\section{Introduction}

Conformal algebras were introduced by Kac \cite{KacValgBeginners}
to formalize the properties of the coefficients of the 
singular part of the operator product expansion  (OPE) 
 in conformal field theory. 
In particular, every vertex algebra \cite{bor} 
is actually a Lie conformal algebra.

Assume $V$ is a vertex algebra with a translation operator 
$\partial$ and a state-field correspondence $Y$. 
Then, as a result of the locality axiom, the operator product expansion of two fields $Y (a, z)$ and $Y(b, z)$, $a, b \in V$, has a finite singular part:
\[
Y (a, w)Y (b, z) =\sum^{N(a,b)-1}_{n=0} Y (c_n, z)\frac{1}{(w-z)^{n+1}}+ (\text{regular part}).
\]
The coefficients of the singular part are determined by the commutator of the fields:
\[
[Y (a, w),Y (b, z)] =\sum^{N(a,b)-1}_{n=0} Y (c_n, z)\frac{1}{n!}\frac{\partial^n\delta(w-z)}{\partial z^n}.
\]
where $\delta(w-z)=\sum_{s\in Z} w^sz^{-s-1} $is the formal delta-function. The correspondence
\[
(a,b) \mapsto c_n \ , \ n\geq 0.
\]
defines an infinite series of bilinear operations ($n$-products) on $V$. Together with the translation operator $\partial$, these operations turn $V$ into a system called {\em conformal Lie algebra}. The structure theory of finite Lie conformal algebras and superalgebras was developed in \cite{DK1998, Fattori}. 
Cohomology theory of conformal algebras was introduced in \cite{BKV}, then developed in a more general context of pseudo-algebras in \cite{BDK}. 

The study of universal structures for conformal algebras was initiated in \cite{Roitman99}. 
The classical theory of Lie and associative algebras often needs universal constructions like free algebras and universal enveloping algebras. This was a motivation 
for the development of combinatorial issues in the theory of conformal algebras, in particular, Gr\"obner--Shirshov bases 
theory \cite{BFK}.

For every conformal Lie algebra $L$ 
one can construct a series of universal enveloping 
associative conformal algebras corresponding to 
different associative locality functions on the 
generators \cite{Roit2000}. 
For example, consider the Virasoro conformal algebra $\mathop {\fam 0 Vir} \nolimits $
which is generated by a single element~$v$.
One may fix a natural number $N$ 
and construct 
the associative conformal algebra $U(N)$ 
generated by the element $v$ such that 
$(v\oo{n} v) = 0$ for $n\ge N$, and 
the commutation relations of $\mathop {\fam 0 Vir} \nolimits $ hold.
Obviously, $U(1)=0$; the algebra $U(2)$ 
is known as the Weyl conformal algebra 
(also denoted $\mathop {\fam 0 Cend} \nolimits_{1,x}$ in \cite{BKL}). 

For every conformal algebra $C$ one may construct 
an ``ordinary'' algebra $\mathcal A(C)$ called a 
coefficient algebra of~$C$, which inherits many properties of $C$ (associative, commutative, Lie, Jordan, etc.).
Every conformal module over $C$ is also a module over 
$\mathcal A(C)$. Moreover, it was proved in \cite{BKV} that the (reduced) cohomology of a conformal algebra $C$ may be calculated via the corresponding cochain complex of its coefficient $\mathcal A(C)$.

It was shown in \cite{Kozlov2017} that the second Hochschild 
cohomology groups $\Homol ^2(U(2), M)$ are trivial for every conformal 
(bi-)module~$M$, but for 
higher Hochschild cohomologies the direct computation becomes too complicated. 
In contrast to the ``ordinary'' Hochschild cohomology, 
if $C$ is an infinite associative conformal algebra 
then 
one cannot reduce the computation of 
$\Homol^n(C,M)$ to $\Homol^{n-1}(C, \mathrm{Chom}(C,M) )$
since the space of conformal homomorphisms 
$\mathrm {Chom}(C,M)$ may not be a conformal module 
over~$C$. Even if $M$ is finite (in this case, 
 the conformal $C$-module 
 $\mathrm {Chom}(C,M)$ is infinite and thus we may just
derive from the result of \cite{Kozlov2017}
that $\Homol^3(U(2),M)=0$ for every finite 
module~$M$.

In this paper we find all higher Hochschild 
cohomology groups $\Homol^n(U(2),M)$, $n\ge 2$,
of the Weyl conformal 
algebra $U(2)$ with 
coefficients in all finite modules~$M$. In order to obtain this result we construct the Anick resolution for its coefficient algebra via the algebraic discrete Morse theory as presented, for example, 
in \cite{JW}.
It is discussed in \cite{Akl} how to adjust this technique for differential algebras to calculate Hochschild cohomologies with coefficients in a trivial module. 
The purpose of our work is to apply the Morse matching method
for calculation of Hochschild cohomologies
of associative conformal algebras with coefficients 
in an arbitrary module.
As a result, we find that all Hochschild cohomology 
groups 
of $U(2)$ with coefficients in a finite module 
are trivial except the first one.
This result is close to the observation of \cite{SuY} 
about cohomologies of the Lie conformal algebra 
$\mathrm{gc}_n$ with coefficients in the natural 
module. 
\section{Preliminaries in conformal algebras}


Throughout the paper, 
$\Bbbk$ is a field of characteristic zero,
$\mathbb Z_+$ is the set of nonnegative integers.

\begin{definition} A conformal algebra \cite{KacValgBeginners} is a linear space $C$ equipped with a linear map 
$\partial:C \rightarrow C$ 
and a family of bilinear operations 
$(\cdot \oo n \cdot) :C\otimes C \rightarrow C$, 
$n \in \mathbb Z_+ $,
satisfying the following  properties:

(C1) for every $a,b \in C$ there exists $N = N(a,b)\in \mathbb Z_+$ such that $(a \oo n b) = 0$ for all $n\geq N$ ;\\
(C2) $(\partial a\oo n b) = -n(a \oo {n-1} b)$;\\
(C3)  $(a \oo n \partial b) = \partial(a \oo n b) + n(a \oo {n-1} b)$.

