = \mbox{Re} \,\textbf{y}_{2} + \textbf{j} \,\mbox{Im}\, \textbf{y}_{1},
\end{equation*}
or equivalently,
\begin{equation}\label{civp1}
\textbf{y}_{1}^\prime = (\textbf{y}_{1} + \textbf{y}_1^* + \textbf{y}_{2} - \textbf{y}_2^*)/2,
\end{equation}
\begin{equation}\label{civp2}
\textbf{y}_{2}^\prime = (\textbf{y}_{2} + \textbf{y}_2^* + \textbf{y}_{1} - \textbf{y}_1^*)/2.\\
\end{equation}
\end{definition}

The notion of coordinate interleaving of two complex variables has been used in \cite{KhR} to obtain single-symbol decodable STBCs with higher rate than the well known complex orthogonal designs. Definition \ref{def1} is an extension of the above technique to two complex vectors. The idea of vector co-ordinate interleaving has been used in \cite{WuB} in order to obtain better diversity results in fast fading MIMO channels.

\indent In this paper, we show that multiple antennas at the relays can be exploited to improve the performance of the network. Towards this end, a single antenna source and a single antenna destination with two antennas at each of the $R$ relays is considered. Also, the two phase protocol as in \cite{JiH2} is assumed where the first phase consists of transmission of a $T$ length complex vector from the source to all the relays (not to the destination) and the second phase consists of transmission of a $T$ length complex vector from each of the antennas of the relays to the destination, as shown in Fig.\ref{model}. The modification in the protocol we introduce is that the two received vectors at the two antennas of a relay during the first phase is coordinate interleaved as defined in Definition \ref{def1}. Then, multiplying the coordinate interleaved vector with the predecided antenna specific $T\times T$ unitary matrices, each antenna produces a $T$ length vector that is transmitted to the destination in the second phase. The collection of all such vectors, as columns of a $T\times 2R$ matrix constitutes a codeword matrix and collection of all such codeword matrices is referred as coordinate interleaved distributed space time code (CIDSTC). The contributions of this paper may be summarized as follows in more specific terms:
\begin{itemize}
\item For $T \geq 4R$, an upper bound on the PEP of our scheme with fully diverse CIDSTC, at large values of the total power $P$ is derived.
\item For $T \geq 2R,$ the PEP of the RDSTC scheme in \cite{JiH2} with fully diverse DSTBC is upper bounded  by the expression given in \eqref{JingPEP}. Comparing this bound, with ours, for equal number of $2R$ antennas, a term $\left[ \mbox{log}_{e}(P)\right]^{R}$ appears in the numerator of the PEP expression of our scheme instead of the term $\left[ \mbox{log}_{e}(P)\right]^{2R}$. This improvement in the PEP comes just by vector co-ordinate interleaving at every relay the complexity of which is negligible.
\item It is shown that CIDSTC scheme provides asymptotic coding gain compared to the corresponding RDSTC scheme.
\item CIDSTC in variables $x_{1}, x_{2} \cdots x_{T}$ is shown not to provide full diversity if the variables $x_{1}, x_{2} \cdots x_{T}$ take values from any 2-dimensional signal set.
\item Multi-dimensional signal sets are shown to provide full diversity for CIDSTCs whose choice depends on the design in use.
\item The number of channel uses needed in the proposed scheme is at least $4R$ where as only $2R$ is needed in an RDSTC scheme. With $T=4R$ for both the schemes, through simulation, it is shown that CIDSTC gives improved BER (Bit Error Rate) performance over that of RDSTC scheme.
\end{itemize}

\textit{Notations:} Through out the paper, boldface letters and capital boldface letters are used to represent vectors and matrices respectively. For a complex matrix \textbf{X}, the matrices $\textbf{X}^*$, $\textbf{X}^T$,  $\textbf{X}^{H}$, $\mbox{det}\left[\textbf{X}\right]$, $||\textbf{X}||_{F}^{2}$, $\mbox{Re}~\textbf{X}$ and $\mbox{Im}~\textbf{X}$ denote the conjugate, transpose, conjugate transpose, determinant, Frobenious norm,  real part and imaginary part of \textbf{X} respectively. $\textbf{I}_T$ and $\textbf{0}_T$ denotes the $ T\times T$ identity matrix and  the $T \times T$ zero matrix respectively. Absolute value of a complex number $x$, is denoted by $|x|$ and $E \left[x\right]$ is used to denote the expectation of the random variable $x.$ A circularly symmetric complex Gaussian random vector $\textbf{x}$ with mean $\mu$ and covariance matrix $\mathbf{\Gamma}$ is denoted by $\textbf{x} \sim \mathcal{CG} \left(\mu, \mathbf{\Gamma} \right)$. The set of all integers and complex numbers are denoted by $\mathbb{Z}$ and $\mathbb{C}$ respectively and $\textbf{j}$ is used to denote $\sqrt{-1}.$ Through out the paper $\mbox{log} (.)$ refers to $\mbox{log}_{e}(.)$.\\

\indent The remaining content of the paper is organized as follows: In Section \ref{sec2}, the signal model and a formal definition of CIDSTC is given along with an illustrative example.
The pairwise error probability (PEP) expression for a CIDSTC is obtained in Section \ref{sec3} using which it is shown that (i) CIDSTC scheme gives asymptotic diversity gain equal to the total number of antennas in the relays and (ii) offers asymptotic coding gain compared to the corresponding RDSTCs. Constructions of CIDSTCs along with conditions on the full diversity of CIDSTCs are provided in Section \ref{sec4}. In Section \ref{sec5}, simulation results are presented to illustrate the superiority of CIDSTC schemes. Concluding remarks and possible directions for further work constitute Section \ref{sec6}.

\section{signal model}
\label{sec2}

\begin{figure}
\centering
\includegraphics[width=2.5in]{rajan_paper-tw-nov-07-1303_fig1}

\caption{Wireless relay network with two-antenna-relays}
\label{model}
\end{figure}

The channel from the source node to the $i$-th antenna of the $j$-th relay is denoted as $f_{ij}$ and the channel from the $i$-th antenna of the $j$-th relay to the destination node is represented by $g_{ij}$ for $i=1,2$ and $j=1,2, \cdots, R$  as shown in Fig.\ref{model}.

