text:
\section{Introduction}
Most classical
(co)homology theories of algebraic
objects such as groups or Lie,
Lie-Rinehart or associative algebras
can be realised as
\begin{equation}\label{anfang}
		  H^\bullet(X,M):=
		  \mathrm{Ext}^\bullet_U(A,M),\quad
		  H_\bullet(X,N):=
		  \mathrm{Tor}^U_\bullet(N,A)
\end{equation}
for an augmented ring $X=(U,A)$
(a ring with a distinguished left module)
that is functorially attached to a
given object.
The cohomology coefficients
are left $U$-modules $M$ and
those in homology
are right $U$-modules $N$.

Our aim here is to clarify the origin
and interplay of multiplicative structures
and dualities between such
(co)homology groups, and to
provide
a unified treatment of results
by Van den Bergh on Hochschild
(co)homology \cite{VdB:ARBHHACFGR}
and by Huebschmann on
Lie-Rinehart (co)homology
\cite{Hue:DFLRAATMC}.
The key
concept involved is that of a $\times_A$-Hopf
algebra introduced by
Schauenburg \cite{Schau:DADOQGHA}.

The main results can be summarised as
follows:
\begin{thm}\label{main}
For any $A$-biprojective $\times_A$-Hopf
 algebra $U$ there is a functor
$$
		\otimes : U\mbox{-}\mathbf{Mod} \times U^\mathrm{op}\mbox{-}\mathbf{Mod}
		  \rightarrow U^\mathrm{op}\mbox{-}\mathbf{Mod}
$$
that induces for
$M \in U\mbox{-}\mathbf{Mod},N \in U^\mathrm{op}\mbox{-}\mathbf{Mod}$
and $m,n \ge 0$
natural products
$$
		  \smallfrown \,\, :
		  \mathrm{Ext}^m_U(A,M) \times
		  \mathrm{Tor}^U_n(N,A)
		  \rightarrow \mathrm{Tor}^U_{n-m}
		  (M \otimes N,A).
$$

If $A \in U\mbox{-}\mathbf{Mod}$ admits a finitely
generated projective resolution of
finite length and
there exists $d \ge 0$ with
$\mathrm{Ext}_U^m(A,U)=0$
for $m \neq d$, then
there is a canonical element
$$
		  [\omega] \in
 \mathrm{Tor}^U_d(A^*,A),\quad
A^*:=\mathrm{Ext}_U^d(A,U)$$
such that
for $m \ge 0$
and $M \in U\mbox{-}\mathbf{Mod}$ with
$\mathrm{Tor}^A_q(M,A^*)=0$ for $q>0$
$$
		  \cdot \smallfrown [\omega] :
		  \mathrm{Ext}_U^m(A,M)
 \rightarrow
		  \mathrm{Tor}^U_{d-m}
		  (M \otimes A^*,A)
$$
is an isomorphism.
\end{thm}

As we will recall below,
$\times_A$-bialgebras
and $\times_A$-Hopf algebras
generalise bialgebras and
Hopf algebras towards
possibly noncommutative base algebras $A$.
Besides Hopf algebras, both the
universal enveloping algebra $U(A,L)$
of a Lie-Rinehart algebra $(A,L)$ and the
enveloping algebra $A^e=A \otimes_k
A^\mathrm{op}$ of an associative algebra
$A$ are $\times_A$-Hopf algebras, see
Section~\ref{secex}.

For any $\times_A$-bialgebra $U$, the
base algebra $A$ carries a left $U$-action
and the category $U\mbox{-}\mathbf{Mod}$ of left $U$-modules
is monoidal
with unit object $A$. But only for
$\times_A$-Hopf algebras one has
a canonical operation
$\otimes$ as in Theorem~\ref{main}
which turns $U^\mathrm{op}\mbox{-}\mathbf{Mod}$ into a module
category over
$(U\mbox{-}\mathbf{Mod},\otimes,A)$
(Lemma~\ref{RechtsLinks}).

Any $\times_A$-Hopf algebra carries two
left and two right actions of the base
algebra, all commuting with each other.
The biprojectivity assumed in
Theorem~\ref{main} refers to the
projectivity of two of these, see
Section~\ref{soco}. Under this condition, we
can use the elegant formalism of suspended
monoidal categories from
\cite{Sua:THHAFTACOCP}
to define for $M,N \in U\mbox{-}\mathbf{Mod}$ and
$P \in U^\mathrm{op}\mbox{-}\mathbf{Mod}$ products
$$
 \smallsmile \,\,:
			H^m(X,M) \times
		  H^n(X,N)  \rightarrow
		  H^{m+n}(X,M \otimes
		  N),
$$
$$
 \smallfrown \,\,: H^n(X,N) \times
		  H_p(X,P) \rightarrow
		 H_{p-n} (X,N \otimes P),
$$
where we again use the abbreviations
from (\ref{anfang})
(cf.~Sections~\ref{absch1} and~\ref{absch2}).

In the last part of Theorem~\ref{main},
$A^*=H^d(X,U)=
\mathrm{Ext}^d_U(A,U)$ is a right
$U$-module via right multiplication
in $U$, and if we define the functor
$$
		  \hat{} \,\,: U\mbox{-}\mathbf{Mod} \rightarrow
		  U^\mathrm{op}\mbox{-}\mathbf{Mod},\quad
		  M \mapsto \hat M:=M \otimes A^*,
$$
then the statement can be rewritten
as an isomorphism
$$
		H^m(X,M) \simeq
		H_{\mathrm{dim}(X)-m}(X,\hat M),\quad
		\mathrm{dim}
		(X):=\mathrm{proj.dim}_U(A)
$$
that is given as in topology by the cap
product with the fundamental class
$[\omega] \in H_{\mathrm{dim}(X)}(\hat A)$ which
corresponds under the duality to
$\mathrm{id}_A \in
H^0(A)=\mathrm{Hom}_U(A,A)$.
For $M=A$ this simply means that the
$H^\bullet(A)$-module
$H_\bullet(A^*)$ is
free with generator $[\omega]$.

Theorem~\ref{main} is well-known
in group and Lie
algebra (co)homology. For $U=A \otimes_k
A^\mathrm{op}$ it reduces to Van den
Bergh's result \cite{VdB:ARBHHACFGR} that stimulated
a lot of recent research, see e.g.~
\cite{BroZha:DCATHCFNHA,Dol:TVDBDATMSOAPV,Gin:CYA,
LauRic:TPDFSQPA}. Note that
we do not need
Van den Bergh's invertibility
assumption about $A^*$, which says
that $\hat {}$ is an
equivalence. However, it is satisfied
for many well-behaved algebras [ibid.]
and implies the condition
$\mathrm{Tor}^A_q(M,A^*)=0$ for
arbitrary $A$-bimodules $M$ (since
invertible bimodules are finitely
generated projective as one-sided
modules from either side).
For Lie-Rinehart
algebras $(A,L)$, Theorem~\ref{main} is due
to Huebschmann
\cite{Hue:DFLRAATMC}, and we find
the general setting helpful for
example to understand the different
roles of left and right modules that
he has observed. As
Huebschmann has showed, the
conditions of Theorem~\ref{main} are
satisfied whenever $L$ is
finitely generated projective over $A$, and $A^*$ coincides as
an $A$-module with $\Lambda^d_AL$ and is
in particular projective, so also here we
have $\mathrm{Tor}^A_q(M,A^*)=0$ for
arbitrary $(A,L)$-modules $M$.

We were also motivated by
the current discussion
of the numerous bialgebroid
generalisations of Hopf algebras, see
\cite{Boe:HA}. Several authors have
raised the question
where Lie-Rinehart algebras fit in. They
were shown in
\cite{Xu:QG,MoeMrc:OTUEAOALRA}
to be $\times_A$-bialgebras, see also
\cite{KhaRan:CCOEHA,Hue:TUHAAWAHLRA}.
Here we remark that they are
in fact always $\times_A$-Hopf algebras,
but not necessarily
Hopf algebroids in the sense of B\"ohm
and Szlach\'anyi (Example~\ref{bumpy},
this answers a question of B\"ohm
\cite{Boe:HA}).
So both these examples and the
applications in homological algebra
clearly demonstrate the relevance of
the intermediate concept of a
$\times_A$-Hopf algebra.

Theorem~\ref{main} could be generalised
to differentially graded $\times_A$-Hopf
algebras, sheaves of such, or suitable
abstract monoidal
categories. One can also drop the
condition $\mathrm{Ext}^n_U(A,U)=0$
for $n \neq d$ and the assumption that
$\mathrm{Tor}^A_q(M,A^*)=0$. Then one obtains
for a bounded below chain complex $M$
over $U\mbox{-}\mathbf{Mod}$ an isomorphism
$\mathrm{RHom}_U(A,M) \simeq
(M \otimes_A^L \mathrm{RHom}_U(A,U))
\otimes_U^L A$.\\

N.K.~is supported by the NWO
through the GQT cluster. U.K.~is supported by the EPSRC
fellowship EP/E/043267/1 and partially by the
Polish Government Grant N201 1770 33. We
thank Andy Baker, Gabriella B\"ohm, Ken
Brown, Henning Krause and
Valery Lunts for discussions.

\section{Preliminaries on
$\times_A$-Hopf algebras}
\subsection{Some conventions}\label{soco}
Throughout this paper, ``ring'' means
``unital and associative ring'', and we fix
a commutative ring $k$.
All other algebras, modules etc.~will
have an underlying structure of a
$k$-module.

Secondly, we fix a
$k$-algebra $A$, i.e., a ring with a
ring homomorphism
$ \eta_A : k \rightarrow Z(A)$ to
its centre.
We denote by $A\mbox{-}\mathbf{Mod}$ the category of
left $A$-modules,
by $A^\mathrm{op}$ the
opposite and by
$A^e:=A \otimes_k A^\mathrm{op}$
the enveloping algebra
of $A$. Thus left
$A^e$-modules are $A$-bimodules
with symmetric action of $k$.

Our main object is finally an algebra $U$
over $A^e$, where we now refer to the
less standard notion of
an algebra over a possibly noncommutative base
algebra:
$U$ is a $k$-algebra with a $k$-algebra
homomorphism $ \eta=\eta_U : A^e \rightarrow
U$. This gives rise to a
forgetful functor
$U\mbox{-}\mathbf{Mod} \rightarrow \amoda$ using which
we consider every
$U$-module $M$ also as an $A$-bimodule
with actions
\begin{equation}\label{brot}
		  a {\,{\scriptstyle\rhd}\,} m {\,{\scriptstyle\lhd}\,} b := \eta (a \otimes_k b)m,\quad
		  a,b \in A,m \in M.
\end{equation}
Similarly, every right $U$-module $N$ is also
an $A$-bimodule via
\begin{equation}\label{salz}
		  a {\,{\scriptstyle\blacktriangleright}\,} m {\,{\scriptstyle\blacktriangleleft}\,} b :=
			n \eta (b \otimes_k a),\quad
		  a,b \in A,n \in N.
\end{equation}
In particular, $U$ itself carries
two left and two right
$A$-actions all commuting
with each other.
Usually
we consider $U$ as an $A^e$-module using
$a {\,{\scriptstyle\rhd}\,} u {\,{\scriptstyle\lhd}\,} b$, and otherwise
we write e.g.~${}_{\,{\scriptstyle\blacktriangleright}\,} U_{{\,{\scriptstyle\lhd}\,}}$ to denote
which actions are considered. Since this
will be  repeatedly a necessary
technical condition, we define:
\begin{dfn}
For an $A^e$-algebra $U$ we call
$M \in U\mbox{-}\mathbf{Mod}$
$A$-biprojective if both ${}_{\,{\scriptstyle\rhd}\,} M \in A\mbox{-}\mathbf{Mod}$
and $M_{\,{\scriptstyle\lhd}\,} \in A^\mathrm{op}\mbox{-}\mathbf{Mod}$ are
projective modules.
\end{dfn}

\subsection{$\times_A$-bialgebras \cite{Tak:GOAOAA}}
Consider an
$A^e$-algebra $U$ as above which
is also a coalgebra
in the monoidal category
$\amoda$. That is, there are maps
$$
		  \Delta : U \rightarrow U \otimes_A U,\quad
		  \varepsilon : U \rightarrow A
$$
satisfying the usual coalgebra
axioms (see e.g.~\cite{Boe:HA}
for the details), where
\begin{equation}\label{daoben}
		  U \otimes_A U=U \otimes_k U/
		  \mathrm{span}_k
		  \{u {\,{\scriptstyle\lhd}\,} a \otimes_k v-u \otimes_k
		  a {\,{\scriptstyle\rhd}\,} v\,|\,
		  a \in A,u,v \in U\}.
\end{equation}