 Every conformal algebra $C$ is a left module over the polynomial algebra $H=\Bbbk [\partial ]$.
The structure of a conformal algebra on an $H$-module $C$
may be expressed by means of
a single polynomial-valued map called $\lambda$-product: 
\[
\begin{gathered}
(\cdot \oo{\lambda} \cdot) : C \otimes C \longrightarrow C[\lambda],
\\
(a \oo{\lambda} b) =\sum_{n=0}^{N(a,b)-1} \frac{\lambda^{n}}{n!}(a \oo{n} b),
\end{gathered}
\] 
where $\lambda $ is a formal variable, 
satisfying the following axioms:
\begin{gather}
    (\partial a\oo\lambda b) = -\lambda (a\oo\lambda b), 
      \label{eq:3/2-lin(1)}\\
    (a\oo\lambda \partial b) = (\partial+\lambda) (a\oo\lambda b).
      \label{eq:3/2-lin(2)}
\end{gather}
The number $N=N(a,b)$ is said to be a {\em locality level} of $a,b\in C$.
\end{definition}


\begin{definition}
 For every conformal algebra $C$ one may construct 
an algebra $A=\mathcal A(C)$ in the following way. 
As a linear space,
\[
A = \Bbbk [t,t^{-1}] \otimes _{\Bbbk [\partial ]} C,
\]
where $\partial $ acts on $\Bbbk[t,t^{-1}]$ as $- d/dt$.
Denote $a(n)=t^n\otimes_{\Bbbk [\partial ]} a$ for $a\in C$, $n\in \mathbb Z$.
The operation on $A$ is given by a well-defined expression
\[
a(n)\cdot b(m) = \sum\limits_{s\ge 0} \binom{n}{s} (a\oo{s} b) (n+m-s).
\]
The algebra $\mathcal A(C)$ is called the {\em coefficient algebra} of~$C$.
\end{definition}

The coefficient algebra $\mathcal A(C)$ has a derivation 
also denoted $\partial $ such that 
$\partial (a(n)) = (\partial a)(n) = -na(n-1)$.
The space $\mathcal A_+(C)$ spanned by all $a(n)$, $n\in \mathbb Z_+$, $a\in C$,
is a subalgebra of $\mathcal A(C)$ which is closed under the derivation~$\partial $.


Conformal algebra $C$ is called associative 
(commutative, Lie, Jordan, etc.) if so is $A(C)$ \cite{Roitman99}. 
For example, $A(C)$ is associative if and only if 
\begin{equation}\label{eq:ConfAss}
a \oo n (b \oo m c) = \sum _{s \geq 0} \binom{n}{s}
(a \oo{n-s} b) \oo {m+s} c
\end{equation}
for all $a, b, c \in C, n,m \in \mathbb Z_+$. 
In terms of the $\lambda$-product, the last relation may be expressed by a single formula
\[
a \oo \lambda (b \oo \mu c) = (a \oo \lambda b) \oo {\lambda+\mu} c, \quad a, b, c \in C,
\]
where $\lambda$ and $\mu$ are independent commuting variables \cite{KacForDistrib}.


In general, an associative conformal algebra $C$ turns into a Lie conformal algebra $C^{(-)}$
when equipped with a new $\lambda $-product $[\cdot\oo\lambda \cdot ]$ defined as follows:
\[
 [a\oo\lambda b] = (a\oo\lambda b) - (b\oo{-\partial - \lambda } a),\quad a,b\in C.
\]


\begin{example}
(1) The 1-generated free $H$-module $C = \Bbbk [\partial ] v$
equipped with 
\[
 (v\oo\lambda v ) = (\partial +2\lambda )v
\]
is a Lie conformal algebra known as the Virasoro conformal algebra.
The coefficient algebra of $\mathop {\fam 0 Vir} \nolimits $ is the Witt algebra
 $\mathcal A(\mathop {\fam 0 Vir} \nolimits) = \mathop {\fam 0 Der} \nolimits \Bbbk [t,t^{-1}]$, 
 and $\mathcal A_+(\mathop {\fam 0 Vir} \nolimits )= \mathop {\fam 0 Der} \nolimits \Bbbk [t]$.