\noindent The following assumptions are made in our system model:

\begin{itemize}
\item All the nodes are subjected to half duplex constraint.
\item Fading coefficients  $f_{ij},g_{ij}$ are i.i.d $ \mathcal{CG} \left(0,1 \right)$ with coherence time interval, $T$.
\item All the nodes are synchronized at the symbol level.
\item Destination knows all the fading coefficients $f_{ij},g_{ij}$.
\end{itemize}

\noindent In the first phase the source transmits a $T$ length complex vector from the codebook $\mathcal{S}$ = $\left\{ \textbf{s}_{1},\, \textbf{s}_{2},\, \textbf{s}_{3},\, \cdots , \textbf{s}_{L} \right\} $ consisting of information vectors $\textbf{s}_{l} \in \mathbb{C}^{T}$ such that $E\left[\textbf{s}_{l}^{H}\textbf{s}_{l}\right]$ = 1 for all $l= 1,\cdots, L$, so that $P_{1}$ is the average transmit power used at the source every channel use. When the information vector $\textbf{s}$ is transmitted, the received vector at the $i$-th antenna of the $j$-th relay is given by
\begin{equation*}\label{rij}
\textbf{r}_{i,j} = \sqrt{P_{1}T}f_{ij}\textbf{s} + \textbf{n}_{ij},~ \textit{i} =1,2 \, \, \mathrm{and} \textit{ j} = 1,2,\cdots, R
\end{equation*}

\noindent where $\textbf{n}_{ij} \sim \mathcal{CG} \left(0,\textbf{I}_{T} \right) $ is the additive noise vector at the $i$-th antenna of the $j$-th relay.

\noindent
In the second phase, all the relay nodes are scheduled to transmit $T$ length vectors to the destination simultaneously. In general, the transmitted signals from the different antennas of the same relay can be designed as a function of the received signals at both the antennas of the relay. We use one such technique which is very simple; every relay manufactures a $\mbox{CIVP}$ using the received vectors $\textbf{r}_{1j}$ and $\textbf{r}_{2j}$ as given in \eqref{civp1} and \eqref{civp2}, i.e., $\left\{ \textbf{r}_{1j}^\prime \textbf{r}_{2j}^\prime \right\}   = \mbox{CIVP}\left\{ \textbf{r}_{1j},\textbf{r}_{2j}\right\}.$ It is straight forward. to verify that $E\left[ (\textbf{r}_{ij}^\prime)^{H}(\textbf{r}_{ij}^\prime)\right] = \left( 1 + P_{1}\right)T.$  Each relay is equipped with a pair of fixed  $T\times T$  unitary matrices $\textbf{A}_{1j}$ and $\textbf{A}_{2j}$, one for each antenna and process the above CIVP as follows: The $1^{st}$ and the $2^{nd}$ antennas of the $j$-th relay are scheduled to transmit
\begin{equation}\label{t1j}
\textbf{t}_{1j} = \sqrt{\frac{P_{2}}{(1 + P_{1})}}\textbf{A}_{1j}\textbf{r}_{1,j} ^\prime \mbox{ and }\textbf{t}_{2j} = \sqrt{\frac{P_{2}}{(1 + P_{1})}}\textbf{A}_{2j}\textbf{r}_{2,j}^\prime
\end{equation}
\noindent respectively. The average power transmitted by each antenna of a relay per channel use is $P_{2}$. The vector received at the destination is given by
\begin{equation}\label{bfy}
\textbf{y} = \sum_{j = 1}^{R} (g_{1j}\textbf{t}_{1j} + g_{2j}\textbf{t}_{2j}) + \textbf{w}
\end{equation}

\noindent where $\textbf{w} \sim \mathcal{CG} \left(0, \textbf{I}_{T} \right)$ is the additive noise at the destination. Using \eqref{t1j} in \eqref{bfy},  \textbf{y} can be written as

\begin{equation*}\label{bfy1}
\textbf{y} = \sqrt{\frac{P_{1}P_{2}T}{(1 + P_{1})}}\textbf{S}\textbf{h} + \textbf{n}
\end{equation*}

\noindent where
\begin{itemize}
\item The additive noise, $\textbf{n}$ in the above equivalent MIMO channel is given by,
\begin{equation*}
\label{bfN}
\textbf{n} = \sqrt{\frac{P_2}{(1 + P_1)}}\left[ \sum_{j=1}^{R} (g_{1j}\textbf{A}_{1j}\textbf{n}_{1j}^\prime + g_{2j}\textbf{A}_{2j}\textbf{n}_{2j}^\prime) \right]  + \textbf{w}
\end{equation*}
with $\left\{  \textbf{n}_{1j}^\prime , \textbf{n}_{2j}^\prime \right\} =  \mbox{CIVP}\left\{ \textbf{n}_{1j}, \textbf{n}_{2j}\right\}$. Since $\textbf{n}_{ij} \sim \mathcal{CG} \left(0, \textbf{I}_{T} \right)$, we have $\textbf{n}_{ij}^\prime \sim \mathcal{CG} \left(0, \textbf{I}_{T} \right).$\\

\item The equivalent channel  \textbf{h} is given by
\begin{equation}
\label{channel}
\textbf{h} = [\textbf{d}_{1} ~~ \textbf{d}_{2} ~~ \cdots ~~ \textbf{d}_{R} ]^T \in \mathbb{C}^{4R}\\
\end{equation}
where $\textbf{d}_{j} = [g_{1j}k_{1j} ~~~  g_{1j}k_{2j} ~~~  g_{2j}k_{1j} ~~~  -g_{2j}k_{2j}]$ for $j = 1, \cdots, R$ and $k_{1j}  = (f_{1j} + f_{2j})/2, ~ k_{2j}  = (f_{1j}^* - f_{2j}^*)/2$.\\

\item The $T \times 4R$ matrix, $$\textbf{S} = \left[  \textbf{A}_{11}\textbf{s}~~ \textbf{A}_{11}\textbf{s}^*~~ \textbf{A}_{21}\textbf{s}~~ \textbf{A}_{21}\textbf{s}^*~~  \ldots ~~ \textbf{A}_{2R}\textbf{s}~~ \textbf{A}_{2R} \textbf{s}^* \right]$$ is the equivalent codeword matrix. Henceforth, by codeword matrix will be meant only this equivalent $T \times 4R$ matrix even though the transmitted vectors from the $2R$ antennas constitute a $T \times 2R$ matrix.
\end{itemize}

\noindent The collection $\mathcal{C}$ of codeword matrices shown below when $\textbf{s}$ runs over the codebook $\mathcal{S}$,
\begin{equation}
\label{code}
\mathcal{C} = \left\{ \left[  \textbf{A}_{11}\textbf{s}~~ \textbf{A}_{11}\textbf{s}^*~~ \textbf{A}_{21}\textbf{s}~~ \textbf{A}_{21}\textbf{s}^*~~  \ldots ~~ \textbf{A}_{2R}\textbf{s}~~ \textbf{A}_{2R} \textbf{s}^* \right] \right\}
\end{equation}
\noindent will be called the Co-ordinate Interleaved Distributed Space-Time code (CIDSTC).