For $A=k$ one calls $U$ a bialgebra if
$\Delta$ and $\varepsilon$ are algebra
homomorphisms, but in general there is no
natural algebra structure on
$U \otimes_A U$. The way out of this problem
was found by Takeuchi \cite{Tak:GOAOAA}
and involves the embedding
\begin{equation}\label{iota}
		  \iota : U \times_A U \rightarrow
		  U \otimes_A U,
\end{equation}
where $U \times_A U$ is the centre of
the
$A$-bimodule ${}_{\,{\scriptstyle\blacktriangleright}\,}  U_{\,{\scriptstyle\lhd}\,}
\otimes_A {}_{\,{\scriptstyle\rhd}\,} U_{\,{\scriptstyle\blacktriangleleft}\,}$:
$$
		  U \times_A U:=
		  \Bigl\{\sum_i u_i \otimes_A v_i \in
		  U \otimes_A U\,|\,
		  \sum_i a {\,{\scriptstyle\blacktriangleright}\,} u_i \otimes_A v_i=
		  \sum_i u_i \otimes_A v_i {\,{\scriptstyle\blacktriangleleft}\,} a
		  \Bigr\}.
$$
The product of $U$ turns this into
an algebra over $A^e$, with
$$
		  \eta_{U \times_A U} : A^e \rightarrow
		  U \times_A U,\quad
		  a \otimes_k b \mapsto
		  \eta (a \otimes_k 1) \otimes_A
		  \eta(1 \otimes_k b).
$$
Similarly,
$A$ is an algebra over $k$, but
not over $A^e$ in general.
To handle this one needs the canonical
map
\begin{equation}\label{pi}
		  \pi : \mathrm{End}_k(A) \rightarrow A,
		  \quad \varphi \mapsto \varphi(1),
\end{equation}
and the fact that
$\mathrm{End}_k(A)$
is an algebra over
$A^e$, with
$$
		  \eta_{\mathrm{End}_k(A)} :
		  A^e \rightarrow \mathrm{End}_k(A),\quad
		  (\eta_{\mathrm{End}_k(A)}
		  (a \otimes b))(c):=acb.
$$
Now it makes sense to require
$ \Delta $ and
$ \varepsilon $ to factor through
$ \iota $ and $ \pi $:
\begin{dfn}\label{takeke}
A (left) $\times_A$-bialgebra is an
algebra $U$ over $A^e$ together with
two homomorphisms
$\hat \Delta : U \rightarrow U \times_A U$
and $\hat \varepsilon : U \rightarrow
\mathrm{End}_k(A)$
of algebras over $A^e$
such that
$U$ is a coalgebra in $\amoda$ via
$\Delta = \iota \circ \hat \Delta$
and $ \varepsilon = \pi \circ \hat
 \varepsilon$.
\end{dfn}

So one has for example for any $\times_A$-bialgebra
$$
		  \Delta (a {\,{\scriptstyle\rhd}\,} u {\,{\scriptstyle\lhd}\,} b)=
		  a {\,{\scriptstyle\rhd}\,} u_{(1)} \otimes_A
		  u_{(2)} {\,{\scriptstyle\lhd}\,} b,\quad
		  \Delta (a {\,{\scriptstyle\blacktriangleright}\,} u {\,{\scriptstyle\blacktriangleleft}\,} b)=
		  u_{(1)} {\,{\scriptstyle\blacktriangleleft}\,} b \otimes_A a {\,{\scriptstyle\blacktriangleright}\,} u_{(2)},
$$
where we started to use Sweedler's shorthand
notation
$u_{(1)} \otimes_A u_{(2)}$
for $\Delta(u)$.

Be aware that the four
$A$-actions are not the only feature
of $\times_A$-bialgebras that disappears
for $A=k$. Another
crucial one is for example that
the counit $ \varepsilon : U \rightarrow A$ is
not necessarily a ring homomorphism.
Note also that many authors write
$s(a):=\eta (a \otimes 1)$ and
$t(a):=\eta (1 \otimes a)$ and formulate
the theory using these so-called source and
target maps rather than $ \eta $.

\subsection{The monoidal category
  $U\mbox{-}\mathbf{Mod}$ \cite{Schau:BONCRAASTFHB}}
Definition~\ref{takeke}
might appear complicated, but it is
the correct concept from several points
of view. An
important one for us is the following result
of Schauenburg \cite[Theorem~5.1]{Schau:BONCRAASTFHB}:
\begin{thm}\label{schaui1}
The $\times_A$-bialgebra
structures on an algebra
$\eta : A^e \rightarrow U$ over
$A^e$ correspond
bijectively to monoidal structures
on $U\mbox{-}\mathbf{Mod}$ for which
the forgetful functor
$U\mbox{-}\mathbf{Mod} \rightarrow \amoda$ induced by
$\eta$ is strictly monoidal.
\end{thm}

Given a $\times_A$-bialgebra structure
on $U$, the monoidal structure on
$U\mbox{-}\mathbf{Mod}$ is defined as for bialgebras:
one takes the tensor product
$M \otimes_A N$ of
the $A$-bimodules underlying
$M,N \in U\mbox{-}\mathbf{Mod}$
and defines a left
$U$-action via $\Delta$,
\begin{equation}\label{roaig}
		  u(m \otimes_A n):=u_{(1)}m
		  \otimes_A
		  u_{(2)}n,\quad u \in U,m \in M,n
		  \in N.
\end{equation}
\begin{dfn}
If $U$ is a $\times_A$-bialgebra and
$M,N \in U\mbox{-}\mathbf{Mod}$ are left $U$-modules, we
denote the left $U$-module
$M \otimes_A N$ with
$U$-action (\ref{roaig}) by
$M \otimes N$.
\end{dfn}
The unit object in $U\mbox{-}\mathbf{Mod}$ is $A$
on which $U$ acts
via
$$
		  \hat \varepsilon (u)(a)=
		  \varepsilon(a {\,{\scriptstyle\blacktriangleright}\,} u)=
		  \varepsilon (u {\,{\scriptstyle\blacktriangleleft}\,} a),
$$
where the last equality
is a consequence of the definition
of a $\times_A$-bialgebra.

There is an analogous notion of
right $\times_A$-bialgebra for which
$U^\mathrm{op}\mbox{-}\mathbf{Mod}$ is monoidal.
However,
for a left $\times_A$-bialgebra there is
in general
no canonical monoidal structure on
$U^\mathrm{op}\mbox{-}\mathbf{Mod}$ or even only
right action of $U$ on $A$.

\subsection{$\times_A$-Hopf algebras \cite{Schau:DADOQGHA}}
Let $U$ be a
$\times_A$-bialgebra and define
\begin{equation}
\label{Galois}
\beta: {}_{\,{\scriptstyle\blacktriangleright}\,} U \otimes_{A^\mathrm{op}} U_{\,{\scriptstyle\lhd}\,} \to
U_{\,{\scriptstyle\lhd}\,} \otimes_A {}_{\,{\scriptstyle\rhd}\,} U, \quad u
\otimes_{A^\mathrm{op}} v \mapsto
u_{(1)}  \otimes_A u_{(2)}v,
\end{equation}
the so-called Galois map of $U$, where
$$
 {}_{\,{\scriptstyle\blacktriangleright}\,} U \otimes_{A^\mathrm{op}} U_{\,{\scriptstyle\lhd}\,} = U
 \otimes_k U/
{{\rm
 span}\{a {\,{\scriptstyle\blacktriangleright}\,} u \otimes_k v - u
 \otimes_k v {\,{\scriptstyle\lhd}\,} a\,|\,
u,v \in U, a \in A \}}.
$$
One could flip the tensor components
in order to avoid taking the tensor
product over $A^\mathrm{op}$, but we
found it more convenient to
keep $\beta$ in the form which is
standard for bialgebras over fields. For
the latter it is
 easily seen that $\beta$ is bijective if
and only if $U$ is a Hopf algebra with
$\beta^{-1}(u \otimes_k v) := u_{(1)}
\otimes S(u_{(2)})v$, where $S$ is the
antipode of $U$. This motivates
the following definition due to
Schauenburg~\cite{Schau:DADOQGHA}:
\begin{dfn}
A $\times_A$-bialgebra $U$ is
a $\times_A$-Hopf  algebra if
$\beta$ is a bijection.
\end{dfn}

Following Schauenburg, we adopt a Sweedler-type notation
\begin{equation}
\label{pm}
u_+ \otimes_{A^\mathrm{op}} u_- := \beta^{-1}( u \otimes_A 1)
\end{equation}
for the so-called translation map
$$
		  \beta^{-1}(\cdot \otimes_A 1) : U \rightarrow
		  {}_{\,{\scriptstyle\blacktriangleright}\,} U \otimes_{A^\mathrm{op}} U_{\,{\scriptstyle\lhd}\,}.
$$
Since substantial for the subsequent
calculations, we
list some
properties of $\beta^{-1}$ as proven in
\cite[Proposition~3.7]{Schau:DADOQGHA}: one
has for all $u, v \in U$, $a \in A$
\begin{eqnarray}
\label{Sch1}
u_{+(1)} \otimes_A u_{+(2)} u_- &=& u \otimes_A 1 \in U_{\,{\scriptstyle\lhd}\,} \otimes_A {}_{\,{\scriptstyle\rhd}\,} U \\
\label{Sch2}
u_{(1)+} \otimes_{A^\mathrm{op}} u_{(1)-} u_{(2)}  &=& u \otimes_{A^\mathrm{op}} 1 \in  {}_{\,{\scriptstyle\blacktriangleright}\,} U
\otimes_{A^\mathrm{op}} U_{\,{\scriptstyle\lhd}\,} \\
\label{Sch3}
u_+ \otimes_{A^\mathrm{op}} u_- & \in
& U \times_{A^\mathrm{op}} U\\
\label{Sch37}
u_+ \otimes_{A^\mathrm{op}} u_{-(1)} \otimes_A
u_{-(2)}
&=& u_{++}
\otimes_{A^\mathrm{op}} u_-
\otimes_A u_{+-}\\
\label{Sch4}
(uv)_+ \otimes_{A^\mathrm{op}} (uv)_- &=& u_+v_+
\otimes_{A^\mathrm{op}} v_-u_-
\\
\label{Sch5}
\eta(a \otimes b)_+ \otimes_{A^\mathrm{op}}
\eta(a \otimes b)_- &=&
\eta(a \otimes 1) \otimes_{A^\mathrm{op}}
\eta(b \otimes 1),
\end{eqnarray}
where in (\ref{Sch3}) we abbreviated
$$
		  U \times_{A^\mathrm{op}} U:=
		  \Bigl\{\sum_i u_i \otimes_{A^\mathrm{op}} v_i \in
		  {}_{\,{\scriptstyle\blacktriangleright}\,} U \otimes_{A^\mathrm{op}} U_{\,{\scriptstyle\lhd}\,}\,|\,
		  \sum_i u_i {\,{\scriptstyle\lhd}\,} a \otimes_{A^\mathrm{op}} v_i=
		  \sum_i u_i \otimes_{A^\mathrm{op}} a {\,{\scriptstyle\blacktriangleright}\,}
		  v_i
		  \Bigr\}
$$
and in (\ref{Sch37}) the tensor product
over $A^\mathrm{op}$ links the first and
third tensor component (cf.~\cite
[Equation (3.7)]{Schau:DADOQGHA}).
By (\ref{Sch1}) and (\ref{Sch3})
one can write
\begin{equation}
\label{pmb}
\beta^{-1}(u \otimes_A v) = u_+ \otimes_{A^\mathrm{op}} u_-v
\end{equation}
 which is easily checked to be
 well-defined over $A$ with (\ref{Sch4})
 and (\ref{Sch5}).

\subsection{Examples}\label{secex}
Clearly, Hopf algebras over $k$
such as universal enveloping algebras of
Lie algebras or group algebras
are $\times_k$-Hopf
algebras. But also the
enveloping algebra of an associative
algebra that governs Hochschild
(co)homology is an example as
pointed out by
Schauenburg \cite{Schau:DADOQGHA}:
\begin{ex}
The enveloping
 algebra $U:=A^e$
of any $k$-algebra $A$ is a
$\times_A$-bialgebra
with $\eta=\mathrm{id}_{A^e}$ and
coproduct and counit
$$
\Delta: U \to U \otimes U, \> a
 \otimes_k b \mapsto (a \otimes_k 1)
\otimes_A
(1 \otimes_k b),\quad
\varepsilon: U \to A, \> a \otimes_k
 b \mapsto ab.
$$
As for the $\times_A$-Hopf algebra
 structure, the tensor product in
question reads
$$
 {}_{\,{\scriptstyle\blacktriangleright}\,} U \otimes_{A^\mathrm{op}} U_{\,{\scriptstyle\lhd}\,} = U
 \otimes_k U/{\rm span}_k\{(a \otimes_k cb)
\otimes_k (a' \otimes_k b')
 -(a \otimes_k b) \otimes_k
(a' \otimes_k b'c)\},
$$
where $cb$ and $b'c$ is understood to be
the product in $A$.
One then easily verifies that
$$
(a \otimes_k b)_+ \otimes_{A^\mathrm{op}}
(a \otimes_k b)_- := (a \otimes_k 1)
\otimes_{A^\mathrm{op}} (b \otimes_k 1)
$$
yields an inverse of the Galois map
defined as in (\ref{pmb}).
\end{ex}

Finally we discuss
Lie-Rinehart algebras which
define for example
Poisson (co)homology. Several authors
\cite{Xu:QG,KhaRan:CCOEHA,MoeMrc:OTUEAOALRA} have shown that
their enveloping algebras are
$\times_A$-bialgebras,
but they are
in fact also $\times_A$-Hopf algebras:

\begin{ex}\label{bumpy}
Let $(A,L)$ be a Lie-Rinehart algebra
over $k$ \cite{Rin:DFOGCA,Hue:PCAQ}. We
denote by $(a,X) \mapsto aX$ the
$A$-module structure on $L$ and
by $(X,a) \mapsto X(a)$ the
 $L$-action on $A$ given by the anchor
 $\hat \varepsilon : L \to \mathrm{Der}_k(A)$.
Its universal enveloping
algebra $U=U(A,L)$ is the
universal $k$-algebra equipped with
two maps
$$
		  \iota_A: A \to U,\quad \iota_L: L \to U
$$
of $k$-algebras and of $k$-Lie algebras,
respectively, and subject to the identities
$$
		  \iota_A(a) \iota_L(X) = \iota_L(aX),\quad
		  \iota_L(X) \iota_A(a) - \iota_A(a) \iota_L(X) =
		  \iota_A(X(a))
$$
for $a \in A,X \in L$; confer
\cite{Rin:DFOGCA} for the precise
construction.
The map
 $\iota_A$ is injective, so we refrain
from further mentioning it. We
will also merely write $X$ when we
mean $\iota_L(X)$ (if $L$ is
 $A$-projective, then $\iota_L$ is injective
 as well).