(2) The infinitely generated free $H$-module 
$C = 
H\otimes \Bbbk\{v,v^2,v^3,\dots  \} \simeq \Bbbk [\partial, v]v$
equipped with 
\[
 (v^n\oo\lambda v^m) = v^n(v+\lambda )^m, \quad n,m\ge 1
\]
is an associative conformal algebra denoted ${U(2)}$.
The coefficient algebra $\mathcal A (U(2))$ 
is known to be the right ideal in the 
associative algebra $\Bbbk \langle p,q,q^{-1} \mid qp-pq=1\rangle $
generated by $p$. The positive part $\mathcal A_+(U(2))$
is isomorphic to $pW_1$, where $W_1$ is the first Weyl algebra 
generated by $p$, $q$ with $[q,p]=1$.
This is why $U(2)$ is called the {\em conformal Weyl algebra} \cite{Roitman99}.
\end{example}

Note that $\mathop {\fam 0 Vir} \nolimits \subset U(2)^{(-)}$, and $U(2)$ is generated 
(as an associative conformal algebra) by the elements of $\mathop {\fam 0 Vir} \nolimits $.
Hence, $U(2)$ is an associative envelope of $\mathop {\fam 0 Vir} \nolimits $. 
Moreover, this envelope is universal in the class 
of all associative envelopes $C$ of $\mathop {\fam 0 Vir} \nolimits $ such that 
$N(v,v)\le 2$ in $C$ \cite{BFK}. 
In fact, for the Virasoro conformal algebra $\mathop {\fam 0 Vir} \nolimits $ one may construct a series of associative conformal algebras $U(N)$, $N\in \mathbb Z_+$. 
Each $U(N)$ is the universal enveloping associative conformal algebra of $\mathop {\fam 0 Vir} \nolimits $ in the class of all associative conformal envelopes $C$
of $\mathop {\fam 0 Vir} \nolimits $ with $N(v,v)\le N$.

\begin{definition}[\cite{ChengKac}] A module $M$ over an associative conformal algebra $C$ is a $\Bbbk[\partial]$-module endowed with the $\lambda$-action $a_{(\lambda)}m$ which defines a map 
$C \otimes M\rightarrow M [\lambda] $ such that:
\begin{gather}
    (\partial a\oo\lambda m) = -\lambda (a\oo\lambda m), 
      \quad 
    (a\oo\lambda \partial m) = (\partial+\lambda) (a\oo\lambda m),
      \label{eq:3/2-lin(mod1-2)}\\
    (a\oo{\lambda } (b\oo\mu m)) = ((a\oo{\lambda } b)\oo{\mu+\lambda } m).
\end{gather}

Similarly, a conformal action of a Lie conformal algebra $L$
on a module $M$ meets \eqref{eq:3/2-lin(mod1-2)} 
and the conformal analogue of the Jacobi identity:
\[
(a\oo\lambda (b\oo\mu m )) - (b\oo\mu (a\oo\lambda m)) =
((a\oo\lambda b)\oo{\lambda+\mu} m),
\]
for $a,b\in L$, $m\in M$.
\end{definition}

\begin{example}
Given a 1-generated free $H$-module  $M = \Bbbk [\partial ]u$
and two scalars $\Delta, \alpha \in \Bbbk$, 
one may define conformal action of $\mathop {\fam 0 Vir} \nolimits $  on $M$ as
\begin{equation}\label{eq:Vir}
(v\oo\lambda u) = (\alpha +\partial +\Delta\lambda )u.
\end{equation}
Denote the conformal $\mathop {\fam 0 Vir} \nolimits$-module obtained by $M_{(\alpha, \Delta)}$.
For $\Delta \ne 0$, this is an irreducible $\mathop {\fam 0 Vir} \nolimits $-module, 
and every finite irreducible $\mathop {\fam 0 Vir} \nolimits $-module is isomorphic 
to an appropriate $M_{(\alpha, \Delta)}$ \cite{ChengKac}.
\end{example}

If $M$ is a module over an (associative or Lie) conformal algebra 
$C$ then $M$ is also a module over the ordinary 
(associative or Lie, respectively)
algebra $\mathcal A_+(C)$. 
Namely, for $a\in C$, $n\in \mathbb Z_+$, $u\in M$ the element 
$a(n)u$ is the coefficient at $\lambda ^{n}/n!$ of $(a\oo\lambda u)$:
\[
a\oo\lambda u  = \sum\limits_{n\ge 0} \dfrac{\lambda^n}{n!} a(n)u.
\]

For every conformal $C$-module $M$ over an associative conformal algebra $C$, the space $M$ is also a $C^{(-)}$-module relative to the same conformal action. 
The converse construction has a restriction due to locality. 
For example, the module $M_{(\alpha,\Delta)}$
over the Virasoro (Lie) conformal algebra is also a module 
over its universal enveloping associative conformal algebra 
$U(2)$
if and only if $\Delta = 0$ or $\Delta =1$.

Indeed, $(v\oo\lambda v) = v^2+\lambda v$ in $U(2)$, so 
\[
(v\oo\lambda v)\oo\mu u = v\oo\lambda (v\oo{\mu-\lambda } u )
=
v\oo\lambda (\alpha +\partial + \Delta(\mu-\lambda ))u
=(\alpha +\partial +\lambda + \Delta(\mu-\lambda ))(\alpha+\partial +\Delta\lambda )u.
\]
The polynomial in the right-hand side is of degree $<2$ 
in $\lambda $ if and only 
if $\Delta=0$ or $\Delta =1$. 

In \cite{BKL}, the algebra $U(2)$ (up to an isomorphism) is denoted 
by $\mathop {\fam 0 Cend} \nolimits_{1,x}$. The classification of finite irreducible modules 
over $\mathop {\fam 0 Cend} \nolimits_{1,x}$ indeed consists of the modules $M_{(\alpha, 1)}$.
The modules $M_{(\alpha,0)}$ are not irreducible, they contain 
submodules $(\partial+\alpha)M_{(\alpha, 0)}$.