\begin{proposition}\label{prop} The random variables $k_{ij}$ for all $i= 1,2$ and $ j = 1,\cdots ,R.$ are independent and also $k_{ij} \sim \mathcal{CG} \left( 0, 1/2\right).$\end{proposition}

\begin{proof}  The proof is straight forward.
\end{proof}

\begin{example}
\label{exple1}
Consider $R=1$ and $T=4.$ Let the relay specific unitary matrices $\textbf{A}_{11}$ and $\textbf{A}_{21}$ be
\begin{equation*}
\textbf{A}_{11} = \left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
\end{array}\right];~~~ \textbf{A}_{21} = \left[\begin{array}{cccc}
0 & \textbf{j} & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 & \textbf{j}\\
0 & 0 & 1 & 0\\
\end{array}\right].
\end{equation*}
The equivalent channel is $\textbf{h} = \left[ g_{11}k_{11} ~~  g_{11}k_{21} ~~ g_{21}k_{11} ~~ -g_{21}k_{21}\right] ^T.$
\noindent The CIDSTC is the collection of 4 $\times$ 4 matrices given by,
\begin{center}
$\mathcal{C} = \left\{ \left[  \textbf{A}_{11}\textbf{s} ~~ \textbf{A}_{11}\textbf{s}^* ~~ \textbf{A}_{21}\textbf{s} ~~ \textbf{A}_{21}\textbf{s}^*\right] ~~ :\textbf{s} \in \mathcal{S} \right\} $
\end{center}
and to be explicit, with $\textbf{s}= \left[ x_1,x_2,x_3,x_4 \right]^{T}$ where $x_1,x_2,x_3,x_4$ are complex variables which may take values from a signal set like QAM, PSK etc.

\begin{equation*}
\mathcal{C} = \left\{ \left[
\begin{array}{rrrr}
x_1 & x_1^* & \textbf{j}x_2 & \textbf{j}x_2^* \\
x_2 & x_2^* & x_1  & x_1^* \\
x_3 & x_3^* & \textbf{j}x_4 & \textbf{j}x_4^* \\
x_4 & x_4^* & x_3 &x_3^*
\end{array}
\right]  \right\}.
\end{equation*}
\end{example}

\section{pairwise error probability}
\label{sec3}
Since the relay specific matrices $\textbf{A}_{ij}$ are unitary, $\textbf{w}$ and $\textbf{n}_{ij}^\prime$ are independent Gaussian random variables and since $g_{ij}$ are known at the receiver, $\textbf{n}$ is a Gaussian random vector with
\begin{center}
 $E\left[ \textbf{n}\right]$  = $ \textbf{0}_{T}$ and  $E\left[ \textbf{n}\textbf{n}^{H}\right]$   = $\left(1 + \frac{P_{2}}{(1 + P_{1})}\sum_{j = 1}^{R}(|g_{1j}|^{2} + |g_{2j}|^{2})\right)\textbf{I}_{T}$.
\end{center}
Assume that $S$ is a codeword in the CIDSTC  given in \eqref{code}. When both $k_{ij} $
and $g_{ij}$ are known, ${\textbf y}|\textbf{S}$ is also a Gaussian random vector with

\begin{center}
$E\left[{\textbf y}|\textbf{S}\right] = \sqrt{\frac{P_{1}P_{2}T}{(1 + P_{1})}}\textbf{S}\textbf{h}$ and $E\left[ \textbf{yy}^{H}| \textbf{S}\right] = (1 + \frac{P_{2}}{(1 + P_{1})}\sum_{j = 1}^{R}(|g_{1j}|^{2} + |g_{2j}|^{2}))\textbf{I}_{T}.$
\end{center}

\noindent
The maximum likelihood (ML) decoding is given by
\begin{equation}
\label{ML}
arg\, \min_{\textbf{S}} || \textbf{y} - \sqrt{\frac{P_{1}P_{2}T}{(1 + P_{1})}}\textbf{S}\textbf{h} ||_{F}^{2}.
\end{equation}

\subsection{Chernoff bound on the PEP.}
\label{subsec3}
\begin{lemma} \label{chf_bound}Assume $\textbf{S}, \textbf{S}^\prime \in \mathcal{C},$ where ${\mathcal C}$ is a CIDSTC. With the ML decoding as in \eqref{ML}, the probability of decoding to $\textbf{S}^\prime$ when  $\textbf{S}$ is transmitted given that $k_{ij},\, g_{ij}$ are known at the destination has the following Chernoff bound \cite{JiH1}:

\begin{equation}
\label{chernoff}
P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right | k_{ij},\, g_{ij}) \leq \,e^{-(P^\prime \mathbf{h}^{H}\textbf{U}\mathbf{h})}
\end{equation}
where
\begin{center}
$\textbf{U} = (\textbf{S} - \textbf{S}^\prime)^{H}(\textbf{S}- \textbf{S}^\prime)$ and
$P^\prime = \frac{P_{1}P_{2}T}{4(1 + P_{1} + P_{2}g)}$
\end{center}
where $g= \sum_{j = 1}^{R}(|g_{1j}|^{2} + |g_{2j}|^{2}).$
\end{lemma}

We refer to a CIDSTC as fully diverse if $\textbf{U}$ is a full rank matrix for every codeword pair.

\begin{lemma}\label{thm2} If $\textbf{U}$ is of full rank and the minimum singular value of $\textbf{U}$ is denoted by $\rho^2$, then the PEP in \eqref{chernoff} averaged over $k_{ij}$ satisfies
\begin{equation}
\label{theorem4eqn}
P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right | g_{ij})  \leq  \prod_{j = 1}^{R} 4\left[ \frac{1}{2 + P^\prime \rho^{2}g_{j}}\right] ^{2}
\end{equation}
where $g_{j} = |g_{1j}|^{2} + |g_{2j}|^{2}.$
\end{lemma}
\begin{proof} See Appendix \ref{append2}.
\end{proof}

The power allocation problem of our model is the same as the one considered in \cite{JiH1} with $2R$ single antenna relays. As introduced in the result of Lemma \ref{chf_bound}, $P^\prime = \frac{P_{1}P_{2}T}{4(1 + P_{1} + P_{2}g)}$  where $g= \sum_{j = 1}^{R}(|g_{1j}|^{2} + |g_{2j}|^{2})$ has the gamma distribution with mean and variance being $2R$. For very large values of $R$, we can make the approximation $g \thickapprox 2R$ and hence $P^\prime = \frac{P_{1}P_{2}T}{4(1 + P_{1} + P_{2}2R)}$. Since for every channel use, the power used at the source and every antenna of a relay are $P_{1}$ and $P_{2}$ respectively, total power $P$ is $P_{1} + 2RP_{2}$. Therefore,
\begin{equation*}
P^\prime = \frac{P_{1}P_{2}T}{4(1 + P_{1} + P_{2}2R)} \leq \frac{P^{2}T}{32R(1 + P)}.
\end{equation*}
Thus, $P^\prime$ achieves the above equality when $P_{1} = \frac{P}{2}$ and  $P_{2} = \frac{P}{4R}$.
Since we have used the approximation $g \thickapprox 2R$, the above power allocation is valid only for large values of $R$ as in \cite{JiH1}. With this optimum power allocation, when $P >>1,$ we have $\left( \mbox{from } \cite{JiH1}\right)$,
\begin{equation*}
P^\prime = \frac{P_{1}P_{2}T}{4(1 + P_{1} + P_{2}g)} \simeq \frac{PT}{8(2R +g )}.
\end{equation*}