Recall now from e.g.\
 \cite{Xu:QG,MoeMrc:OTUEAOALRA}
that $U$ carries the
structure of a
$\times_A$-bialgebra: the maps
 $\eta(- \otimes 1)$ and $\eta(1 \otimes
 -)$
are equal and given by $\iota_A$. The
prescriptions
\begin{equation}
\label{LCoProdVL1}
\Delta (X) = 1 \otimes_A X + X \otimes_A
1, \quad \Delta (a)
= a \otimes_A 1
\end{equation}
which map $X \in L$ and
$a \in A$ into $U \times_A U$ can
be extended by the universal
property
to a coproduct $\hat\Delta: U \to U \times_A U$.
The counit is similarly
given by the extension of the
anchor $\hat \varepsilon$ to $U$.
The bijectivity of the Galois
map is seen in the same way:
the translation map is given on
generators as
\begin{equation}
\label{LieRineIsHopfX}
a_+ \otimes_{{A^\mathrm{op}}} a_- :=
a \otimes_{A^\mathrm{op}} 1,\quad
X_+ \otimes_{A^\mathrm{op}} X_- :=
X \otimes_{A^\mathrm{op}} 1 - 1 \otimes_{A^\mathrm{op}} X.
\end{equation}
These maps stay in
 $U \times_{A^\mathrm{op}} U$ which is an algebra
through the product of $U$ in the
 first and its opposite
in the second tensor
 factor. By universality
 we obtain a map $U \to U \times_{A^\mathrm{op}} U$,
 and then $\beta^{-1}$ is defined using
 (\ref{pmb}).

On the other hand, $U$ is not
necessarily a Hopf algebroid in the
sense of \cite{Boe:HA} (this also
 answers B\"ohm's question asked therein
 whether any $\times_A$-Hopf algebra is a Hopf
 algebroid). This structure
assumes the existence of an antipode
$S : U \rightarrow U^\mathrm{op}$
satisfying certain axioms. As a result,
the left $U$-action on $A$ yields by
composition with $S$ also a right
 $U$-module structure. However, there
might be an obstruction for this. For
example, take
$L=\Gamma(T^{1,0}S^2)$, where
$T^{1,0}S^2 \oplus T^{0,1}S^2=TS^2
 \otimes \mathbb{C}$ is the
 decomposition of the complexified
 tangent bundle of the 2-sphere $S^2
 \subset \mathbb{R}^3$ into the
 holomorphic and antiholomorphic part
 with respect to the standard complex
 structure. This defines together with
$A=C^\infty(S^2,\mathbb{C})$ a
Lie-Rinehart algebra, where the action
of $L$ on $A$ is the usual action of a
vector field on a smooth function and
the action of $A$ on $L$ is given by
fibrewise multiplication. We know by work
of Huebschmann \cite{Hue:LRAGAABVA}
that the right $U$-module structures on
$A$ correspond bijectively to left
$U$-module structures on
$L$ itself (in general on its top
exterior power  over $A$, but here this
is $L$ because $T^{1,0}S^2$ is only a
line bundle).
Such a left $U$-action
corresponds precisely to a flat
connection $\nabla$ on the complex line bundle
$T^{1,0}S^2$, with $X \in L$ acting on
 sections of $T^{1,0}S^2$ by the covariant
 derivative $\nabla_X$ (see
\cite{Hue:LRAGAABVA} for the details).
But the
curvature of any connection
represents the first Chern
class of the bundle which is nonvanishing
since $T^{1,0}S^2$ is not
trivial. Therefore, there is no flat
connection aka left $U$-action on $L$ and hence no right
$U$-action on $A$.
\end{ex}

\section{Multiplicative structures}
\subsection{$\mathcal{D}^-(U)$ as a suspended
monoidal category \cite{Sua:THHAFTACOCP} }
For any ring $U$, we denote by
$\mathcal{D}^{-}(U)$ the derived category of
bounded above cochain
complexes of left $U$-modules. As
usual, we identify any
$M \in U\mbox{-}\mathbf{Mod}$ with a complex in
$\mathcal{D}^-(U)$ concentrated in degree $0$, and
any bounded below chain complex
$P_\bullet$ with a
bounded above cochain complex by putting
$P^n:=P_{-n}$.

If $U$ is an
$A$-biprojective $\times_A$-bialgebra,
then any projective
$P \in U\mbox{-}\mathbf{Mod}$ is $A$-biprojective. Hence
the monoidal structure of $U\mbox{-}\mathbf{Mod}$
extends to a monoidal
structure on $\mathcal{D}^-(U)$
with unit
object still given by $A$ and product
being the total tensor product
$\otimes^\mathrm{L}=\otimes^\mathrm{L}_A$ (the
$A$-biprojectivity of $U$-projectives is
needed for example to have
\cite[Lemma~10.6.2]{Wei:AITHA}).

Together with the shift functor
$T : \mathcal{D}^-(U) \rightarrow \mathcal{D}^-(U)$,
$(TC)^n=C^{n+1}$, $\mathcal{D}^-(U)$ becomes what
is called a suspended monoidal category
in \cite{Sua:THHAFTACOCP}. This just means
that for all $C,D \in \mathcal{D}^-(U)$, the canonical
isomorphisms
$$
		  TC \otimes^\mathrm{L} D \simeq T(C \otimes^\mathrm{L} D)
		  \simeq C \otimes^\mathrm{L} TD
$$
given by the obvious renumbering make
the diagrams
$$
\xymatrix{A \otimes^\mathrm{L} TC \ar[d]\ar[r] & TC
\ar[dl]\\
T(A \otimes^\mathrm{L} C)}\quad\quad
\xymatrix{TC \otimes^\mathrm{L} A \ar[d]\ar[r] & TC
\ar[dl]\\
T(C \otimes^\mathrm{L} A)}
$$
commutative and the diagram
$$
\xymatrix{TC \otimes^\mathrm{L} TD \ar[d]\ar[r] &
T(C \otimes^\mathrm{L} TD)
\ar[d]\\
T(TC \otimes^\mathrm{L} D) \ar[r] & T^2(C \otimes^\mathrm{L} D)}
$$
anticommutative (commutative up to a sign
$-1$).

\subsection{$\smallsmile$ and
$\circ$ \cite{Sua:THHAFTACOCP} }\label{absch1}
As a special case of the constructions
from \cite{Sua:THHAFTACOCP}, we
define
for any $A$-biprojective
$\times_A$-bialgebra $U$
and $L,M,N \in U\mbox{-}\mathbf{Mod}$ the cup product
$$
		  \smallsmile \,\,:
		  \mathrm{Ext}^m_U(A,M) \times
		  \mathrm{Ext}^n_U(A,N)
		  \rightarrow
		  \mathrm{Ext}^{m+n}_U(A,M \otimes
		  N)
$$
and the classical Yoneda product
$$
		  \circ :
		  \mathrm{Ext}^m_U(N,M) \times
		  \mathrm{Ext}^n_U(L,N)
		  \rightarrow
		  \mathrm{Ext}^{m+n}_U(L,M).
$$
The latter is just the composition
of morphisms in $\mathcal{D}^-(U)$ if one
identifies
$$
		  \mathrm{Ext}_U^n(L,N)
		  \simeq \mathrm{Hom}_{\mathcal{D}^-(U)}(L,T^nN),
$$
and
$$
		  \mathrm{Ext}_U^m(N,M)
		  \simeq \mathrm{Hom}_{\mathcal{D}^-(U)}(N,T^mM)
		  \simeq
		  \mathrm{Hom}_{\mathcal{D}^-(U)}(T^nN,T^{m+n}M).
$$
The former is obtained as follows: given
$$
		  \varphi \in \mathrm{Ext}_U^m(A,M)
		  \simeq
		  \mathrm{Hom}_{\mathcal{D}^-(U)}(A,T^mM),
$$
$$
		  \psi \in \mathrm{Ext}_U^n(A,N)
		  \simeq \mathrm{Hom}_{\mathcal{D}^-(U)}(A,T^nN),
$$
one defines $\varphi \smallsmile \psi$
as the composition
\begin{eqnarray}
&& A \simeq
		 A \otimes A \nonumber\\
&\xymatrix{\ar[r]^{\varphi
		 \otimes \psi} &}&
 		 T^mM \otimes^\mathrm{L} T^nN \simeq T^m(M
		 \otimes^\mathrm{L} T^nN) \simeq
		 T^{m+n}(M \otimes^\mathrm{L} N)
		 \nonumber\\
&\xymatrix{\ar[r] &}& T^{m+n}(M \otimes N),
		 \nonumber
\end{eqnarray}
where the last map is the augmentation
$M \otimes^\mathrm{L} N \rightarrow H^0(M \otimes^\mathrm{L}
N) \simeq \mathrm{Tor}^A_0(M,N) \simeq M
\otimes N$, or rather $T^{m+n}$ applied
to this morphism in $\mathcal{D}^-(U)$.

A straightforward extension of
Theorem~{1.7} from
\cite{Sua:THHAFTACOCP} now gives:
\begin{thm}\label{suarez}
If $U$ is an $A$-biprojective
$\times_A$-bialgebra,
then we have
$$
		  \psi \circ \varphi = \varphi
		  \smallsmile \psi = (-1)^{mn}\psi \smallsmile
		  \varphi,\quad
		  \varphi \in
		  \mathrm{Ext}_U^m(A,A),
		  \psi \in \mathrm{Ext}_U^n(A,M)
$$
as elements of
$
		  \mathrm{Ext}^{m+n}_U(A,M) \simeq
		  \mathrm{Ext}_U^{m+n}(A,A \otimes
		  M) \simeq
 \mathrm{Ext}_U^{m+n}(A,M \otimes A).
$

In particular, $\mathrm{Ext}_U(A,A)$
 becomes through either of the products
 a graded commutative algebra over
the commutative subring $\mathrm{Hom}_U(A,A)$.
\end{thm}
\begin{pf}
This is proven exactly as in
 \cite{Sua:THHAFTACOCP}.
For the reader's convenience we include
one of the diagrams involved. The
 unlabeled arrows are
canonical maps coming from
the suspended monoidal structure.
$$
\xymatrix@=38pt{
A \ar[d]_\varphi \ar[r]
& A \otimes A \ar[d]_{\mathrm{id}
 \otimes \varphi} \ar[rd]^{\psi \otimes
 \mathrm{id}}\\
T^m A \ar[dr] \ar[r]\ar[dd]_{\mathrm{id}} & A \otimes
 T^m A \ar[d] \ar[dr]^{\psi \otimes
 \mathrm{id}}& T^n M
 \otimes^\mathrm{L} A \ar[d]^{\mathrm{id} \otimes
 \varphi}\\
& T^m(A \otimes A) \ar[dl]
 \ar[d]_{T^m(\psi \otimes \mathrm{id})\!\!}&
T^n M \otimes^\mathrm{L} T^m A \ar[d] \ar[dl]\\
T^m A \ar[d]_{T^m(\psi)} & T^m(T^n M
 \otimes^\mathrm{L} A) \ar[dl] \ar[d]& T^n(M
 \otimes^\mathrm{L} T^mA)\ar[dl]\\
T^{m+n}M & T^{m+n}(M \otimes^\mathrm{L} A) \ar[l]
}
$$
The morphism
$\psi \circ \varphi \in \mathrm{Hom}_{\mathcal{D}^-(U)}(A,T^{m+n}M)$
is the path going straight down
from $A$ to $T^{m+n}M$, and $\psi
\smallsmile \varphi$ is the one which
goes clockwise round the whole diagram. All
faces of the
diagram commute except the
 lower right square
which introduces a sign $(-1)^{mn}$,
so we get
$\psi \circ \varphi = (-1)^{mn} \psi
 \smallsmile \varphi$. The other
 identity is shown with a similar diagram.
\end{pf}
{}\\[-10mm]

\subsection{Tensoring
projectives}\label{barreso}
This paragraph is a small excursus about
the projectivity of the tensor product of two
projective objects of a monoidal
category.
For example, $U \otimes U \in U\mbox{-}\mathbf{Mod}$
is not necessarily projective even for a
bialgebra $U$ over a field $A=k$
(so the $A$-projectivity of $U$ or the
exactness of $\otimes$ does not
help). Here is a simple example
(for a detailed study of examples
of categories
of Mackey functors see
\cite{Gau:WPDNIFAOHA}):
\begin{ex}\label{bsp}
Consider the bialgebra
$U=\mathbb{C}[a,b,c]$ over
$A=k=\mathbb{C}$ with
$$
		  \Delta (a)=a \otimes a,\quad
		  \Delta (b)=a \otimes b+b \otimes c,\quad
		  \Delta (c) = c \otimes c,
$$
$$
		  \varepsilon (a) =1,\quad
		  \varepsilon (b) = 0,\quad
		  \varepsilon (c) = 1.
$$
Geometrically, this is the coordinate ring
of the complex algebraic semigroup
$G$ of upper triangular $2 \times 2$-matrices,
and $ \Delta $ and $ \varepsilon $ are dual
to the semigroup law $G \times G \rightarrow G$
and the embedding of the
identity matrix into $G$.