The explicit formula for the action of $vh(\partial, v)\in U(2)$ on 
$M_{(a,1)}=Hu$, $H=\Bbbk [\partial ]$,
is a consequence of \eqref{eq:Vir} and associativity \eqref{eq:ConfAss}:
\[
vh(\partial, v)\oo\lambda f(\partial )u = 
h(-\lambda , \partial+\alpha )(\lambda +\partial +\alpha )
f(\partial +\lambda )u, 
\quad f(\partial )\in H.
\]
For example, the element $v(n)f(\partial)u$, $n\in \mathbb Z_+$, 
$f(\partial )\in H$,  is given by
\begin{equation}\label{eq:v(n)-Action}
v(n) f(\partial ) u  = (\partial+\alpha ) f^{(n)}(\partial )u+n f^{(n-1)}(\partial )u,
\end{equation}
where $f^{(n)}(\partial )$ 
stands for the $n$th derivative of $f(\partial )$.

Even for finite conformal modules over a simple associative conformal algebra,
there is no complete reducibility. 
For example, if $M_{(\alpha, 1)}$ and $M_{(\beta, 1)}$ are two irreducible 
modules over the Weyl conformal algebra $U(2)$ then 
there exists a non-split extension 
\[
0\to M_{(\alpha, 1)}=Hu \to E \to M_{(\beta , 1)}=Hw \to 0,
\]
e.g., 
\[
(v\oo\lambda w) = (\lambda +\partial +\beta)w + \gamma u,
\]
for $\gamma \in \Bbbk $.
Indeed, all extensions like that for the Virasoro (Lie) conformal algebra 
were described in \cite{ChengKac}, it is enough to choose those 
of limited locality.


\subsection{Hochschild cohomology for associative conformal algebras}\label{2.2}

Let $C$ be an associative conformal algebra, 
and let $M$ be a conformal module over~$C$.
The {\em basic Hochschild complex} \cite{BKV} 
$\tilde \C^\bullet (C,M)$
consists of the cochain spaces $\tilde \C^n (C,M)$, $n=1,2,\ldots$,
each of them is the space of all maps 
\[
\varphi_{\bar \lambda }: 
C^{\otimes n}\to M[\bar \lambda ],
\]
where $\bar\lambda = (\lambda_1,\ldots, \lambda_{n})$,
satisfying the conformal anti-linearity condition: 
\[
 \varphi_{\bar \lambda }(a_1,\ldots,\partial a_i, \ldots , a_n) =
 -\lambda_i \varphi _{\bar \lambda }(a_1,\ldots, a_n),\quad i=1,\ldots, n.
\]
The Hochschild differential 
$\dcobound_n : \tilde \C^n(C,M) \to \tilde \C^{n+1}(C,M)$
on the basic complex is given by
\begin{multline}\nonumber
(\dcobound_n\varphi)_{\bar\lambda }
(a_1,\ldots, a_{n+1}) = 
a_1 \oo{\lambda_1} 
\varphi_{\bar\lambda_0} (a_2, \ldots , a_{n+1})  
 + \sum\limits_{i = 1}^{n} (-1)^i\varphi_{\bar\lambda_i}
  (a_1,\ldots , a_i \oo{\lambda_i} a_{i+1},\ldots , a_{n+1}),
\end{multline}
for 
$\bar \lambda = (\lambda_1, \ldots, \lambda_{n+1})$, 
$\bar\lambda_0 = (\lambda_2, \ldots, \lambda_{n+1})$, 
$\bar\lambda_i = (\lambda_1, \ldots, \lambda_i+\lambda_{i+1},
\ldots , \lambda_{n+1} )$, $i=1,\dots, n$. The cohomology of the  basic Hochschild complex is called {\em the basic Hochschild cohomology} $\tilde H^\bullet (C, M)$. 

For every $n\ge 1$, the cochain space $\tilde \C^n(C,M)$ 
is a left  $\Bbbk[\partial]$-module:
\[
(\partial\varphi)_{\bar \lambda}(a_1,\ldots,a_n)=(\partial+\sum_{i=1}^{n}\lambda_i)\varphi_{\bar \lambda}(a_1,\ldots,a_n).
\]
For every $n\ge 1$, the map $\dcobound_n$ commutes with $\partial$. 
The quotient complex
\[
\C^\bullet (C,M)= 
\tilde \C^\bullet (C, M)/\partial \tilde \C^\bullet (C,M)\]
is called the {\em reduced Hochschild complex} and its cohomology is called the {\em reduced  Hochschild cohomology} $H^\bullet (C, M)$.

Another approach to the definition of Hochschild cohomologies for 
associative conformal algebras was considered in \cite{BDK}, 
see also \cite{Dolg2007}. The Hochschild cohomology groups 
defined there coincide with the reduced Hochschild cohomologies 
when written in terms of $\lambda $-operations.

Consider the ``ordinary'' Hochschild complex 
$\C^\bullet (\mathcal A_+(C),M)$
The maps
\[
\partial_n^* : \C^n( \mathcal A_+(C),M) \to \C^n( \mathcal A_+(C) ,M)
\]
given by
\[
(\partial_n^* f)(\alpha_1,\ldots, \alpha_n)
= \partial f(\alpha_1,\ldots, \alpha_n)
- \sum\limits_{i=1}^n 
f(\alpha_1,\ldots, \partial\alpha_i, \ldots, \alpha_n),
\]
and $\partial a(n) = -n a(n-1)$ for $a\in C$, $n\ge 0$, 
commute with the Hochschild differentials.