\subsection{Derivation of Diversity order for Large R}

The upper-bound on the PEP in \eqref{theorem4eqn} needs to be averaged over $g_{1j}$'s and $g_{2j}$'s to obtain the diversity order of the CIDSTC scheme. A simple approximate derivation of the diversity order considering large number of relays in the network is presented. When $R$ is large, $g \simeq 2R$ with high probability and $P^\prime  \simeq \frac{PT}{32R}$.

\begin{theorem}\label{thm3} Assume $T \geq 4R$ and the CIDSTC is fully diverse. For large total transmit power $P$, the probability of decoding to $\textbf{S}^\prime$  when  $\textbf{S}$ is transmitted is upper bounded as

\begin{equation}
\label{theorem41}
P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right) \leq \left[\frac{64R}{T\rho^{2}}\right]^{2R}\left[\frac{(\mbox{log}(P))^{R}}{P^{2R}}\right].
\end{equation}
\end{theorem}

\begin{proof} See Appendix \ref{append3}.
\end{proof}

\indent The term $\frac{(log(P))^{R}}{P^{2R}}$ in the right hand side of \eqref{theorem41} can be written as $P^{-2R(1-\frac{log(log(P))}{2log(P)})}$. Hence, the diversity of the wireless relay network with CIDSTC is
$$2R\left(1-\frac{\mbox{log}(\mbox{log}(P))}{2\mbox{log}(P)}\right)$$
whereas the diversity of the scheme in \cite{JiH2} is
$$2R\left(1-\frac{\mbox{log}(\mbox{log}(P))}{\mbox{log}(P)}\right).$$
Asymptotically, both the expressions $$1-\frac{\mbox{log}(\mbox{log}(P))}{\mbox{log}(P)} \mbox{ and } 1-\frac{\mbox{log}(\mbox{log}(P))}{2\mbox{log}(P)}$$ can be taken to be equal, and hence the diversity gain is approximately $2R$ in both the schemes. However, for moderate values of $P,$ the second term is larger than the first one and this difference depends on $P.$ So, our scheme performs better than the one in \cite{JiH2} by an amount that depends on $P.$

\noindent The PEP of the scheme in \cite{JiH2} for large $P$ when specialized to $M = N = 1$ with 2$R$ antennas at relays is upper-bounded by \eqref{JingPEP}.

\noindent Using \eqref{JingPEP} and \eqref{theorem41}, the fractional change in PEP of CIDSTC with respect the one in \cite{JiH2} can be written as
\begin{equation}
\label{fraction}
\left(1 - \left(\frac{2\rho^\prime}{\rho}\right)^{2R}\left(\frac{1}{\mbox{log}(P)}\right)^{R}\right).
\end{equation}
For a specified PEP, the following scenarios may occur: The total power, $P,$ required by the CIDSTC may be smaller than that of the RDSTC or vice-verse. In the former case, since we have already shown that the PEP of CIDSTC drops at a faster rate than RDSTC, the value of $P$ required to achieve a PEP below the specified PEP will be lesser for CIDSTC compared to RDSTC.
In the event of the latter case, from \eqref{fraction} we see that, depending on the value of $\frac{\rho^\prime}{\rho}$ the corresponding value of $P$ for CIDSTC for the specified PEP may be more or less than that of the value for RDSTC. However, at large $P$, $\left(\mbox{log}(P)\right)^{R}$ dominates the above ratio and hence the expression in \eqref{fraction} increases with increase in $P$.\\
\indent Upper-bounds on the PEP in \eqref{JingPEP} and \eqref{theorem41} are plotted for $R = 5$, $T = 20$, using different values of $\rho^\prime$ and $\rho$ in Figure \ref{more}, Figure \ref{less} and Figure \ref{equal} over several values of the total power, $P$. Figures \ref{more} - \ref{equal} provide useful information on the PEP behavior of RDSTC and CIDSTC for different values of $\rho$ and $\rho^{\prime}$. In particular, these figures provide information on the power levels beyond which CIDSTC starts out performing RDSTCs and the power levels below which RDSTCs outperforms CIDSTCs. It is to be noted that the power level at which the crossover in the performance between the two schemes takes place depends on the values of $\rho$ and $\rho^{\prime}$. It is also interesting to observe that irrespective of the values of $\rho$ and $\rho^{\prime}$, there exists a sufficiently large total power $\hat{P}$ such that for $P > \hat{P}$, CIDSTC outperforms RDSTC. However, the plot shows that for all practical purposes, asymptotic coding gain provided by CIDSTC is meaningful only for the case when $\rho^{\prime} < \rho$.\\
\indent In Figures \ref{more}, \ref{less} and \ref{equal}, the upper-bounds on the PEP in \eqref{JingPEP} and \eqref{theorem41} are compared at lower values of $P$ also. Since the upper-bounds on the PEP is derived assuming a large value of $P$, the above plots may not provide actual behavior of our scheme at lower values of $P$, which corresponds to PEP in the range of $10^{-1}$ to $10^{-5}$. Plots in the above figures show that RDSTC outperforms CIDSTC at lower values of $P$ for the cases when $\rho^{\prime} > \rho$ and $\rho^{\prime} = \rho$, but we caution the reader once again to note that these plots may not provide the correct information since the derived bound is no longer valid at lower power values.

\begin{figure}
\centering
\includegraphics[width=3in]{rajan_paper-tw-nov-07-1303_fig2}

\caption{PEP comparison : R = 5, T = 20, $\rho^\prime = 2$, $\rho = 1.5$}
\label{more}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3in]{rajan_paper-tw-nov-07-1303_fig3}

\caption{PEP comparison : R = 5, T = 20, $\rho^\prime = 1.5$, $\rho = 2.$ }
\label{less}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=3in]{rajan_paper-tw-nov-07-1303_fig4}

\caption{PEP comparison : R = 5, T = 20, $\rho^\prime = 2$, $\rho = 2.$}
\label{equal}
\end{figure}

\subsection{Receiver complexity of CIDSTC}
\label{subsec3.3}
\indent From the results of Theorem \ref{thm3}, a necessary condition on the full diversity of CIDSTC is $T \geq 4R$. This implies that the number of complex variables transmitted from the source to relays in the first phase is at least twice the total number of antennas at all the relays. Therefore, CIDSTC is a design in at least 4$R$ complex variables where as a RDSTC for the same setup is a design in at least 2$R$ variables. With this, the ML decoder for CIDSTC has to decode at least 4$R$ complex variables every codeword use where as the ML decoder of RDSTC has to decode at least 2$R$ symbols every codeword use. Thus CIDSTC increases the ML decoding complexity at the receiver even though the additional complexity in performing co-ordinate interleaving of symbols at the relays is very marginal.