We prove that
$U \otimes U \in U\mbox{-}\mathbf{Mod}$
is not projective by
considering the fibres
of the semigroup law $G \times G \rightarrow G$.
The fibre over a generic and hence invertible
element is 3-dimensional, but over $0$ it is
 4-dimensional, and this will imply our
claim. We can use for example \cite[Theorem 19 on p.~79]{Mat:CA}:
\begin{thm}
Let $U \subset V$
be a flat extension of
commutative Noetherian rings,
$\mathfrak{p} \subset V$ be a prime
 ideal and
$\mathfrak{q} :=
U \cap \mathfrak{p}$. Then
$$
		  \mathrm{dim} (V_\mathfrak{p}) =
		  \mathrm{dim} (U_\mathfrak{q}) +
		  \mathrm{dim} (V_\mathfrak{p} \otimes_U
		  U(\mathfrak{q})),
$$
where $ \mathrm{dim} $ denotes the Krull
dimension of a ring, $V_\mathfrak{p}$ is
the localisation of $V$ at
$\mathfrak{p}$ and
$U(\mathfrak{q}):=U_\mathfrak{q} /
\mathfrak{q} U_\mathfrak{q}$
is the
residue field of the localisation
$U_\mathfrak{q}$.
\end{thm}

Apply this to our example
$U \simeq \Delta (U) \subset V:=U
 \otimes U$:
let $\mathfrak{p}$ be the ideal of $V$
generated by $a \otimes_\mathbb{C} 1$,
$1 \otimes_\mathbb{C} a$, $b \otimes_\mathbb{C} 1$,
$1 \otimes_\mathbb{C} b$, $c \otimes_\mathbb{C} 1$,
$1 \otimes_\mathbb{C} c$. Geometrically, $V$ is
the coordinate ring of $\mathbb{C}^6$ and
$V_\mathfrak{p}$ is the local ring in
$0$, so $\mathrm{dim} (V_\mathfrak{p})=6$.
Since $1 \notin \mathfrak{p}$,
$\mathfrak{q}=U \cap \mathfrak{p}$
is proper, and it contains the ideal
generated by
$\Delta(a)=a \otimes_\mathbb{C} a$,
$\Delta(b)=a \otimes_\mathbb{C} b+b \otimes_\mathbb{C} c$,
$\Delta (c)=c \otimes_\mathbb{C} c$ which is maximal in
$U$, so
$\mathfrak{q} \subset U$ is the ideal generated
by $a,b,c$, and $U_\mathfrak{q}$
is the local ring of
$\mathbb{C}^3$ at $0$ with
$\mathrm{dim} (U_\mathfrak{q})=3$.
The field $U(\mathfrak{q})$ is
obviously $\mathbb{C}$, and we can write
$V_\mathfrak{p} \otimes_U U(\mathfrak{q})$
also as $V_\mathfrak{p}/
\Delta(\mathfrak{q})V_\mathfrak{p}$.
Since $ \Delta (\mathfrak{q}) V_\mathfrak{p}$
is contained in the ideal $\mathfrak{r}$
generated in $V_\mathfrak{p}$ by the
elements $a \otimes_\mathbb{C} 1,1 \otimes_\mathbb{C} c$,
we have
$\mathrm{dim} (V_\mathfrak{p}/
\Delta(\mathfrak{q})V_\mathfrak{p})
\ge
\mathrm{dim} (V_\mathfrak{p} /\mathfrak{r})$.
Now $V_\mathfrak{p}/\mathfrak{r}$
is the local ring of
$\mathbb{C}^4 \subset \mathbb{C}^6$
at $0$ and hence
$\mathrm{dim}
(V_\mathfrak{p} /\mathfrak{r})=4$.
In total, we get the strict inequality
$3+\mathrm{dim} (V_\mathfrak{p}/
\Delta(\mathfrak{q})V_\mathfrak{p})
\ge 3+4=7>6$,
and hence $V$ is not flat over
$U$ and in particular not projective.
\end{ex}

For $\times_A$-Hopf algebras the
situation is, however, much simpler:
notice that
$$
{}_{\,{\scriptstyle\blacktriangleright}\,} U \otimes_{A^\mathrm{op}}
M {\,{\scriptstyle\lhd}\,} := U \otimes_k M/{{\rm
 span}\{a {\,{\scriptstyle\blacktriangleright}\,} u \otimes_k m -
u \otimes_k m {\,{\scriptstyle\lhd}\,} a\,|\,u \in U,
a \in A,m \in M\}}
$$
is for any $\times_A$-bialgebra $U$ and
$M \in U\mbox{-}\mathbf{Mod}$ a left $U$-module by
left multiplication
on the first factor. Just as for $M=U$,
there is a Galois map
$$
		  \beta_M : {}_{\,{\scriptstyle\blacktriangleright}\,} U \otimes_{A^\mathrm{op}}
		  M_{\,{\scriptstyle\lhd}\,} \to  U \otimes M, \quad
		  u \otimes_{A^\mathrm{op}} m \mapsto u_{(1)} \otimes_A
     	  u_{(2)} m,
$$
and we have:
\begin{lem}
\label{resomod}
For any $\times_A$-bialgebra $U$, the
generalised Galois map $\beta_M$ is a
morphism of $U$-modules. If $U$ is a
 $\times_A$-Hopf algebra, then
$\beta_M$ is bijective.
\end{lem}
\begin{pf}
The $U$-linearity of $\beta_M$ follows
immediately from the fact that
$\hat\Delta : U \rightarrow U \times_A U$
is a homomorphism of algebras over $A^e$.
Furthermore, if $\beta$ is a bijection,
 then $\beta_M$ is so as well
since we can identify $\beta_M$ with
$\beta \otimes_U {\rm id}_M$, and then
the inverse is simply
given by
$\beta_M^{-1}(u \otimes_A m) = u_+
 \otimes_{A^\mathrm{op}} u_-m$.
\end{pf}
{}\\[-7mm]

Using this one now gets:
\begin{thm}\label{uuu}
If $U$ is a $\times_A$-Hopf algebra
and $U_{\,{\scriptstyle\lhd}\,} \in A^\mathrm{op}\mbox{-}\mathbf{Mod}$ is projective, then
$P \otimes Q \in U\mbox{-}\mathbf{Mod}$ is
projective for all projectives $P,Q \in
 U\mbox{-}\mathbf{Mod}$.
\end{thm}
\begin{pf}
By assumption, any projective module
over $U$ is also projective over
 $A^\mathrm{op}$, and
if $\varphi : R \rightarrow S$ is
any ring map, then
$S \otimes_R \cdot : R\mbox{-}\mathbf{Mod} \rightarrow S\mbox{-}\mathbf{Mod}$
maps projectives to projectives.
This shows that
${}_{\,{\scriptstyle\blacktriangleright}\,} U \otimes_{A^\mathrm{op}} U_{\,{\scriptstyle\lhd}\,}$
and hence (Lemma~\ref{resomod})
$U \otimes U$ is projective.
Since $\otimes=\otimes_A$ commutes
with arbitrary direct sums,
$P \otimes Q$ is projective for
all projectives $P,Q$.
\end{pf}
{}\\[-9mm]
\begin{cor}
If $U$ is as in Theorem~\ref{uuu} and
$P \in \mathcal{D}^-(U)$ is a
projective resolution of $A \in U\mbox{-}\mathbf{Mod}$,
then so is $P \otimes P:=\mathrm{Tot}(P_\bullet \otimes
 P_\bullet)=P \otimes^\mathrm{L} P$.
\end{cor}

This leads to the traditional
construction of $\smallsmile\,$
given for $A=k$ in
\cite[Chapter XI]{CarEil:HA}:
one fixes a projective resolution
$P$ of $A$, and by the above,
$ \mathrm{Ext}_U(A,M \otimes N)$
is the total (co)homology of the double (cochain)
complex
$$
C^2_{mn}:=
\mathrm{Hom}_U(P_m \otimes P_n,
M \otimes N).
$$
Then $\smallsmile$ is given as the composition of the canonical map
\begin{eqnarray}
&& \bigoplus_{m+n=p}
		  \mathrm{Ext}^m_U(A,M) \otimes_k
		  \mathrm{Ext}^n_U(A,N)\nonumber\\
&\simeq&
		  \bigoplus_{m+n=p}
		  H^m(\mathrm{Hom}_A(P_\bullet,M))
		  \otimes_k
 H^n(\mathrm{Hom}_A(P_\bullet,N))\nonumber\\
&\rightarrow& H^p(\bigoplus_{m+n=\bullet}
		  \mathrm{Hom}_A(P_m,M)
		  \otimes_k
 \mathrm{Hom}_A(P_n,N))
= H^p(\mathrm{Tot}(C^1_{\bullet\bullet}))
		  \nonumber
\end{eqnarray}
where $C^1_{mn}:=\mathrm{Hom}_U(P_m,M)
\otimes_k \mathrm{Hom}_U(P_n,N)$, with the map
$$
		  H(\mathrm{Tot}(C^1_{\bullet\bullet})) \rightarrow
		  H(\mathrm{Tot}(C^2_{\bullet\bullet}))
		  \simeq
		  \mathrm{Ext}_U(A,M \otimes N)
$$
that is induced by
the morphism of double complexes
$$
		  C^1_{mn} \ni \varphi \otimes_k
		  \psi \mapsto
		  \{x \otimes y \mapsto
		  \varphi(x) \otimes \psi(y)\}
		  \in C^2_{mn}.
$$

For the sake of completeness let us
finally remark that as for $A=k$ one can in particular use
the bar construction
to obtain a canonical resolution:
\begin{lemma}
For any $\times_A$-bialgebra $U$,
the complex of left $U$-modules
$$
		  C^\mathrm{bar}_n:=
		  ({}_{\,{\scriptstyle\blacktriangleright}\,}
		  U_{\,{\scriptstyle\lhd}\,})^{\otimes_{A^\mathrm{op}} n+1},\quad
		  u(v_0 \otimes_{A^\mathrm{op}} \cdots \otimes_{A^\mathrm{op}}
		 v_n):=uv_0 \otimes_{A^\mathrm{op}} \cdots \otimes_{A^\mathrm{op}} v_n
$$
whose boundary map is given by
\begin{eqnarray}
b' : u_0 \otimes_{A^\mathrm{op}} \cdots \otimes_{A^\mathrm{op}} u_n
&\mapsto& \sum_{i=0}^{n-1} (-1)^i
 		  u_0 \otimes_{A^\mathrm{op}} \cdots \otimes_{A^\mathrm{op}}
		  u_iu_{i+1} \otimes_{A^\mathrm{op}} \cdots \otimes_{A^\mathrm{op}}
		  u_n \nonumber\\
&& + (-1)^n u_0 \otimes_{A^\mathrm{op}} \cdots
		  \otimes_{A^\mathrm{op}} \varepsilon(u_n)
		  {\,{\scriptstyle\blacktriangleright}\,} u_{n-1}
		  \nonumber
\end{eqnarray}
is a contractible resolution of $A \in U\mbox{-}\mathbf{Mod}$ with
augmentation
$$
		  \varepsilon : C^\mathrm{bar}_0=U \rightarrow
		  A=:C^\mathrm{bar}_{-1},
$$
and if $U_{\,{\scriptstyle\lhd}\,} \in A^\mathrm{op}\mbox{-}\mathbf{Mod}$ is
 projective, then $C^\mathrm{bar}_n \in U\mbox{-}\mathbf{Mod}$
is projective.
\end{lemma}
\begin{pf}
All claims are straightforward:
there is a contracting homotopy
$$
		  s : C^\mathrm{bar}_n \rightarrow C^\mathrm{bar}_{n+1},\quad
		  u_0 \otimes_{A^\mathrm{op}} \cdots \otimes_{A^\mathrm{op}} u_n
		  \mapsto
		  1 \otimes_{A^\mathrm{op}} u_0 \otimes_{A^\mathrm{op}} \cdots
		  \otimes_{A^\mathrm{op}} u_n,\quad n \ge 0,
$$
$$
		  s : A=C^\mathrm{bar}_{-1} \rightarrow U=C^\mathrm{bar}_0,\quad
		  a
		  \mapsto
		  \eta (a \otimes 1),
$$
and the
projectivity of $C^\mathrm{bar}_n$
follows as in the proof of
Theorem~\ref{uuu}.
\end{pf}
{}\\[-10mm]