The following statement describes the relations between 
Hochschild cohomologies of $C$ and $\mathcal A_+(C)$ 
with coefficients on the same module~$M$.

\begin{proposition}[{\cite[Theorem 6.1, Corollary 6.1]{BKV}}]\label{prop:MainTool}
\begin{gather*}
\C^\bullet (C,M)\cong 
\C^\bullet ( \mathcal A_+(C), M)/\partial^*_\bullet \C^\bullet ( \mathcal A_+(C),M)\\
\tilde H^\bullet (C,M)= 
H^\bullet ( \mathcal A_+(C), M)
\end{gather*}
\end{proposition}

So we can calculate (reduced) Hochschild cohomologies of 
a conformal algebra $C$ via the Hochschild complex of its coefficient algebra $\mathcal A_+(C)$ and its quotient. 
To do that, we construct the Anick resolution for $\mathcal A_+(C)$ by means of the Morse matching method. 
We will use the homotopy maps constructed in this way 
to transfer the map $\partial^*_\bullet $
to the dual complex obtained from the Anick resolution,
as explained in the following sections.  



\section{Morse matching and Anick resolution}\label{sec:Morse}

\subsection{The basics of algebraic discrete Morse theory}	
Suppose $\Bbbk $ is a field and $\Lambda $ is a unital associative  $\Bbbk$-algebra.
Let $\B_\bullet =(\B_n,\dcobound_n)_{n\ge 0}$ be a chain complex of free left $\Lambda $-modules, and let $X_n$ be a basis 
of $\B_n$ over $\Lambda $. One may represent the complex with a weighted oriented graph
$\Gamma (\B_\bullet )$
whose vertices are $\bigcup\limits_{n} X_n$ and there is an edge 
$\mathbf x \stackrel{\lambda} \longrightarrow \mathbf y $
from $\mathbf x\in X_n$ to $\mathbf y\in X_{n-1}$ 
of weight $\lambda $, $0\ne \lambda \in \Lambda $, 
if the distribution of $\dcobound_n(\mathbf x)\in \B_{n-1}$ as a linear 
combination of  $X_{n-1}$ contains $\lambda \mathbf y$.


A subset $M \subseteq E$ of the set of edges is called
Morse matching, if it satisfies the following three conditions:
\begin{itemize}
    \item  $M$ is a particular matching in the graph $\Gamma (\B_\bullet )$, i.e., a subset of edges such that 
neither of vertices belongs to more than one edge from $M$. 
\item The weights of edges from $M$ are invertible central elements of $\Lambda $. Then transform the graph $\Gamma(\B_\bullet )$ in the following way: invert the direction of all edges from $M$
and replace their weights $\lambda $ with $-\lambda^{-1}$.
\item Resulting graph $\Gamma_M(\B_\bullet )$ has no directed cycles.
\end{itemize}

 The vertices that do not belong to  edge from $M$ are said to be critical cells.

For a Morse matching graph $\Gamma_M(\B)$, one may construct another chain complex 
of free $\Lambda $-modules $(\A_m,\delta_m)_{m\ge 0}$, where $\A_m$ is spanned over 
$\Lambda $ by the set $X_{(m)}\subseteq X_m$ 
of all critical cells from $\B_m$ and the calculation of $\delta_m$ 
is based on the consideration of all paths in $\Gamma_M(\B)$  \cite{formancell,formanguide,JW}.

Namely,
\begin{equation}\label{eq:Anick}
\delta_m(\mathbf x)=\sum\limits_{\mathbf y\in  X_{(m-1)}}\Gamma(\mathbf x,\mathbf y)\mathbf y,\quad  \mathbf x\in  X_{(m)}
\end{equation}
where $\Gamma(\mathbf x,\mathbf y)$ is the sum of path weights in the Morse matching graph $\Gamma_M(\B)$.
The new complex $(\A_m,\delta_m)_{m\ge 0}$ is homotopy equivalent to the initial one, 
but it is smaller since we choose only critical cells as generators of the modules $\A_m$.

The homotopy maps $\mathrm f_n: \B_n \to \A_n$, $\mathrm g_n:\A_n \to \B_n$ are given by
\begin{align*}
     & \mathrm{f}_n(\mathbf {b}_n) = \sum_{\mathbf {a}_n\in \A_n} \Gamma_{\B_\bullet}(\mathbf {b}_n,\mathbf{a}_n) \mathbf{a}_n, \\
     & \mathrm{g}_n(\mathbf{a}_n) = \sum_{\mathbf{b}_n \in \B_n} \Gamma_{\A_\bullet}(\mathbf{a}_n,\mathbf{b}_n)\mathbf{b}_n.
 \end{align*}
If we are given a chain map $\varphi_\bullet : \B_\bullet \to \B_\bullet $
then for every $n\ge 1$ the map 
$\tilde \varphi _n = \mathrm f_n\varphi_n \mathrm  g_n$
is a chain map on $\A_\bullet$.
We have the following commutative diagram
\[
  \xymatrix{
   \A_n \ar@{->}[r]^{\delta_n} \ar@{->}[d]_{\mathrm{g}_n} & \A_{n-1} \\
   \B_n \ar@{->}[r]_{\dcobound_n} & \B_{n-1} \ar@{->}[u]_{\mathrm{f}_{n-1}}
  }
 \]
Therefore

\begin{equation}\label{com.}
\delta_n 
= \mathrm f_{n-1}\dcobound_n \mathrm g_n:
    \A_n\to \A_{n-1}.
\end{equation}