\section{on the full diversity of CIDSTC}\label{sec4}
In this section, we provide conditions on the signal set such that the CIDSTC in variables  $x_{1}, x_{2}, \cdots x_{T}$ is fully diverse. First, we show that CIDSTC in variables $x_{1}, x_{2}, \cdots x_{T}$ is not fully diverse if the variables take values from any 2-dimensional signal set. Towards that end, let $\textbf{X}_{\mbox{RDSTC}}$ be a fully diverse RDSTC for $2R$ relays and $T = 4R$ given by,

\begin{equation*}
\textbf{X}_{\mbox{RDSTC}} = \left[  \textbf{A}_{1}\textbf{s}~~ \textbf{A}_{2}\textbf{s}~~  \cdots ~~ \textbf{A}_{2R}\textbf{s} \right]_{4R \times 2R}
\end{equation*}

where $\textbf{s} = \left[x_{1}, x_{2}, \cdots x_{T}\right]^{T}$ and $x_{i}$ take values from some 2-dimensional signal set. Using the above RDSTCs, we can construct a CIDSTC for $R$ relays each having two antennas by assigning the relay matrices of RDSTC to every antenna of our setup.
Therefore, CIDSTC is of the form,

{\small
\begin{equation*}
\textbf{X}_{\mbox{CIDSTC}} = \left[  \textbf{A}_{11}\textbf{s}~  \textbf{A}_{11} \textbf{s}^*~ \textbf{A}_{21}\textbf{s}~ \textbf{A}_{21}\textbf{s}^*~ \cdots ~ \textbf{A}_{2R}\textbf{s}~ \textbf{A}_{2R}\textbf{s}^* \right]_{4R \times 4R}
\end{equation*}
}

\noindent where $\textbf{s}^{*} = \left[x_{1}^{*}, x_{2}^{*}, \cdots x_{T}^{*}\right]^{T}$. By permuting the columns, $\textbf{X}_{\mbox{CIDSTC}}$ can be written as,

\begin{equation}
\label{rep_cidstc}
\textbf{X}_{\mbox{CIDSTC}} = \left[\textbf{X}_{\mbox{RDSTC}} ~ \textbf{X}_{\mbox{RDSTC}}^{\prime}\right]
\end{equation}

where, $\textbf{X}_{\mbox{RDSTC}}^{\prime} = \left[  \textbf{A}_{1}\textbf{s}^{*}~~ \textbf{A}_{2}\textbf{s}^{*}~~  \cdots ~~ \textbf{A}_{2R}\textbf{s}^{*} \right]_{4R \times 2R}$.\\

\noindent From \eqref{rep_cidstc}, every codeword $\textbf{S}$ of $\textbf{X}_{\mbox{CIDSTC}}$ is of the form $\textbf{S} = \left[\textbf{S}_{1} ~ \textbf{S}_{2}\right]$ where $\textbf{S}_{1} \in \textbf{X}_{\mbox{RDSTC}}$ and $\textbf{S}_{2} \in \textbf{X}_{\mbox{RDSTC}}^{\prime}$. From Section \ref{sec3}, a CIDSTC, $\mathcal{C}$ is said to be fully diverse if $\Delta \textbf{S} = \textbf{S} - \textbf{S}^{\prime}$ is full rank, for every $\textbf{S}, \textbf{S}^{\prime} \in \mathcal{C},$ such that $\textbf{S} \neq \textbf{S}^{\prime}$.\\
The difference matrix of two codewords is given by,
\begin{equation}
\label{delta}
\Delta \textbf{S} = \left[\Delta \textbf{S}_{1} ~ \Delta \textbf{S}_{2}\right].
\end{equation}
where $\Delta \textbf{S}_{1} = \textbf{S}_{1} - \textbf{S}_{1}^{\prime}$ and  $\textbf{S}_{1}, \textbf{S}_{1}^{\prime} \in \textbf{X}_{\mbox{RDSTC}}$ such that $\textbf{S}_{1} \neq \textbf{S}_{1}^{\prime}$. Also, $\Delta \textbf{S}_{2} = \textbf{S}_{2} - \textbf{S}_{2}^{\prime}$ and  $\textbf{S}_{2}, \textbf{S}_{2}^{\prime} \in \textbf{X}_{\mbox{RDSTC}}^{\prime}$ such that $\textbf{S}_{2} \neq \textbf{S}_{2}^{\prime}$. Since $\textbf{X}_{\mbox{RDSTC}}^{\prime}$ and $\textbf{X}_{\mbox{RDSTC}}$ are fully diverse, $\Delta \textbf{S}_{1}$ and $\Delta \textbf{S}_{2}$ are full rank.\\
\begin{equation}
\label{delta1}
\Delta \textbf{S}_{1} = \left[  \textbf{A}_{1}(\Delta \textbf{s})~~ \textbf{A}_{2}(\Delta \textbf{s})~~  \cdots ~~ \textbf{A}_{2R}(\Delta \textbf{s}) \right]_{4R \times 2R}.
\end{equation}
\begin{equation}
\label{delta2}
\Delta \textbf{S}_{2} = \left[  \textbf{A}_{1}(\Delta \textbf{s})^{*}~~ \textbf{A}_{2}(\Delta \textbf{s})^{*}~~  \cdots ~~ \textbf{A}_{2R}(\Delta \textbf{s})^{*} \right]_{4R \times 2R}
\end{equation}
where $\Delta \textbf{s} = \left[\Delta x_{1}, \Delta x_{2}, \cdots \Delta x_{4R} \right]^{T}$.\\

The following proposition shows that $\Delta \textbf{S}$ is not full rank even if $\Delta \textbf{S}_{1}$ and $\Delta \textbf{S}_{2}$ are full rank.

\begin{proposition}
\label{2_d_non_full_diversity}
If variables $x_{i}$'s take value from a 2-dimensional signal set, then CIDSTC is not fully diverse.
\end{proposition}

\begin{proof}
Suppose complex variables $x_{i}$'s take value from any 2-dimensional signal set, then $\Delta \textbf{s}$ can possibly take values such that $\Delta x_{1} = \Delta x_{2} \cdots = \Delta x_{4R} \in \mathbb{C}$. Since $\Delta \textbf{S}_{1}, \Delta \textbf{S}_{2} \in \mathbb{C}^{4R \times 2R}$, some of the rows of $\Delta \textbf{S}_{1}$ are linearly dependent. Also, identical rows of $\Delta \textbf{S}_{2}$ will also be linearly dependent. Therefore, $\Delta \textbf{S}_{1}$ and $\Delta \textbf{S}_{2}$ together make the corresponding rows of $\Delta \textbf{S}$ linearly dependent.
\end{proof}

\begin{example}
For the CIDSTC in Example \ref{exple1}, if $\Delta x_{1} = \Delta x_{2} \cdots = \Delta x_{4} = \Delta x$, then $\Delta \textbf{S}$ is given by,

\begin{equation*}
\Delta \textbf{S} = \left[
\begin{array}{rrrr}
\Delta x & \Delta x^* & \textbf{j}\Delta x & \textbf{j}\Delta x^* \\
\Delta x & \Delta x^* & \Delta x  & \Delta x^* \\
\Delta x & \Delta x^* & \textbf{j}\Delta x & \textbf{j}\Delta x^* \\
\Delta x & \Delta x^* & \Delta x & \Delta x^*
\end{array}\right]
\end{equation*}

\noindent It can be observed that the first and the third row of $\Delta \textbf{S}$ are linearly dependent and hence CIDSTC in Example \ref{exple1} is not fully diverse.
\end{example}

\indent From the results of the Proposition \ref{2_d_non_full_diversity}, full diversity of CIDSTC can be obtained by making the real variables $x_{iI}, x_{iQ}$ for $i = 1, 2, \cdots T$ take values from an appropriate multi-dimensional signal set. In particular, the signal set needs to be chosen such that such that $\Delta \textbf{S}$ is full rank for every pair of codewords. The determinant of $\Delta \textbf{S}$ will be a polynomial in variables $\Delta x_{iI}, \Delta x_{iQ}$ for $i = 1, 2, \cdots T$. Therefore, a signal set has to be chosen to make determinant of $\Delta \textbf{S}$ non-zero for every pair of codewords. A particular choice of the signal set depends on the design in use. However, it is to be noted that, more than one multi dimensional signal set can provide full diversity for a given design.