\subsection{The functor
$\otimes : U\mbox{-}\mathbf{Mod} \times U^\mathrm{op}\mbox{-}\mathbf{Mod}
  \rightarrow U^\mathrm{op}\mbox{-}\mathbf{Mod}$}
Now we introduce the functor $\otimes$
mentioned in Theorem~\ref{main}.
\begin{lem}
\label{RechtsLinks}
Let $U$ be a $\times_A$-Hopf algebra and
$M \in U\mbox{-}\mathbf{Mod}$, $P \in U^\mathrm{op}\mbox{-}\mathbf{Mod}$ be left and right $U$-modules, respectively. Then the formula
\begin{equation}
\label{pfeffer}
(m \otimes_A p)u:= u_-m \otimes_A pu_+, \quad\quad u \in U, \ m
\in M, p \in P,
\end{equation}
defines a right $U$-module structure on
the tensor product
\begin{equation}
\label{TensLR}
M \otimes_A P := M \otimes_k P/{{\rm
 span}\{m {\,{\scriptstyle\lhd}\,} a \otimes_k p -
m \otimes_k a {\,{\scriptstyle\blacktriangleright}\,} p\,|\,
a \in A\}}.
\end{equation}
If $N$ is any other (left) $U$-module,
then the canonical isomorphism
\begin{equation}\label{luton}
		  (M \otimes N) \otimes_A P
			\simeq
			M \otimes_A (N \otimes_A P)
\end{equation}
of $A$-bimodules is also an isomorphism
in $U^\mathrm{op}\mbox{-}\mathbf{Mod}$. Finally, the tensor flip
$$
		  (M \otimes_A P) \otimes_U N
		  \rightarrow
		  P \otimes_U (N \otimes_A M),\quad
		  m \otimes_A p \otimes_U n
		  \mapsto
		  p \otimes_U n \otimes_A m
$$
is an isomorphism of $k$-modules.
\end{lem}
\begin{pf}
To show firstly that
(\ref{pfeffer}) is well-defined
over $A$, we compute
\begin{equation*}
\begin{split}
\big(m \otimes_A (a {\,{\scriptstyle\blacktriangleright}\,} p)\big)u &=
u_-m \otimes_A p\eta(1 \otimes a)u_+
= u_-m \otimes_A p(u_+ {\,{\scriptstyle\lhd}\,} a) \\
&= (a {\,{\scriptstyle\blacktriangleright}\,} u_-)m  \otimes_A pu_+
= u_- \big(\eta(1 \otimes a)m\big)
 \otimes_A pu_+  \\
&= \big((m {\,{\scriptstyle\lhd}\,} a) \otimes_A p\big)u,
\end{split}
\end{equation*}
where (\ref{Sch3}) and the action
 properties were used.
Together with (\ref{TensLR}) this also proves the
well-definedness with respect to the presentation
of $ u_+ \otimes_{A^\mathrm{op}} u_-$.
With the help of (\ref{Sch4}) one sees
immediately
that for $u, v \in U$ we have
$$
\big(m\otimes_A p\big)(uv) =  (uv)_-m \otimes_A p(uv)_+ =  v_-u_- m
\otimes_A pu_+v_+
= \big((m\otimes_A p)u\big)v,
$$
since $P$ and $M$ were right and left $U$-modules, respectively.
As a conclusion, $M \otimes_A P \in U^\mathrm{op}\mbox{-}\mathbf{Mod}$.
Equation (\ref{luton}) is a direct
consequence of the associativity of the
tensor product of $A$-bimodules
and of (\ref{Sch37}).

For the last part one has to check that the flip
is well-defined: we have
\begin{eqnarray}
 \eta (1 \otimes a)m \otimes_A p
		  \otimes_U n
&\mapsto& p \otimes_U
		  n \otimes_A \eta (1 \otimes a)m
= p \otimes_U
		  \eta (1 \otimes a)
		  (n \otimes_A m) \nonumber\\
&=& p \eta (1 \otimes a) \otimes_U
		  (n \otimes_A m), \nonumber
\end{eqnarray}
which is what $m \otimes_A p \eta (1 \otimes a)
 \otimes_U n$
gets mapped to. And secondly, we have
\begin{eqnarray}
 m \otimes_A p
		  \otimes_U u n
&\mapsto& p \otimes_U
		  u n \otimes_A m
= p \otimes_U (u_+)_{(1)} n \otimes_A
		  (u_+)_{(2)} u_- m \nonumber\\
&=& p \otimes_U u_+(n \otimes_A
		  u_-m)
= p u_+ \otimes_U n \otimes_A
		  u_-m,\nonumber
\end{eqnarray}
which is what
$u_- m \otimes_A p u_+ \otimes_U
	  n = (m \otimes_A p)u \otimes_U
 n$ gets mapped to.
\end{pf}
{}\\[-9mm]
\begin{dfn}
We denote the above
constructed $U^\mathrm{op}$-module
by $M \otimes P$.
\end{dfn}

Thus an unadorned $\otimes$ refers
from now on either to the monoidal product on
$U\mbox{-}\mathbf{Mod}$ or to the just defined action of
$U\mbox{-}\mathbf{Mod}$ on $U^\mathrm{op}\mbox{-}\mathbf{Mod}$. For
example, (\ref{luton}) would now simply
be written as
$ (M \otimes N) \otimes P \simeq
M \otimes (N \otimes P)$.
\begin{ex}
Let $(A,L)$ be a Lie-Rinehart algebra and $M$ be a left and $N$ a
right $U(A,L)$-module, respectively (or,
 in the terminology of \cite{Hue:PCAQ,
 Hue:DFLRAATMC}, left
and right $(A,L)$-modules). Using
 (\ref{LieRineIsHopfX}), one gets the
right $U(A,L)$-module
structure on $M \otimes_A N$
from formula (2.4) in \cite[p.~112]{Hue:DFLRAATMC}:
$$
(m\otimes_A n)X = m \otimes_A nX - Xm
 \otimes_A n, \qquad m \in M, \ n \in N,
 \ X \in L.
$$
\end{ex}

If we assume again
that $U$ is $A$-biprojective, then
the above results extend directly to the derived
category $\mathcal{D}^-(U^\mathrm{op})$:
we obtain a functor
$$
		  \otimes^\mathrm{L} = \otimes^\mathrm{L}_A :
		  \mathcal{D}^-(U) \times
		  \mathcal{D}^-(U^\mathrm{op}) \rightarrow
		  \mathcal{D}^-(U^\mathrm{op})
$$
and we have for all $M,N \in \mathcal{D}^-(U)$, $P \in
\mathcal{D}^-(U^\mathrm{op})$ canonical isomorphisms
\begin{equation}\label{weihnacht}
		  (M \otimes^\mathrm{L} N) \otimes^\mathrm{L} P \simeq
		  M \otimes^\mathrm{L} (N \otimes^\mathrm{L} P),\quad
		  (M \otimes^\mathrm{L} P) \otimes^\mathrm{L}_U N
		  \simeq
		  P \otimes^\mathrm{L}_U (N \otimes^\mathrm{L} M).
\end{equation}

\subsection{$\smallfrown$ and $\bullet$}\label{absch2}
These products are dual to $\smallsmile$
and $\circ$. The first one is
$$
		  \bullet : \mathrm{Ext}_U^m(L,M)
		  \times \mathrm{Tor}^U_n(N,L)
		  \rightarrow \mathrm{Tor}^U_{n-m}(N,M)
$$
which exists for
a ring $U$
and $L,M \in U\mbox{-}\mathbf{Mod}$, $N \in U^\mathrm{op}\mbox{-}\mathbf{Mod}$:
an element
$$
		  \varphi \in
		  \mathrm{Ext}_U^m(L,M) \simeq
		  \mathrm{Hom}_{\mathcal{D}^-(U)}(L,T^mM)
$$
defines a morphism in $\mathcal{D}^-(\mathbb{Z})$
$$
		  N \otimes^\mathrm{L}_U L \rightarrow N
		  \otimes^\mathrm{L}_U T^mM,\quad
 		  x \otimes_U y \mapsto x \otimes_U
		  \varphi (y),
$$
and $\varphi \bullet \cdot$ is the
induced map in (co)homology
\begin{eqnarray}
&&\mathrm{Tor}_n^U(N,L) \simeq
   	H^{-n}(N \otimes^\mathrm{L}_U L) \nonumber\\
&\xymatrix{\ar[r]^{H^{-n}(\mathrm{id}
 \otimes \varphi)}&}&
		  H^{-n}(N \otimes^\mathrm{L}_U T^mM) \simeq
		  H^{m-n}(N \otimes^\mathrm{L}_U M) \simeq
		  \mathrm{Tor}^U_{n-m}(N,M). \nonumber
\end{eqnarray}

For $M \in U\mbox{-}\mathbf{Mod},N \in U^\mathrm{op}\mbox{-}\mathbf{Mod}$ as before, the cap product
$$
		  \smallfrown \,\,:
		  \mathrm{Ext}^m_U(A,M) \times
		  \mathrm{Tor}^U_n(N,A)
		  \rightarrow
		  \mathrm{Tor}^U_{n-m}(M \otimes
		  N,A)
$$
involves the functor
$\otimes$ from the previous paragraph,
so for this we want $U$ to be an $A$-biprojective $\times_A$-Hopf
algebra again. Similarly as for
$\bullet$,
$$
		  \varphi \in
		  \mathrm{Ext}_U^m(A,M) \simeq
		  \mathrm{Hom}_{\mathcal{D}^-(U)}(A,T^mM)
$$
defines a morphism in $\mathcal{D}^-(k)$
\begin{eqnarray}
&& N \otimes^\mathrm{L}_U A \simeq
		  N \otimes^\mathrm{L}_U (A \otimes A)
		  \nonumber\\
&\xymatrix{\ar[r]^{\mathrm{id} \otimes
 \mathrm{id} \otimes \varphi} &} & N \otimes^\mathrm{L}_U (A \otimes^\mathrm{L} T^mM) \simeq
		  N \otimes^\mathrm{L}_U (T^mA \otimes^\mathrm{L} M) \simeq
		  (M \otimes^\mathrm{L} N) \otimes^\mathrm{L}_U T^mA
		  \nonumber\\
&\xymatrix{\ar[r]&}& (M \otimes N)
 \otimes^\mathrm{L}_U T^mA,\nonumber
\end{eqnarray}
where the last $\simeq$ in the second
line is induced
by the tensor flip as in the
derived version (\ref{weihnacht}) of
Lemma~\ref{RechtsLinks}, and the morphism from
the second to the third line is
similarly as in the definition of
$\smallsmile$ induced by the morphism
$M \otimes^\mathrm{L} N \rightarrow M \otimes N$
in $\mathcal{D}^-(U^\mathrm{op})$ that takes
zeroth cohomology. Passing now to
cohomology we get
$\varphi \smallfrown \cdot :
\mathrm{Tor}^U_n(N,A)
\rightarrow \mathrm{Tor}_{n-m} (M
\otimes N,A)$.

More explicitly, if $P \in \mathcal{D}^-(U)$ is a
projective resolution of $A$,
then $\smallfrown$ is
induced by the morphism
$$
		  B^1_{ij} \ni n \otimes_U
		  (x \otimes_A y) \mapsto
		  \{\varphi \mapsto
		  (\varphi (y) \otimes_A n) \otimes_U
		  x\} \in B^2_{ij}
$$
from the double complex
$$
		  B^1_{ij}:=N \otimes_U
		  (P_i \otimes_A P_j)
$$
whose total homology is
$\mathrm{Tor}^U(N,A)$ to the
double complex
$$
		  B^2_{ij}:=\mathrm{Hom}_k(
		  \mathrm{Hom}_U(P_j,M),
		  (M \otimes N) \otimes_U P_i)
$$
whose homology has a natural map to
$\mathrm{Hom}_k(\mathrm{Ext}_U(A,M),
\mathrm{Tor}^U(M \otimes N,A))$.