\subsection{The Anick resolution for associative algebras.}

Suppose $\Lambda $ has an an augmentation $\varepsilon: \Lambda \to \Bbbk $, 
$A$ is a set of generators for $\Lambda$. 
Then $\Lambda $ is a homomorphic image of the free associative algebra 
$\Bbbk \langle A\rangle$ generated by~$A$.
Assume  $\le$ is a monomial order on the free monoid $A^{*}$,
and $\mathrm{GSB}_\Lambda$ is a Gr\"obner---Shirshov basis of $\Lambda $.
The latter may be considered as a confluent set of defining relations 
for the algebra~$\Lambda $, each of relations is of the form $u-f$, where $u\in A^*$, 
$f\in \Bbbk \langle A\rangle$, $u\ge \bar f$, $\bar f$ is the leading monomial of $f$
relative to $\le $.
Denote by $V$ the set of all leading terms $u$ of relations from $\mathrm{GSB}_\Lambda$, $V$ is called the set of {\em obstructions}.

Following Anick \cite{Anick1983}, a word $v=x_{i_1}\ldots x_{i_t}$ 
is an $n$-{\em prechain} if and only if there exist $a_j,b_j \in \mathbb{Z}$, $1 \le j \le n$, satisfying the following conditions:
\begin{itemize}
    \item $1=a_1<a_2 \le b_1<\ldots<a_n \le b_{n-1}<b_n=t$;
    \item $ x_{i_{a_j}}\ldots x_{i_{b_j}} \in V$ for $1 \le j \le n$.
\end{itemize}
An $n$-prechain $x_{i_1}\ldots x_{i_t}$ is an {\em $n$-chain} 
if only if the integers $a_j,b_j$ can be chosen in such a way that  $x_{i_1}\ldots x_{i_t}$  is not an $m$-prechain for neither 
$s < b_m$, $1 \le m \le n$.

The set of all $n$-chains is denoted $V^{(n)}$.

The cokernel of  $\varepsilon: \Bbbk \to \Lambda$ is denoted by $\Lambda\slash \Bbbk$.
The set of all non-trivial words in $A^*$ that do not contain a word from $V$
as a subword (i.e., the set of $V$-reduced words \cite{Bok72})
forms a linear basis of $\Lambda/\Bbbk $. This is one of the equivalent 
conditions in the Composition-Diamond Lemma about Gr\"obner---Shirshov bases 
for associative algebras (see, e.g., \cite{Bokut}).
The resolution  $\B_\bullet =(\B_m,\dcobound_m)_{m\ge 0}$ is an 
exact sequence of $\Lambda$-modules where
$$
\B_m:=\Lambda \otimes(\Lambda/\Bbbk)^{\otimes_{m}}.
$$
and differential
$\dcobound_n : \B_n
   \to \B_{n-1}$.
   
We will use the standard convention denoting $1\otimes \alpha_1\otimes \dots \alpha_{m}\in \B_m $ 
 by $[\alpha_1|\ldots |\alpha_{m}]$ for $\alpha_i\in \Lambda/\Bbbk$ so differential is defined as follows:
\[
\dcobound_m([a_1| \ldots | a_n]) = a_1 
[a_2| \ldots |a_m]
+  \sum\limits_{i=1}^{m-1}(-1)^{i}   [a_1| \ldots |N(a_ia_{i+1})| \ldots | a_m].
\]
Here $N(a_ia_{i+1})$ is the corresponding Gr\"obner--Shirshov normal form of the product $a_ia_{i+1}$.

The {\em Anick resolution}  $\A_\bullet =(\A_m, \delta_m)_{m\ge 0}$ is an 
exact sequence of $\Lambda$-modules where
$$
\A_m:=\Lambda \otimes \Bbbk V^{(m-1)}.
$$
The role critical cells $X_{(m)}$ is played by  $(m-1)$-chain and it is not difficult to find it by $\mathrm{GSB}_\Lambda$. The computation of differentials in the Anick resolution  according to 
the original Anick algorithm described in \cite{Anick1983} is extremely hard.
In order to visualize the computation of differentials it is possible to use the discrete algebraic
Morse theory\cite{JW, formancell, formanguide}  based on the concept of a Morse matching.
For word $w$, let $\Lambda_{w,p}$ be set of all the vertices 
$[w_1| \ldots | w_n]$ in 
$\Gamma(\B_\bullet(\Lambda,\Bbbk))$ such that $w = w_1 \cdots w_n$ 
and~$p$ is the largest integer 
$p \ge -1$ for which $w_1\cdots w_{p+1} \in 
\Lambda^{(p)}$ is an Anick $p$-chain.
Let 
$\Lambda_w \coloneqq  \bigcup\limits_{p \ge -1}\Lambda_{w,p}$.

Define a partial matching $\mathcal{M}_w$ on 
$\Gamma(\B_\bullet(\Lambda,\Bbbk))|_{\Lambda_w}$ by letting 
$\mathcal{M}_w$ consist of all edges
\[
[w_1| \ldots| w'_{p+2}|w''_{p+2}| \ldots| w_n] \to [w_1| \ldots| w_{p+2}| 
\ldots| w_m]
\]
where 
$w'_{p+2}w''_{p+2} = w_{p+2}$, 
$[w_1|  \ldots| w_m] \in \Lambda_{w,p}$, 
and $[w_1| \ldots| w_{p+1}|w'_{p+2}] \in \Lambda^{(p+1)}$ is an Anick 
$(p+1)$-chain.