\indent In the rest of this section, we provide a multi-dimensional signal set, $\Lambda$ for the CIDSTC, $\mathcal{C}$ in Example \ref{exple1}  such that, when the variables $x_{1I}, x_{1Q}, \cdots x_{4I}, x_{4Q}$ take values from $\Lambda$, the CIDSTC is fully diverse. Towards that direction, real and imaginary components of \mbox{det}$\left[\Delta \textbf{S}\right]$ for any pair of codewords is given in \eqref{realdet} and \eqref{imagdet} respectively.

\begin{figure*}

\begin{equation}
\label{realdet}
\mbox{Re }\mbox{det} \left[\Delta \textbf{S}\right] = 4\left(\Delta x_{1I}\Delta x_{3Q} - \Delta x_{1Q}\Delta x_{3I}\right)^{2} - 4\left(\Delta x_{2Q}\Delta x_{4I} - \Delta x_{2I}\Delta x_{4Q}\right)^{2} \mbox{ and }
\end{equation}
\begin{eqnarray}
\label{imagdet}
 \mbox{Im } \mbox{det} \left[\Delta \textbf{S}\right] =  -4\left(\Delta x_{1I}\Delta x_{4Q} - \Delta x_{1Q}\Delta x_{4I}\right)^{2} - 4\left(\Delta x_{2I}\Delta x_{3Q} - \Delta x_{3I}\Delta x_{2Q}\right)^{2} \nonumber \\ +
8\left(\Delta x_{1I}\Delta x_{2Q} - \Delta x_{1Q}\Delta x_{2I}\right)\left(\Delta x_{3I}\Delta x_{4Q} - \Delta x_{3Q}\Delta x_{4I}\right).
\end{eqnarray}

\end{figure*}

\indent Full diversity for the code in Example \ref{exple1} can be obtained by using a signal set which is carved out of a rotated $\mathbb{Z}^{8}$ lattice such that either the real or the imaginary components of \mbox{det}$\left[\Delta \textbf{S}\right]$ is non zero for any pair of codewords \cite{Vit}. In general, the variable $z_{i}$'s of the vector $\textbf{z} = \left[z_{1}, z_{2} \cdots z_{8} \right]^{T} \in \mathbb{Z}^{8}$ can take values from $\Xi$ say, a M - PAM set where M is any natural number. The $8$-dimensional real vector $\textbf{z}$ is rotated using the generator, $\textbf{G}$ of a rotated $\mathbb{Z}^{8}$ lattice to generate a lattice point $\textbf{l} = \left[ x_{1I} ~ x_{1Q} \cdots ~x_{4I} ~ x_{4Q}\right]$ as $\textbf{l} = \textbf{G}\textbf{z}$ using which a complex vector $\left[x_{1}~x_{2}~ x_{3}~ x_{4}\right]$ is transmitted to all the relays. The signal set $\Lambda$ is identified using computer search as $\left(\textbf{G}, ~\Xi \right)$. As an example, $z_{i}$'s is allowed to take values from $\Xi = \left\lbrace 1, -1\right\rbrace$ and the generator of the lattice, $\textbf{G}$ is found to be in \eqref{latt_gen_1}.

\begin{figure*}
{\small
\begin{equation}
\label{latt_gen_1}
\textbf{G} = \left[\begin{array}{rrrrrrrr}
-0.4081  &  0.4726  &  0.1809  & -0.3955  & -0.1556  & -0.2860  & -0.2408  &  0.5070\\
-0.3256  & -0.0526  & -0.6611  &  0.3368  & -0.5730 &  -0.0934  & -0.0510  &  0.0329\\
-0.3481  &  0.0745  &  0.1031  &  0.3303  &  0.2050  &  0.0771  &  0.7672  &  0.3421\\
-0.3844  & -0.0969  & -0.2606  & -0.6933 &   0.0365 &   0.2074 &   0.3121 &  -0.3905\\
-0.3905  & -0.1319  &  0.6135  &  0.1962  & -0.3501  & -0.2271  &  0.0040  & -0.4909\\
-0.3832  & -0.0759  &  0.1271  &  0.1629  &  0.0892 &   0.7841  & -0.4096  &  0.1193\\
-0.2931  &  0.4104  & -0.2385  &  0.2690  &  0.5999  & -0.2462  & -0.2212  & -0.3834\\
-0.2710  & -0.7532  & -0.0468 &  -0.0540 &   0.3372 &  -0.3654  & -0.1917 &   0.2649
\end{array}\right].
\end{equation}
}
\hrule
\end{figure*}

The matrix \textbf{G} in \eqref{latt_gen_1} is obtained using computer search. Through simulations, it has been verified that if the vector $\left[ x_{1I} ~ x_{1Q} \cdots ~x_{4I} ~ x_{4Q}\right]$ takes value from the above signal set $\left( \textbf{G}, ~\Xi \right) $, then the determinant of $\Delta \textbf{S}$ is non zero for any pair of codewords of $\mathcal{C}$ and hence $\mathcal{C}$ is fully diverse.
In general, for CIDSTCs of any dimension, appropriate signal sets needs to be designed so as to make the code fully diverse.

\section{simulations}
\label{sec5}

\begin{figure}
\centering
\includegraphics[width=3in]{rajan_paper-tw-nov-07-1303_fig5}

\caption{BER comparison of CIDSTC with RDSTC for R = 2 and T = 8.}
\label{fig_comp1}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=3in]{rajan_paper-tw-nov-07-1303_fig6}

\caption{BER comparison of CIDSTC with RDSTC for R = 1 and T = 4.}
\label{fig_comp2}
\end{figure}

In this section, we provide simulation results for the performance comparison of CIDSTC and RDSTC for a wireless network with two relay nodes (Figure \ref{fig_comp1}) and a single relay node (Figure \ref{fig_comp2}). Optimal power allocation strategy discussed in Subsection \ref{subsec3} has been used in our simulation setup though the strategy is not optimal for smaller values of R. Even though the power allocation used is not optimal, CIDSTCs are found to perform better than their corresponding RDSTCs. We have used the Bit Error Rate (BER) which corresponds to errors in decoding every bit as error events of interest. For the network with 2 relays, since we need $T \geq 4R$, for CIDSTC, we use the channel coherence time of $T=8$ channel use for both the schemes.