In direct analogy with
Theorem~\ref{suarez} we get:
\begin{thm}\label{suarez2}
If $U$ is an $A$-biprojective
$\times_A$-Hopf algebra,
then we have
$$
		  \varphi \bullet (x \otimes_U y) = \varphi
		  \smallfrown (x \otimes_U y),\quad
		  \varphi \in
 \mathrm{Ext}_U^m(A,A),
x \otimes_U y \in N \otimes^\mathrm{L}_U A
$$
as elements of
$
		  N \otimes^\mathrm{L}_U A \simeq
		  (A \otimes N) \otimes^\mathrm{L}_U A.
$
\end{thm}

\section{Duality and the proof of Theorem~\ref{main}}
\subsection{The underived case}
In the special case
that $A$ is finitely
generated projective itself,
Theorem~\ref{main} reduces to
standard linear algebra. We go through
this case first since it is
both instructive and used in the
proof of the general case.
For the reader's convenience
we include full proofs.

\begin{lemma}\label{cardiff}
Let $U$ be a ring, $A \in U\mbox{-}\mathbf{Mod}$ be finitely
generated projective, and
$A^*$ be $\mathrm{Hom}_U(A,U)$
with its canonical
 $U^\mathrm{op}$-module structure.
\begin{enumerate}
\item[1.] $A^*$ is
finitely generated projective, and
if
$e_1,\ldots,e_n $ are generators of $A$, then there
exist generators
$e^1,\ldots,e^n \in A^*$
with
$$
		  \sum_i e^i(a)e_i=a,\quad
		  \sum_i e^i \alpha (e_i)=\alpha
$$
for all $a \in A$ and $\alpha \in A^*$.
The element
$$
		  \omega:=\sum_i e^i \otimes e_i \in A^*
		  \otimes_U A
$$
is independent of the choice of
the generators $e_i,e^j$.
\item[2.] For all
$U^\mathrm{op}$-modules $M$,
the assignment
$$
		  \delta (m \otimes a)(\alpha):=
		  m \alpha (a),\quad
		  m \in M,a \in A,\alpha \in
		  A^*
$$
extends uniquely to an
isomorphism of abelian groups
$$
		  \delta : M
		  \otimes_U A \rightarrow
		  \mathrm{Hom}_{U^\mathrm{op}}(A^*,M).
$$
\item[3.] One has $(A^*)^* \simeq A$
and $A^* \otimes_U M \simeq
\mathrm{Hom}_U(A,M)$
for $M \in U\mbox{-}\mathbf{Mod}$.
\end{enumerate}
\end{lemma}
\begin{pf}
Since $A$ is projective, there is a
splitting $\iota : A \rightarrow U^n$
of
$$ \pi : U^n \rightarrow A,\quad
(u_1,\ldots,u_n) \mapsto \sum_i
 u_ie_i.
$$ Hence
$U^n  \simeq A \oplus A_\perp$ for some
$A_\perp \in U\mbox{-}\mathbf{Mod}$. Dually this
gives $A^* \oplus (A_\perp)^* =
(U^n)^* \simeq U^n$, whence
$A^*$ is finitely generated projective.
The $e^i$ can be defined as the
composition of $\iota$ with the
projection of $U^n$ on the $i$-th summand.
This proves the first parts of 1. For
2.~just note that
$$
		  \mathrm{Hom}_{U^\mathrm{op}}
		  (A^*,M) \ni
		  \varphi \mapsto
		  \sum_i \varphi (e^i) \otimes e_i
		  \in M \otimes_U A
$$
inverts $\delta$. Since
$\omega =
\delta^{-1}(\mathrm{id}_{A^*})$,
it does indeed not
depend on the choice of generators.
3.~now follows from 1.~and 2.
\end{pf}

As in the introduction, let us
abbreviate in the situation
of this theorem
$$
		  H^0(M):=\mathrm{Hom}_U(A,M),\quad
		  H_0(N):=N \otimes_U
		  A
$$
for $M \in U\mbox{-}\mathbf{Mod}$,
$N \in U^\mathrm{op}\mbox{-}\mathbf{Mod}$, and call
$\omega \in H_0(A^*)$
the fundamental class
of $(U,A)$. Then 3.~says for $M=A$
that we have an isomorphism
\begin{equation}\label{sonn}
		  \cdot \bullet \omega :
		  H^0(A) \rightarrow
		  H_0(A^*),\quad
		  \varphi \mapsto \sum_i
		  e^i \otimes \varphi (e_i).
\end{equation}
Using Lemma~\ref{RechtsLinks} we can upgrade this
to the underived case of Theorem~\ref{main}:
\begin{lemma}\label{ravel}
Let $U$ be a $\times_A$-Hopf algebra
and assume $A$ is finitely generated
projective as a $U$-module. Then the
cap product with the fundamental class
$\omega \in H_0(A^*)=A^* \otimes_U A$
defines for all $M \in U\mbox{-}\mathbf{Mod}$ an isomorphism
$$
		  \cdot \smallfrown \omega :
		  H^0(M) \rightarrow H_0(M
 \otimes
 A^*).
$$
\end{lemma}
\begin{pf}
We have
$\varphi \smallfrown \omega
=\sum_i  (\varphi (1) \otimes_A e^i)
		  \otimes_U e_i$,
and Lemma~\ref{RechtsLinks} identifies
$$
		  H_0(M \otimes A^*)=
		  (M \otimes A^*) \otimes_U A
 \simeq
		  A^*\otimes_U (A \otimes M)
 \simeq A^* \otimes_U M.
$$
In this chain of identifications, $\varphi \smallfrown
 \omega$ is mapped to
\begin{eqnarray}
 \varphi \smallfrown \omega
&\mapsto&\sum_i  e^i \otimes_U
		  (e_i \otimes_A \varphi (1))
\mapsto\sum_i  e^i \otimes_U
		  (e_i\varphi (1))
		  = \sum_i e^i \otimes_U
		  \varphi (e_i) \nonumber
\end{eqnarray}
which is identified with
$\varphi $ under the isomorphism
$\mathrm{Hom}_U(A,M) \simeq
A^* \otimes_U M$ given by
$
		  \varphi \mapsto \sum_i e^i
 \otimes_U \varphi (e_i)
$
as in (\ref{sonn}). The claim follows.
\end{pf}
{}\\[-9mm]

\subsection{The derived case}
It remains to throw in some
homological algebra to obtain
Theorem~\ref{main} in general. To shorten
the presentation, we define:

\begin{dfn}
A module $A$ over a ring $U$
is perfect if it admits a finite
resolution by finitely generated
projectives. We call such a module a
duality module if there exists $d \ge 0$
such that $ \mathrm{Ext}^n_U(A,U)=0$
for all $n \neq d$. We abbreviate in
this case $A^*:=\mathrm{Ext}^d_U(A,U)$
and call $d$ the dimension of $A$.
\end{dfn}

The main remaining step is to prove a
derived version of
Lemma~\ref{cardiff}. One could use a
result of Neeman by which $A \in U\mbox{-}\mathbf{Mod}$
is perfect if and only if
$\mathrm{Hom}_U(A,\cdot)$
commutes with direct sums
\cite{Kel:ODGC,Nee:TC}, or
the Ischebeck spectral
sequence
which degenerates at $E^2$ if $A$ is a
duality module
\cite{Isch:EDZDFEUT,Kra:PDIHC,Skl:PDARTFEAT}.
However, we
include a more elementary and
self-contained proof.

\begin{thm}\label{swansea}
Let $A \in U\mbox{-}\mathbf{Mod}$
be a duality module of dimension $d$.
\begin{enumerate}
\item[1.] The projective dimension
of $A \in U\mbox{-}\mathbf{Mod}$ is $d$.
\item[2.] $A^*$ is a duality module
of the same dimension $d$.
\item[3.] If $P_\bullet \rightarrow A$ is a finitely
generated projective resolution of
length $d$, then
$P^*_{d-\bullet}=
\mathrm{Hom}_U(P_{d-\bullet},U)$
is a finitely generated projective
resolution of $A^*$ and
the canonical isomorphism
$$
		 \delta :  M \otimes_U P_i \rightarrow
		\mathrm{Hom}_U(P^*_i,M),\quad
		m \otimes_U p \mapsto \{\alpha
		\mapsto m \alpha (p)\}
$$
induces for all $U^\mathrm{op}$-modules $M$
a canonical isomorphism
$$
		  \mathrm{Tor}^U_{i}(M,A)
		\rightarrow
		\mathrm{Ext}^{d-i}_{U^\mathrm{op}}(A^*,M).
$$
\item[4.] There is a canonical isomorphism
$(A^*)^* \simeq A$.
\end{enumerate}
\end{thm}
\begin{pf}
Let $P_\bullet \rightarrow A$ be a
finitely generated projective
resolution of finite length
$m \ge 0$ (which exists since
$A$ is perfect).
Then the (co)homology of
$$
		  0 \rightarrow P_0^* \rightarrow \ldots
		  \rightarrow P_m^* \rightarrow 0,\quad
		  P_n^*=\mathrm{Hom}_U(P_n,U)
$$
is
$ \mathrm{Ext}_U^\bullet(A,U)$, so by
assumption we have $m \ge d$
and the above complex is
exact except at $P^*_d$ where the homology is
$A^*=\mathrm{Ext}_U^d(A,U)$. Furthermore,
all the $P_n^*$ are finitely generated
projective since the $P_n$ are so
(Lemma~\ref{cardiff}).

Let $\pi_i$ be the map
$P^*_i \rightarrow P^*_{i+1}$
and put $K:=\mathrm{ker}\, \pi_{d+1}$.
By construction,
\begin{equation}\label{auflnull}
0 \rightarrow K \rightarrow
P^*_{d+1} \rightarrow \ldots
 \rightarrow P^*_m \rightarrow 0
\end{equation}
is exact. If one compares this
exact sequence with the sequence
$$
		  \ldots \rightarrow 0 \rightarrow
 0 \rightarrow P^*_m \rightarrow
 P^*_m \rightarrow 0
$$
using Schanuel's lemma
(see \cite[7.1.2]{McCRob:NcNR}), one obtains that
$K$ is projective.

The exactness of $P_\bullet^*$ at
$P^*_{d+1}$ gives
$K=\mathrm{im}\, \pi_d$, and
by the projectivity of $K$, the map
$\pi_d : P^*_d \rightarrow K \subset
P^*_{d+1}$ splits so that
$P^*_d \simeq K \oplus K_\perp$,
$K_\perp:=\mathrm{ker}\, \pi_d$.
In particular, both
$K$ and $K_\perp$ are
finitely generated.

It follows from all this that the complex
\begin{equation}\label{auflastern}
		  0 \rightarrow P^*_0 \rightarrow
 \ldots
 \rightarrow P^*_{d-1} \rightarrow
 K_\perp
\rightarrow 0
\end{equation}
is a finitely generated projective
 resolution of $A^*$: since
$\mathrm{im}\, \pi_{d-1} \subset
P^*_d$ is contained in
$\mathrm{ker}\, \pi_d = K_\perp$,
the above complex is still
exact at $P^*_{d-1}$, and the homology
 at $K_\perp$ is the homology of
 $P^*_\bullet$ at $P^*_d$, that is,
 $A^*$.

Since (\ref{auflnull}) is a finitely
generated projective resolution of $0$
and
$P^*_{d-\bullet}$ is as a complex a
direct sum of (\ref{auflastern}) and (a
 shift of) (\ref{auflnull}) we also know
that
 $\mathrm{Ext}^\bullet_{U^\mathrm{op}}
(A^*,M)$ is
for any $M \in U^\mathrm{op}\mbox{-}\mathbf{Mod}$ the (co)homology of
$\mathrm{Hom}_U(P^*_{d-\bullet},M)$.
By Lemma~\ref{cardiff},
this is isomorphic as a chain complex to
$M \otimes_U P_{d-\bullet}$
via the isomorphism given in 3.,
and the homology of this
complex is
$\mathrm{Tor}^U_{d-\bullet}(M,A)$. This
proves 3. The special case
$M=U$ implies the remaining claims.
\end{pf}

Assume finally
that in the situation of the above
theorem, $U$ is an $A$-biprojective $\times_A$-Hopf
algebra. Since
$P$ is a projective resolution, we have
$M \otimes_U P \simeq M \otimes^\mathrm{L}_U P$
and $\mathrm{Hom}_{U}(P^*,M) \simeq
\mathrm{RHom}_U(P^*,M)$, and $\delta$
gives an isomorphism between them.
The fundamental class is defined
to be
$$
		  \omega := \delta^{-1}
		  (\mathrm{id}_{A^*}) \in A^*
		  \otimes^\mathrm{L}_U A \simeq
		  P^* \otimes_U A \simeq
		  A^* \otimes_U P,
$$
and Theorem~\ref{swansea} gives immediately:
\begin{cor}
If $e_1,\ldots,e_n$ and
$\tilde e^1,\ldots\tilde e^n$
are generators of $A$ and of
$A^*$, respectively, then
there are
$e^1,\ldots,e^n \in P^*_0$ and
$\tilde e_1,\ldots,\tilde e_n \in P_d$ such that
$$
		  \omega = \sum_i e^i \otimes_U e_i
		  = \sum_i \tilde e^i \otimes_U
		  \tilde e_i,
$$
and $\delta$ is given by the Yoneda
 product $\cdot \bullet \omega$.
\end{cor}

Theorem~\ref{main} follows now
as in the underived case
(Lemma~\ref{ravel}) working with
$\mathrm{RHom}_U(A,M)$ and
$(M \otimes^\mathrm{L} A^*) \otimes^\mathrm{L}_U A$ instead of
$H^0(M)=\mathrm{Hom}_U(A,M)$ and
$H_0(M \otimes A^*)=(M \otimes A^*)
\otimes_U A$: using
Theorem~\ref{suarez2} and (\ref{weihnacht})
one gets
\begin{eqnarray}
 (M \otimes^\mathrm{L} A^*) \otimes^\mathrm{L}_U A
&\simeq&
		  A^* \otimes^\mathrm{L}_U (A \otimes^\mathrm{L} M)
		  \simeq
		  A^* \otimes^\mathrm{L}_U M \nonumber\\
&\simeq&  P^* \otimes^\mathrm{L}_U M \simeq
		  \mathrm{RHom}_U(P,M) \nonumber\\
&\simeq&
		  \mathrm{RHom}_U(A,M),\nonumber
\end{eqnarray}
where we hide the
reindexing of the complexes for the sake
of better readability (so $P^*$ stands
for
$P^*_{d-\bullet}$, and
$\mathrm{RHom}_U(P,M)$ and
$\mathrm{RHom}_U(A,M)$ are reindexed in
the same way).
This leads to a convergent spectral
sequence
$$
\mathrm{Tor}^U_p(\mathrm{Tor}^A_q(M,A^*),A)
\Rightarrow
\mathrm{Ext}_U^{d-p-q}(A,M),
$$
and under the last assumption
of Theorem~\ref{main}
($\mathrm{Tor}^A_q(M,A^*)=0$ for $q>0$)
this spectral sequence degenerates to the claimed
isomorphism.