\section{The Anick complex for $\mathcal A_+(U(2))$}

\subsection{The Morse matching graph for $\mathcal A_+(U(2))$}\label{anick}
Let us write down the Gr\"obner--Shirshov basis of 
$\mathcal A_+ = \mathcal A_+(U(2))$ 
as of an associative algebra 
over a field $\Bbbk$ generated by
elements $v(n)$, $n\ge 0$.
The defining relations of $U(2)$ reflect 
the Virasoro commutator
$[v(n),v(m)]=(n-m)v(n+m-1)$ and
locality $N(v,v)=2$:
$v(n+2)v(m)-2v(n+1)v(m+1)+v(n)v(m+2)=0$, 
$n,m\ge 0$.

The Gr\"obner--Shirshov basis of $\mathcal A_+$
is easy to derive. It consists of the relations
\begin{equation}\label{eq:u3-gsb}
   v(n)v(m)=v(0)v(n+m)+nv(n+m-1),\quad n \ge 1, m \ge 0.
\end{equation}
Indeed, the linear basis of $\mathcal A_+ \simeq pW_1$
is given explicitly by the monomials $p^{k+1}q^n$, $k,n\ge 0$, 
that represent the reduced forms 
$v(0)^{k+1}v(n)$, $n\ge 0$, relative to \eqref{eq:u3-gsb}.


Throughout the rest of the paper 
$\Lambda $ stands for 
the augmented algebra 
$\Lambda = \mathcal A_+ \oplus \Bbbk $. 


To find the Anick complex, we need two steps.
First, we have to find the set of obstructions 
for $\mathcal A_+ $ 
relative to the given Gr\"obner---Shirshov basis (the set of leading terms in $\mathcal A_+(U(2))$ ) 
and the set of $n$-chains. 
Next, build a Morse graph and calculate the path weights 
$\Gamma(w,w')$ for every $n$-chain $w$ and $(n-1)$-chain $w'$ for all $n\ge1$. For all $n\ge2$ we have
\[
V^{(n)}=\{v(i_1)v(i_2)\ldots v(i_{n+1});\quad i_1,i_2,\ldots,i_n\ge1,i_{n+1}\ge0\}.
\]

For $n=1$ we have the following set of obstructions
\[
V^{(1)}=\{v(n)v(m);\quad n\ge1,m\ge0\}.
\]
Let us  
evaluate $\delta_2: \A_{2}\to \A_{1}$ by means of the Morse graph is shown on Figure \ref{FIG:1}.

Hence,
\begin{figure} 
\centering
\input fig1.tex
    \caption{}
    \label{FIG:1}
\end{figure}
\[
\delta_2[v(n)|v(m)]=v(n)[v(m)]-v(0)[v(n+m)]-n[v(n+m-1)]
\]
For $n=2$ we have
\[
V^{(2)}=\{v(n)v(m)v(p);\quad n,m\ge1,p\ge0\}.
\]
and $\delta_3: \A_{3}\to \A_{2}$ is given by the following rule:
\[
\begin{aligned}
  \delta_3[v(n)v(m)v(p)]{} = v(n)[v(m)v(p)]- n[v(n+m-1)v(p)] - v(0)[v(n+m)v(p)] \\+ v(0)[v(n)v(m+p)] + n[v(n-1)v(m+p)]
			+ m[v(n)v(m+p-1)];\quad n,m\ge1,p\ge0.
\end{aligned}
\]

\begin{figure} 
\centering
\input fig3.tex
    \caption{}
    \label{fig:3}
\end{figure}
\begin{figure} 
\centering
\input fig4.tex
    \caption{}
    \label{fig:4}
\end{figure}
we can evaluate Anick differential $\delta_{n}: \A_{n}\to \A_{n-1}$ as we found $\delta_3$ by repeating the steps \cite{A2022}, which are shown in the two Figure \ref{fig:3}, \ref{fig:4} 
then $\delta_n$ is given by the following rules:
\[
\begin{aligned}
	&\delta_{n}[v(i_1)v(i_2)|\ldots|v(i_{n-1})v(i_n)]= v(i_1)[v(i_2)v(i_3)\ldots v(i_{n-1})v(i_n)]\\
	&+\sum\limits_{j=1}^{n-1}(-1)^ji_j[v(i_1)|v(i_2)|\ldots|v(i_j+i_{j+1}-1)|\ldots|v(i_n)] +\\ &+\sum\limits_{j=1}^{n-1}(-1)^{j}v(0)[v(i_1)|v(i_2)|\ldots|v(i_j+i_{j+1})|\ldots|v(i_n)] + \\
	&+\sum\limits_{j=2}^{n-1}\sum\limits_{k=1}^{j-1}(-1)^{j}i_k[v(i_1)|\ldots|v(i_k-1)|\ldots|v(i_j+i_{j+1})|\ldots|v(i_n)];\quad i_1,\ldots,i_{n-1}\ge1,i_n\ge0.
	\end{aligned}
\]


\subsection{Anick resolution for differential algebras}

Let $C$ be an associative conformal algebra. Throughout the rest of the paper 
$\Lambda $ stands for 
the augmented algebra  
$\Lambda = \mathcal A_+ \oplus \Bbbk 1$,
where 
$\mathcal A_+ = \mathcal A_+(C)$, 
and the augmentation is given by
$\varepsilon (\mathcal A_+)=0$.
Then $\mathcal A_+=\Lambda /\Bbbk $, 
and
$\Lambda $ has a derivation $\partial $ such that
$\partial (a(n)) = -na(n-1)$, $\partial (1)=0$. 