\indent The real and imaginary parts of information symbols are chosen equiprobably from a 2- PAM signal set $\{-1, 1\}$ and are appropriately scaled to maintain the unit norm condition. Simulations are carried out using the linear designs in variables $x_{1}$, $x_{2} \cdots x_{8}$ as given in \eqref{has_design} and \eqref{CI_design} for RDSTC and CIDSTC respectively. It can be verified that design in \eqref{CI_design} is of the required form given in \eqref{code}.\\
\indent The design in \eqref{has_design} is four group decodable, i.e., the variables can be partitioned into four groups and the ML decoding can be carried out for each group of variables independently of the variables in the groups and the variables of each groups need to be jointly decoded \cite{RaR}. The corresponding four groups of real variables are, $\{x_{1\textit{I}}, x_{4Q}, x_{5\textit{I}}, x_{8Q}\}, \,\{x_{1Q}, x_{4\textit{I}}, x_{5Q}, x_{8\textit{I}}\},\, \{x_{2\textit{I}}, x_{3Q}, x_{6\textit{I}}, x_{7Q}\},\\ \{x_{3\textit{I}}, x_{2Q}, x_{7\textit{I}}, x_{6Q}\}$. We use sphere decoding algorithm for ML decoding \cite{ViB}. Though the design in \eqref{has_design} is four group decodable, the design in \eqref{CI_design} is not four group ML decodable. Full diversity is obtained by making every group of real variables choose values from a rotated $\mathbb{Z}^{4}$ lattice constellation \cite{Vit} whose generator given by,
\begin{equation*}
\textbf{G} = \left[\begin{array}{rrrr}
-0.4316 &  -0.2863  &  0.5857 &  -0.6234\\
-0.6856  & -0.4520  & -0.5445  & 0.1707\\
-0.4479  &  0.8285  & -0.2068  & -0.2647\\
-0.3782  &  0.1649  &  0.5636  &  0.7157\\
\end{array}\right].
\end{equation*}
{\small
\begin{equation}
\label{has_design}
\left[\begin{array}{rrrr}
x_{1} & x_{4}^{*} & x_{5} & x_{8}^{*}\\
x_{2} & x_{3}^{*} & x_{6} & x_{7}^{*}\\
x_{3}^{*} & -x_{2} & x_{7}^{*} &- x_{6}\\
-x_{4}^{*} & x_{1} & -x_{8}^{*} & x_{5}\\
x_{5} & x_{8}^{*} & x_{1} & x_{4}^{*}\\
x_{6} & x_{7}^{*} & x_{2} & x_{3}^{*}\\
x_{7}^{*} & -x_{6} & x_{3}^{*} &- x_{2}\\
-x_{8}^{*} & x_{5} & -x_{4}^{*} & x_{1}\\
\end{array}\right]
\end{equation}
\begin{equation}
\label{CI_design}
\left[\begin{array}{rrrrrrrr}
x_{1} & x_{1}^{*} & x_{4}^{*} & x_{4} & x_{5} & x_{5}^{*} & x_{8}^{*} & x_{8} \\
x_{2} & x_{2}^{*} &  x_{3}^{*} & x_{3} & x_{6} & x_{6}^{*} & x_{7}^{*} & x_{7}\\
x_{3}^{*} & x_{3} & -x_{2} & -x_{2}^{*} & x_{7}^{*} & x_{7} & -x_{6} & -x_{6}^{*}\\
-x_{4}^{*} & -x_{4} &  x_{1} & x_{1}^{*} & -x_{8}^{*} & -x_{8} & x_{5} & x_{5}^{*}\\
x_{5} & x_{5}^{*} & x_{8}^{*} & x_{8} & x_{1} & x_{1}^{*} & x_{4}^{*} & x_{4}\\
x_{6} & x_{6}^{*} &  x_{7}^{*} & x_{7} & x_{2} & x_{2}^{*} & x_{3}^{*} & x_{3}\\
x_{7}^{*} & x_{7} & -x_{6} & -x_{6}^{*} & x_{3}^{*} & x_{3} & -x_{2} & -x_{2}^{*}\\
-x_{8}^{*} & -x_{8} &  x_{5} & x_{5}^{*} & -x_{4}^{*} & -x_{4} & x_{1} & x_{1}^{*}\\
\end{array}\right]\\
\end{equation}
}

\noindent BER comparison of the two schemes using the above designs is shown in Figure \ref{fig_comp1}. The plot shows that CIDSTC in \eqref{CI_design} performs better than the RDSTC in \eqref{has_design} by close to 1.5 to 2 db.\\
\indent Simulation results comparing the BER performance of CIDSTC in Example \ref{exple1} with its corresponding RDSTC is shown in Figure \ref{fig_comp2}. Full diversity is obtained by choosing a rotated $\mathbb{Z}^{8}$ lattice constellation (Section \ref{sec4}) whose generator is given by \eqref{latt_gen_1}. The plot shows the superiority of the design in Example \ref{exple1} over its RDSTC counterpart by 1 db. Simulation results comparing the BER performance of CIDSTC in Example \ref{exple1} with RDSTC from random coding is also shown in Figure \ref{fig_comp2} which shows the superiority of CIDSTC by 1.75 - 2 db at larger values of P.

\section{Discussion}
\label{sec6}
The technique of co-ordinate interleaved distributed space-time coding at the relays was introduced for wireless relay networks having $R$ relays each having two antennas. For $T \geq 4R$, we have shown that CIDSTC provides coding gain compared to the scheme when transmit and receive signals at different antennas of the same relay are processed independently. This improvement is at the cost of only a marginal additional complexity in processing at the relays. Condition on the full diversity of CIDSTCs is also presented. Some of the possible directions for future work is to
extended the above technique to relay networks where the source and the destination nodes have multiple antennas.
Also, if the relays have more than two antennas then a general linear processing need to be employed in the place of CIVP and new performance bounds need to be derived.

\appendices

\section{Proof of Lemma \ref{thm2}}\label{append2}

The channel \textbf{h} as given in \eqref{channel} can be written as the product $\textbf{G}\textbf{k}$ of  $\textbf{G}$ and $\textbf{k}$ where
\begin{equation*}
\textbf{G} = \mbox{diag}\left\{ g_{11}, g_{11}, g_{21}, g_{21}, g_{12}, \ldots ,g_{1R}, g_{1R}, g_{2R}, g_{2R}\right\}
\end{equation*}
and
\begin{equation*}
\textbf{k} = \left[ k_{11}, k_{21}, k_{11},- k_{21}, k_{12} \ldots , k_{1R}, k_{2R}, k_{1R},- k_{2R}\right]^T.
\end{equation*}
Since $\textbf{U}$ is Hermitian and positive definite, we can write $\textbf{U} = \textbf{V}^{H}\textbf{D}\textbf{V}$, where $\textbf{D}$ is diagonal matrix containing the eigen values of $\textbf{U},$ in \eqref{chernoff}. Since, $\textbf{U}$ is of full rank, the right hand side of the resulting following PEP expression
\begin{center}
$P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right | g_{ij})\, \leq \,E \,e^{-(P^\prime \textbf{k}^{H}\textbf{G}^{H}\textbf{V}^{H}\textbf{D}\textbf{V}\textbf{G}\textbf{k})}.$ \\
\end{center}
can be upper-bounded by replacing $\textbf{D}$ by $\rho^{2}\textbf{I}_{4R},$  where $\rho^2$ is the minimum singular value of $\textbf{U}.$ Then, we have,
\begin{center}
$P^\prime  \rho^{2}\textbf{k}^{H}\textbf{G}^{H}\textbf{G}\textbf{k} = \sum_{j = 1}^{R} \left[ P^\prime \rho^{2}g_{j}(|k_{1j}|^{2} + |k_{2j}|^{2})\right] $
\end{center}