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}

redpajama_set_name: RedPajamaArXiv


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Rated 5 out of 5 by Mr Grec from Great phone Wish I would have upgraded sooner from the LGL41C. I like the features that I didn't have on my old phone. The larger screen and call waiting a big plus. Don't like the camera on this phone it doesn't take good photos like my other cellphone. Disappointed that it doesn't have the settings like my old cellphone. Re: ISO, White balance and Brightness ect. I also don't like the fact that you can't block in coming calls before hand. On the LGL41C you could set the cellphone to receive only in coming calls that are on your contact list. Maybe there is a way to set this up on this cellphone I just can't find it. My wife also ordered this cellphone and loves it. Great value if your looking for a new cellphone.
Rated 2 out of 5 by Rena Rena from Not the easiest to use. I buy tracfones every year and this one is the hardest one to use that I have had.I had to send it back because it locked me out .I spent almost an hour with a support person trying to help get it unlocked to no avail .I am trying one more if I can get one but would not recommend to any older and not too tech savoy folks.
Rated 5 out of 5 by Piesncakes from Fits my needs If you just need a cell phone for emergencies, this fits the bill. We rarely use up 300 minutes in a year, so this is perfect.


What are Best Sites to learn Oracle SQL/PlSQL?
Playlist - https://www.youtube.com/playlist?list=PLKCk3OyNwIzvkz21ulI8ZzyG-i1JJVV14 A panel discussion about SQL at SQL Summit 2016 at Oracle HQ. https://nocoug.wildapricot.org/event-2126471 ================================= For more information, see http://www.oracle.com/goto/oll Copyright © 2016 Oracle and/or its affiliates. Oracle is a registered trademark of Oracle and/or its affiliates. All rights reserved. Other names may be registered trademarks of their respective owners. Oracle disclaims any warranties or representations as to the accuracy or completeness of this recording, demonstration, and/or written materials (the "Materials"). The Materials are provided "as is" without any warranty of any kind, either express or implied, including without limitation warranties or merchantability, fitness for a particular purpose, and non-infringement.
Steven Feuerstein, Oracle Developer Advocate for PL/SQL, and Rob Lockard, database security expert and winner of a 2015 Oracle Database Developer Choice award, discuss the top security issues facing Oracle developers today and how they can harden their applications against attack. ================================= For more information, see http://www.oracle.com/goto/oll Copyright © 2016 Oracle and/or its affiliates. Oracle is a registered trademark of Oracle and/or its affiliates. All rights reserved. Other names may be registered trademarks of their respective owners. Oracle disclaims any warranties or representations as to the accuracy or completeness of this recording, demonstration, and/or written materials (the "Materials"). The Materials are provided "as is" without any warranty of any kind, either express or implied, including without limitation warranties or merchantability, fitness for a particular purpose, and non-infringement.
Oracle Solaris is a great choice for running your DevOps stack. Learn how to implement it right now -- and about the features that makes Oracle Solaris brings an easy decision.
Hi guys today we learn how to print number in fibonacci series(0,1,1,2,3,5,8,13,21,34 etc ) #FIBONACCISERIES in PLSQL language Oracle database Unbeatable,Unbreakable Platform..
http://www.perftuning.com/ - Some people choose to hire a professional for the process of tuning SQL Oracle for them. Performance tuning in SQL is essential to protect one's database system from future problems and errors. One can obtain a server performance tuning service from an IT company online today.
In this video i'm going to demonstrate you how to work on SQL Developer on windows 7 using oracle 11g R2 step by step.


text:
package de.onyxbits.raccoon.gui;

import java.awt.Toolkit;
import java.awt.event.ActionEvent;
import java.awt.event.KeyEvent;

import javax.swing.AbstractAction;
import javax.swing.Action;
import javax.swing.KeyStroke;

import de.onyxbits.weave.Globals;
import de.onyxbits.weave.LifecycleManager;

/**
 * An action for shutting the application down.
 *
 * @author patrick
 *
 */
public class QuitAction extends AbstractAction {

	private Globals globals;

	/**
	 *
	 */
	private static final long serialVersionUID = 1L;

	/**
	 * Create a new action and bind an accelerator key to it.
	 *
	 * @param globals
	 *          registry
	 */
	public QuitAction(Globals globals) {
		this.globals = globals;
		putValue(Action.ACCELERATOR_KEY, KeyStroke.getKeyStroke(KeyEvent.VK_Q,
				Toolkit.getDefaultToolkit().getMenuShortcutKeyMask()));
	}

	@Override
	public void actionPerformed(ActionEvent e) {
		globals.get(LifecycleManager.class).shutdown();
	}

}

redpajama_set_name: RedPajamaGithub


text:
Assam Co-op Societies Act 2007
About Jute
The Assam Co-operative Jute Mills Ltd. is situated on the south bank of the river Brahmaputra at Silghat in the Nagaon district of Assam, located at about a distance of 8 km from Jakhalabandha. This is the only Jute Mill in the Country in Co-operative Sector. It was registered in the year 1959 under the Assam Co-operative Societies Act and after much struggle it was commissioned in the year 19'70 with the initial investment of Rs. 150 lakh. But, the Mill after running for about 14 years in rough weather had to close during March 1984 for acute financial crisis and other infrastructural problems and remain closed for about 2 years.
Again, the Mill was reopened on 1st January 1986, under the terms of the Historic Assam Accord with the financial assistance from the Central Govt. as well as State Govt. But, in spite of sufficient Govt. assistance, financial depression developed again due to imbalance in the income and the expenditure. However, after continuous struggle and efforts, the Mill has started improving its economy from the year 1992-93 and since then making continuous progressive net profits. Now, the Mill is completely free from all loans and liabilities and now it is in a stable economic position.
The Mill is implementing the scheme for renovation/modernization of its plant &machinery from its own resources to increase capacity utilization and quality improvement of its product.
Turnover for previous years
18,67,24,777/- (Including Freight Amount)
42,44,15,000/- (Excluding Freight Amount)
The Assam Co-operative Jute Mills Ltd.
P.O. Silghat
Dist : Nagaon
Email:silghatjutemill@gmail.com
Telephone: (03672) 272006/ 272180
FAX: 03672)276775
© 2014 Assam Co-Operative Jute Mills Ltd. Powered by redpanchi.

redpajama_set_name: RedPajamaCommonCrawl


Like everyone else in the beginning, I have struggled with perfecting the eye brow fashion arch. I have a natural arch so I have all but given up. However, this simple illustration has saved many. Enjoy!
I have so much trouble getting my eyebrows to look good. Thanks for the post, this will help.


Texas Tech has landed transferring St. John's forward Tariq Owens, his father, Renard, told ZAGSBLOG.
Owens, who also considered Maryland, is eligible immediately.
St. John's ended the season with a 16-17 (4-14 Big East) record and finished ninth in the Big East.
In his junior season, Owens averaged 8.4 points, 5.9 rebounds, 2.8 blocks and 30.3 minutes per game.
Sedee Keita, a forward from South Carolina, and Mikey Dixon, a guard from Quinnipiac, will join the Red Storm next season after seating out this season due to NCAA transfer regulations. Both have three years of eligibility remaining.


text:
Stephen V. King , Autism In Motion, and TPC to be recognized at IVCA's 2022 Awards Dinner, sponsored by Kirkland & Ellis LLP, on December 5th
By IVCA Admin | October 26, 2022
2022 IVCA Stanley C. Golder Award will be awarded to Stephen V. King, Co-Founder of Prairie Capital, who epitomizes what the IVCA Golder Award is all about: an exceptional leader who upholds the strongest ethical standards, an astute investor with a strong track record of success, and a civic-minded philanthropist who uses his platform to make a difference.
Steve King and Bryan Daniels founded Prairie with Prairie Capital Fund I in 1997. The firm currently is investing out of Fund VII with a 2021 Vintage year.
Steve's many civic/philanthropic involvements include:
Dean's Business Council for the Gies College of Business at University of Illinois
Board Chair of Arc of Hope (an organization that established and oversees a primary school in Uganda)
Board of Directors and Executive Committee of the Big Shoulders Fund
Board of Catholic Charities
Board of the Academy at St. Joan of Arc
Past Board Chair of Boys Hope/Girls Hope
Past Board of Trustees of Loyola Academy
Outside the Prairie Capital portfolio, Steve has had several other key professional affiliations, including serving as past Chairman of NASBIC (now the Small Business Investor Alliance), being a Leadership Greater Chicago Fellow, and sitting on the Board of Directors of First Busey Corporation (NASDAQ: BUSE).
2022 IVCA Venture-Backed Portfolio Company of the Year will be awarded to Autism In Motion Clinics (AIM Clinics). AIM's mission is to bring world class autism therapy to geographic areas where unmet need is highest. AIM provides one-on-one, gold standard ABA therapy to children at an average of 20+ hours per client per week and is fully accredited by the Behavioral Health Center of Excellence (BHCOE). AIM was launched in Chicago where its headquarters remains. The first clinics were in Arkansas, Oklahoma, Alabama and Tennessee. AIM is a past winner of the Social New Venture Challenge at UChicago Booth. AIM has helped thousands of families with life changing therapy since it was founded in 2018.
The Company was founded by George Boghos who immigrated to the US from Syria while in middle school without knowing any English. George's mother is a pediatrician and George has always had an interest in healthcare. George saw the prevalence of autism continue to grow and the lack of access to care for children with autism, he launched AIM Clinics in his first year of business school at Booth.
The company received Seed stage funding and two follow on rounds of funding from Breakout Capital. Breakout exited the company in 2021 at a 20X multiple of initial capital invested.
2022 IVCA PE-Backed Portfolio Company of the Year will be awarded to TPC. TPC is a provider of industrial skills and workplace safety training, performance management, and compliance software solutions for environmental, health, & safety (EHS). The business model combines a proprietary Learning Management System with an industry-leading content library approved by governmental agencies for required certification and continuing education.
Frontenac acquired Buffalo Grove-based TPC in March 2018. Prior to Frontenac's acquisition, the company's core competency centered around live, instructor-led industrial skills training, with a small but growing base of digital training offerings. TPC sought a partner that could help transform the business to a predominantly digital platform with streamlined delivery of an expanded set of service offerings and increased focus on technical and safety training content.
Frontenac worked with the leadership team and a value-added Board of Directors to expand the Company's customer base (Fortune 500 customers), channels (B2C compliance), technology products (simulations and workforce management software), and end markets (food), while identifying and supporting the acquisition and integration of five M&A targets. The resulting business mix shifted from less than 45% digital delivery to more than 85% digital delivery.
Under Frontenac's ownership (2018-2021), TPC's enterprise value and base of employees increased markedly.
Date: Monday, December 5, 2022Location: The Four Seasons Hotel120 E Delaware Street, ChicagoTime:
5:30pm – Cocktail Reception sponsored by White & Case
6:30pm – Dinner/Awards Presentation
8:15pm – Dessert Reception
Posted in Newsletter Articles


text:
Competition to perform at Louth festival launches
Drogheda band Modern Love performing on the Firestone stage last year
Tia Clarke
tia.clarke@dundalkdemocrat.ie
Firestone Ireland has launched the voting for its 2019 "Road to the Main Stage" band competition in conjunction with Vantastival Music and Campervan Festival.
Voting began on April 1 for the 12 bands selected by the festival organisers. The online vote will reduce the 12 bands to 5 and they will play in a live final in Odd Mollies music venue in Drogheda on May 2, 2019. Voting is from April 1 to April 21, 2019 and is at www.roadtothemainstage. firestonetyre.ie/
The competition offers the competing bands the chance to win a coveted prize of music equipment and recording time with a combined value of €4,000, and the opportunity to feature as the headline act on the Firestone Music Station stage in 2019 as well as a Main Stage act at Vantastival 2020.
Colm Conyngham, Marketing Manager for Firestone Ireland said: "We are very proud to support new and emerging musical talent in Ireland and provide them with this opportunity to show off their skills, in conjunction with Vantastival.
"We hope that by creating this platform, we will be able to shine a spotlight on the great musical talent that Ireland has to offer and we would encourage Ireland's new music fans to support them with a vote."
Meanwhile, Vantastival director Louise Tangney said: "We are delighted to be involved with Firestone in their support for emerging music talent in Ireland and we look forward to an entertaining final in May and a great Firestone stage at Vantastival."
The Road to the Main Stage competition represents an extension of Firestone's authentic association with music that dates back to 1928 when Founder Harvey Firestone brought together musicians and music-lovers on the Voice of Firestone radio show in the USA, which gave known and unknown musicians a platform to perform on, and an audience to perform for. Supporting emerging "grass roots" talent is deeply rooted in Firestone DNA. "At Firestone we support these rising talents on their road to the main stage just like our tyres support drivers on the road," added Colm Conyngham, Firestone Ireland.
This year's competing bands are:
Stolen City
Don't Fear the Natives
Molto Vivo
The Burma
Kiosma
Grainne Cotter
The Last Vinci
The Crayon Set
Sick Love
Vantastival is a family-friendly festival which takes place over the June Bank Holiday Weekend (May 31st to June 2nd) in Beaulieu House, Drogheda will see a host of live musical acts perform, while a focus on the campervan lifestyle brings an extra dimension to the only festival of its kind in Ireland. See www.vantastival.com


text:
Q: Python loop: break and setting loop indice to max value yield different results I'm trying to solve a problem on codeforces, but ran into an issue:
The following codes don't yield the same results (value of nb). Only the 4th line before the end differ: replaced break by j=i-1.
n = int(input())
a = list(map(int, input().split()))
nb = 0
s1 = 0
k = 0
for i in range(n):
    s1 += a[i]
    s3 = s1
    for j in range(k, i):
        if s3 == 0:
            k = i
            nb += 1
            s1 = a[i]
            break
        s3 -= a[j]

print(nb)

second code:
n = int(input())
a = list(map(int, input().split()))
nb = 0
s1 = 0
k = 0
for i in range(n):
    s1 += a[i]
    s3 = s1
    for j in range(k, i):
        if s3 == 0:
            k = i
            nb += 1
            s1 = a[i]
            j = i - 1
        s3 -= a[j]

print(nb)

Moreover, both yield the same values on the 16 first tests. The second fails at the 17th.

redpajama_set_name: RedPajamaStackExchange


text:
Q: ajax sucess: i can not check if the returned information is true or false I have a problem when I try to check if the returned information is true or false. In console.log returns true but always enters else ...
$.ajax({
                    url: "ajax.php",
                    type: 'POST',
                    data: 'act=login&email=' + email + '&password=' + password + '&remember=' + remember,
                    success: function(data)
                    {
                        console.log(data);
                        if (data.success === true)
                        {
                            $("#login-form")[0].reset();
                            window.location.replace('dashboard.php');
                        }
                        else
                        {
                            $(".error-message span").html('Please, insert valid data.');
                            $('.error-message').fadeIn('slow', function () {
                                $('.error-message').delay(5000).fadeOut('slow');
                            });
                        }
                    }
                });

Thanks everyone.

A: Console.log prints data.
Your IF statement checks `data.success'. Those are 2 different elements.
In what format are you sending the data back from ajax.php?
You can't assume that data is an array or JSON object, you have to parse it first.
json = JSON.parse(data);

if (json.success === true) {} //or if (json === true) depends on the response from ajax.php

Per your comment, if you returned data is only true or false, simply use
if (data == true) {//I use double and not triple equality because I do not know if you are returning a string or a boolean.
}

Your have 2 problems:

*

*You are trying to access data.success instead of just data.

*You are using a strict comparison (===) and I am really not sure if your returned data is boolean and not string. @Pekka's answer mentions it as well (but it's not going to work since data.success does not exist).

Hope this helps!

A: The return value is likely to be a string true , not a JavaScript boolean true.
Because you are using the strict === comparison, which requires the same variable type, it will always fail.
Compare against the string:
if (data.success == "true")

A: If your data looks like {"success": true} then it could be a problem with data parsing where the ajax handler is not parsing the response as a json data.. instead is just passing a string.
So try setting the dataType property to json to tell ajax that you are expecting a json response.
$.ajax({
    url: "ajax.php",
    type: 'POST',
    data: 'act=login&email=' + email + '&password=' + password + '&remember=' + remember,
    dataType: 'json',
    success: function (data) {
        console.log(data);
        if (data.success === true) {
            $("#login-form")[0].reset();
            window.location.replace('dashboard.php');
        } else {
            $(".error-message span").html('Please, insert valid data.');
            $('.error-message').fadeIn('slow', function () {
                $('.error-message').delay(5000).fadeOut('slow');
            });
        }
    }
});

A: Have you tried ?

if (data.success == 'true'){......}

or if it comes in "Json" format you need to convert to object like this in "success" event. (included json.js)

var JSONfoo = JSON.stringify(data);
obj = eval('(' + JSONfoo + ')');
if(obj != null){
alert(obj.success );

}

A: yesterday in the form that was build the solution json = JSON.parse (data); worked, but now another form gives me an error (Uncaught SyntaxError: Unexpected token <) on line json = JSON.parse (data);
I do not understand the reason for working on a form and not another!
In ajax.php i return an echo "true";
thank you all


text:
1.7*1.7 ft Handmade Silk Rugs
2*3 ft Handmade Silk Rugs
2.5*6 ft Handmade Silk Rugs
2.5*12 ft Handmade Silk Rugs
3*4.5 ft Handmade Silk Rugs
3*5 ft Handmade Silk Rug
4x4 Ft Handmade Silk Rug
7*7 Ft Handmade Silk Carpet
8*10 ft Handmade Silk Rug
9x9 Ft Handmade Silk Carpet
10*10 ft Handmade Silk Rug
12x12 Ft Hand Made Square Silk Rug
Silk Tapestry
Custom-made Carpets
hotselling carpets
new arrival carpets
moments of our team activities
Carpet Introduction
The carpet is a famous traditional handicraft in China. The Chinese carpet has a history of more than 2,000 years. It is famous for hand-made carpets. It has written records dating back more than 3,000 years. The poet Bai Juyi of the Tang Dynasty wrote in the poem "Red Line Blanket": The famous name of "Takes clothes for lichens" has physical examinations and has been used for more than 2,000 years. According to historical records, there are more and more varieties of carpets in the Tang and Song dynasties and the Ming and Qing dynasties. The carpets made are often made from cotton, wool, linen and paper ropes. Weaving carpets produced in China use veil strands with extremely high strength as warp and ground weft yarns, and weave the colored coarse hair weft yarns on the warp yarns according to the pattern, and weaving them through shearing, brushing and other processes. The front is covered with plush, solid texture, and elasticity. In particular, the rugs produced in the Hetian area of Xinjiang are more valuable and have the reputation of "oriental carpets". The carpets produced there are not only of good quality but also large in output. A 1250 kg carpet woven by the Xinjiang Hall of the Great Hall of the People was praised as "the king of carpets". In 1966, a cotton blanket was unearthed in Lanxi, Zhejiang Province. It was 2.51 meters long and 1.18 meters wide. It was a plain weave.
Rugs (5 photos)
Cashmere, smooth and thick. After identification, the material used was straw fiber. This cotton carpet was woven into the Southern Song Dynasty. This is the earliest known cotton blanket. It can be seen that the carpeting process has reached a very high level at that time. At present, Xinjiang carpets have gradually promoted the new process of carpet ironing, which has improved the quality of carpets. The 540 and 720 or more carpets they developed were exquisitely crafted with high artistic value. It began in the northwest highland pastoral area. In order to meet the needs of nomadic life, the local ethnic minority made use of the rich local wool crepe to weave colorful mats, tapestries and carpets. Thanks to the co-creation of the peoples of the Uyghur, Mongolian and Tibetan communities, and through the Silk Road, the Middle East has gradually formed an outstanding ancient Chinese carpet art.
Previous: Hunting Silk Rugs Are Very Hot These Days
Next: Carpet
Sunshine Road, Industry Cluster District, Nanzhao County, Nanyang City, Henan Province, China
export@yuxiangcarpets.com
http://m.yuxiangrug.com/
Copyright © Nanyang Yuxiang Arts & Crafts Co.,Ltd All Rights Reserved.


text:
Q: Best way to debug and edit third party library in java? In javascript, python, golang or rust usually you have the third party source code available in your development environment such that you can edit it and debug it.
I am a beginner in java.
What is the best recommended work flow to debug third party library?

A: If you're using any kind of package management, instruct it to download the source jar along with the binary jars. Your IDE should then be able to use the downloaded source to debug the library.
Using this method, you aren't really able to modify the library, but you shouldn't be doing that anyway.

A: If you are using Eclipse, you should try the Enhanced Class Decompiler, which can be installed by doing Help>Eclipse Marketplace> and then look for "Enhanced Class Decompiler".
You can then debug your app, and while stepping into the third party code, it will decompile bytecode back to Java language and show it into the IDE.

A: If absolutely necessary you can modify the library, by various techniques:

*

*extending a non-final class and creating an instance of that instead of the original class. This only works when your application controls the relevant points at which the instances are created. But this is actually the most risky approach, due to various problems around the fragility of implementation inheritance. I would not recommend this approach except for the simplest of needs.

*copying and pasting some source code from the library into your code and modifying it. Be sure to abide by all the relevant requirements in the license if you do this.

*checking out the original source code (e.g. from GitHub), tweaking it, rebuilding it and publishing the modified version to your corporate Maven repository under a modified organisation (com.your.company) and/or version. Then tweak your dependency specifications to use this modified jar instead of the original.


Based on Android 6.0.1 SM-N930F Note 7 firmware, CrissCross ROM brings you all of the Note 7 goodies right there on your Note 5 including new S-Pen features, Smart Select, Screen Write, Translate, Magnify, and much more.
Of course, Note 7 also features full Edge Screen Panels and this ROM has all of that also. Now, this ROM will give you pretty much every Note 7 feature except the ones that need hardware such as IRIS scanner and water-proofing. Otherwise everything works out of the box and this is a great way to experience Samsung's new features without buying a new phone.
This ROM is supported on all international models including SM-N920C, SM-N920I, SM-N920G, SM-N920K, SM-N920L, and SM-N920S. You can also install it on T-Mobile Note 5 SM-N920T or Canadian SM-N920W8 if you use SkyHigh Kernel but calls with not work (no audio) but this should be fixed soon and I should have an update on that also.
Installation is straightforward, just install the ROM files and reboot, you can watch our step-by-step video tutorial above.
Xposed for Marshmallow – Need cool Xposed modules to make your custom ROM complete? Check out our step-by-step tutorial! With this module, you can do cool stuff like GPS-Spoofing using Pokemon Go Hack for Android.
Are you able to do this with the sprint variant of the note 5?
I love it except calls dont work on SM-N920C either.
Call not sent is what I get. I even tried the new V5 still no call.
Now I am trying RiverRom and hope that it works.
hey man VoLte is not suported to this rom,rest everythings working fine, any fix for the Volte support?
Is there S-Pen translate for Samsung Galaxy Note 4 or a way to makes Note 4 S-Pen translates?
is there rom for SM-N9208 Dual Sim ??
AT&T variant?? If not should I be hopeful for the near future?
Hey thank you very much for this Rom!
Btw my battery is draining rapidly after installing it. I think a custom kernel will get me out of this.
So, could you recommend (or link) me a kernel for your Rom.
Is there a ROM for the SM-N920A yet??
Ok so this is the second ROM I've tried on my Canadian Note 5.
Both times I have lost the ability to hear anything through the speaker and no one can hear me when I talk.
Any one can help me, can run costom tom Galaxy note 5 SM-N9208 Tnx can I convert SM -N9208 to S7 edge?
Any one can help me, can i run custom Rom Galaxy note 5 SM-N9208 Tnx can I convert SM -N9208 to S7 edge?
Next story Galaxy S8 Conversion for Galaxy Note 5 or S6 Edge Plus!
Previous story Sick as HELL ROM for Galaxy Note 5!


In April 2013 reforms came into effect pursuant to the Legal Aid, Sentencing and Punishment of Offenders Act 2012 ("LASPO") resulting in a ban on recovering after-the-event insurance premiums and success fees from losing parties, save in respect of insolvency proceedings and mesothelioma claims.
The exemption in relation to insolvency proceedings was due to be lifted in April but last week the government announced that it would "for the time being" delay applying the provisions of LASPO to insolvency proceedings.
Accordingly so called "no-win, no fee" agreements will continue to operate on a pre-LASPO basis for insolvency proceedings with any success fees and after-the-event insurance premiums remaining recoverable from the losing party.


Swords to Plowshares and Bay Area Council are partnering to provide more career pathways to well-paid biotech positions in the Bay Area. There is an abundance of biotech positions in the Bay Area, ranging from manufacturing to research.
We encourage veterans to reach out to us at [email protected] to be connected to opportunities in the field.
Since 1974 we've helped veterans in the San Francisco Bay Area address barriers to employment and secure employment after their military service. At Swords to Plowshares, veterans work one-on-one with our Employment & Training Program staff to address their individual needs, overcome obstacles to employment, and determine a long-term career plan. We provide job placement assistance and vocational training programs to help veterans translate their skills and train for new careers in high-growth, high-wage careers in the civilian sector.
Must be a veteran and have a DD214, Social Security card & valid state ID. Visit one of our Bay Area locations during drop-in hours or schedule an appointment.
If you are interested in working with Swords to Plowshares to hire veterans, visit our Hire a Veteran page, email or call us.