\noindent where $g_{j} = |g_{1j}|^{2} + |g_{2j}|^{2}.$\\

\noindent Since the set of random variables $|k_{ij}|^{2}$ are independent (from Proposition \ref{prop}) and distributed as  $2e^{-2|k_{ij}|^{2}},$ we have,
\begin{center}
$P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right | g_{ij})\, \leq  \prod_{j = 1}^{R}\prod_{i = 1}^{2} E \, \left[ e^{-P^\prime \rho^{2}g_{j}(|k_{ij}|^{2})}\right],$
\end{center}
and hence
$$P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right | g_{ij})\, \leq  \prod_{j = 1}^{R}\prod_{i = 1}^{2} \int_{0}^{\infty} 2e^{-(2+ P^\prime \rho^{2}g_{j})|k_{ij}|^{2}}d|k_{ij}|^{2}$$
leading to
\begin{center}
$P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right | g_{ij})  \leq  \prod_{j = 1}^{R} 4\left[ \frac{1}{2 + P^\prime \rho^{2}g_{j}}\right] ^{2}.$
\end{center}
This completes the proof.

\section{Proof of Theorem \ref{thm3}}\label{append3}
From \eqref{theorem4eqn} we have
\begin{center}
$P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right | g_{ij})  \leq  \prod_{j = 1}^{R} 4(\frac{1}{2 + P^\prime \rho^{2}g_{j}})^{2}$\\
\end{center}
\noindent where $g_{j} = |g_{1j}|^{2} + |g_{2j}|^{2}.$

\noindent Since $|g_{ij}|^{2}$ are exponentially distributed independent random variables, the random variable $g_{j} =  |g_{1j}|^{2} + |g_{2j}|^{2}$ has the Gamma distribution, $p(g_{j}) = g_{j}e^{-g_{j}}.$

\noindent Since $g_{j}$ are independent, we omit the subscript $j$ and from \eqref{theorem4eqn} we get\\
\begin{center}
$P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right) \, \leq    \left[ 4E\, (\frac{1}{2 + P^\prime \rho^{2}g})^{2}\right] ^{R}$\\
\end{center}
\begin{equation}\label{eqna}
P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right) \, \leq  \left[4 \int_{0}^{\infty} \left( \frac{1}{2 + P^\prime \rho^{2}g}\right) ^{2}ge^{-g}dg\right] ^{R}.
\end{equation}
Let  $\alpha = P^\prime \rho^{2}.$ By change of variables in the integral in \eqref{eqna} as $t = 2 + \alpha g$
we have
\begin{equation*}
P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right) \, \leq  \left[\frac{4}{\alpha^{2}}e^{\frac{2}{\alpha}}\right] ^{R}\left[ \int_{2}^{\infty} \frac{1}{t}e^{-\frac{t}{\alpha}}dt - \int_{2}^{\infty} \frac{1}{t^{2}}e^{-\frac{t}{\alpha}}dt \right]^{R}\\
\end{equation*}
and further, changing $-\frac{t}{\alpha}\:$ to $\:  t, $
\begin{equation}
\label{fullone}
P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right) \, \leq \left[\frac{4}{\alpha^{2}}e^{\frac{2}{\alpha}}\right]^{R} \left[  \int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t}e^{t}dt + \frac{2}{\alpha}\int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t^{2}}e^{t} dt \right]^{R}.\\
\end{equation}
Using the chain rule of integration, for any integer $m$ we can write the recursive relation,

\begin{equation*}
 \int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t^{m + 1}}e^{t}dt  = \frac{1}{m}\left[\int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t^m}e^{t}dt + \left( \frac{\alpha}{2}\right) ^{m}\left(-1 \right)^{m}  \right].
\end{equation*}

\begin{equation}
\label{recursive}
 \int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t^{2}}e^{t}dt  = \left[\int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t}e^{t}dt - \frac{\alpha}{2}\right].
\end{equation}
Applying \eqref{recursive} in \eqref{fullone}, we have

{\small
\begin{equation*}
P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right) \, \leq \left[\frac{4}{\alpha^{2}}e^{\frac{2}{\alpha}}\right]^{R} \left[  \int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t}e^{t}dt + \frac{2}{\alpha}\left[\int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t}e^{t}dt - \frac{\alpha}{2}\right] \right]^{R}.
\end{equation*}
}

\begin{equation}\label{eqnb}
\leq \left[\frac{4}{\alpha^{2}}e^{\frac{2}{\alpha}}\right]^{R} \left[  \int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t}e^{t}dt + \frac{2}{\alpha}\int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t}e^{t}dt - 1\right]^{R}.
\end{equation}
We know that
\begin{equation*}
-Ei( -\frac{2}{\alpha} ) = \int_{-\frac{2}{\alpha}}^{-\infty}\frac{1}{t}e^{t}dt \\
\end{equation*}
is the exponential integral function \cite{GrR}, \cite{JiH1}.\\

As in \cite{JiH1}, for large $P$, $-Ei( -\frac{2}{\alpha} ) = -Ei( -\frac{64R}{PT\rho^{2}}) =  \mbox{log}(P) + O(1) \simeq \mbox{log}(P)$ and  $e^{\frac{2}{\alpha}} \simeq 1.$
Hence, \eqref{eqnb} can be written as
\begin{equation*}
P\left(\textbf{S}\rightarrow \textbf{S}^\prime\right) \, \leq  \left[\frac{64R}{PT\rho^{2}}\right] ^{2R}\left[ \mbox{log}(P) + \frac{64R}{PT\rho^{2}}\mbox{log}(P) - 1\right] ^{R}
\end{equation*}

\begin{equation}\label{eqnc}
 \leq  \left[\frac{64R}{PT\rho^{2}}\right] ^{2R}(\mbox{log}(P))^{R}\left[ 1 + \frac{64R}{PT\rho^{2}}\right]^{R}.\\
\end{equation}

\noindent Considering the most significant term of $P$ in \eqref{eqnc},
\begin{equation}
P\left( \textbf{S}\rightarrow \textbf{S}^\prime\right) \leq \left[\frac{64R}{T\rho^{2}}\right]^{2R}\left[\frac{(\mbox{log}(P))^{R}}{P^{2R}}\right].
\end{equation}
This completes the proof.