For every $n\ge 1$, there is a $\Bbbk$-linear map 
\[
\partial_n: \B_n\to \B_n,
\]
\[
\partial_n(\beta [\alpha_1|\dots |\alpha_n])
= \partial(\beta) [\alpha_1|\dots |\alpha_n]
+\sum\limits_{i=1}^n 
 \beta [\alpha_1 | \dots |\partial(\alpha_i)| \dots |\alpha_n],
\]
$\alpha_i\in \mathcal A_+$, $\beta \in \Lambda $. 
This is not a morphism of complexes, 
but it induces a morphism of dual cochain complexes
which can be transferred to the Anick resolution 
via homotopy.
Namely, suppose $M$ is a left conformal $C$-module. Thus, 
$M$ is an ``ordinary'' $\Lambda $-module. 
Denote
\[
\mathrm C_{\B}^n = \Hom _\Lambda (\B_n, M) \simeq 
\Hom_\Bbbk (\A^{\otimes n}, M), 
\]
this is the space of Hochschild cochains. 
The Hochschild differential 
\[
\Delta_{\B}^n  : \C_{\B}^n \to \C_{\B}^{n+1} 
\]
is expressed as 
\[
\big (\Delta_{\B}^n\varphi\big )(\mathbf x) = \varphi \dcobound_{n+1}(\mathbf x), \quad \varphi\in \C_{\B}^{n},
\  \mathbf x\in \B_{n+1}.
\]

Note that the $\Lambda $-module $M$ is equipped with a derivation 
also denoted $\partial $ (the same as in the definition of a conformal module), so that 
$\partial (a(n)u) = -na(n-1)u +a(n)\partial(u)$, for $a\in \C$, $u\in M$, 
$n\in \mathbb Z_+$.

Then for every $n\ge 1$ the map
\[
D_{\B}^n : \C_{\B}^n \to \C_{\B}^n 
\]
given by
\[
(D_{\B}^n \varphi)(\mathbf x) = \partial(\varphi (\mathbf x))
-\varphi (\partial_n(\mathbf x)),
\quad 
\varphi\in \mathrm C_{\B}^{n},
\  \mathbf x\in \B_{n+1},
\]
is a morphism of complexes:
\[
D_{\B}^{n+1}\Delta^n_{\B} 
=  \Delta^n_{\B} D_{\B}^{n}.
\]

Let us translate the mapping $D^\bullet _{\B}$ 
from 
$\C_{\B}^\bullet $
to the complex 
$\C_{\A}^\bullet $
constructed on the spaces
\[
\C_{\A}^n = \Hom_\Lambda (\A_n, M)\simeq \Hom_{\Bbbk }
(\Bbbk V^{(n-1)}, M)
\]
with the differential $\Delta_{\A}^n:  \C_{\A}^n \to \C_{\A}^{n+1}$ given by (see \ref{com.})
\[
\Delta^n_{\A}(\psi) = \psi\delta_{n+1}
= \psi \mathrm f_n\dcobound_{n+1} \mathrm g_{n+1}
=\Delta_{\B}^n(\psi \mathrm f_n) g_{n+1}.
\]
The homotopy equivalence between $\A_\bullet $
and $\B_\bullet $
turns $D_{\B}^n$
into 
\[
D_{\A}^n : \C_{\A}^n \to \C_{\A}^{n}
\]
such that 
\[
D_{\A}^n\psi = D_{\B}^n(\psi \mathrm f_n)\mathrm g_n.
\]
For every $\mathbf a\in \A_n$, 
we have
\[
(D_{\mathrm A}^n\psi)(\mathbf a)
= \partial (\psi \mathrm f_n (\mathrm g_n (\mathbf a)))
-(\psi\mathrm f_n)(\partial_n(\mathrm g_n(\mathbf a)))
= \partial (\psi (\mathbf a))
-(\psi\mathrm f_n)(\partial_n(\mathrm g_n(\mathbf a))).
\]

Note that the Anick chains for $\mathcal A_+=\mathcal A_+(U(2))$
constructed in Section~\ref{anick}
have the following property: if $\mathbf x = [\alpha_1|\dots |\alpha_n]\in 
\mathrm B_n$
is not an Anick chain then $\partial_n(\mathbf x)$ does not contain 
summands that are Anick chains, i.e.,
$\mathrm f_n(\mathbf x)=0$ implies
$\mathrm f_n(\partial_n(\mathbf x)) = 0$.
Hence, we may evaluate $D_{\mathrm A}^n\psi $ on $\mathbf a\in \mathrm A_n $
in the same way as 
$D_{\mathrm B}^n(\psi)$ on $\mathbf a$, 
simply removing those terms from $\mathrm B_n$ that do not represent 
Anick chains.

\begin{example}
Let $\psi \in \mathrm C_{\mathrm A}^3$ for $A=\mathcal A_+(U(2))$.
Denote
$\psi([v(n_1)|v(n_2)|v(m)]) = \psi(n_1,n_2,m)\in M$,
for $n_1,n_2\ge 1$, $m\ge 0$.
Let us calculate 
$(D^3_{\mathrm A}\psi )[v(2)|v(1)|v(1)]$.
\end{example}

According to the Morse matching graph in Sec \ref{sec:Morse},
we have 
\begin{multline*}
\mathrm g_3[v(2)|v(1)|v(1)] = [v(2)|v(1)|v(1)]-[v(0)|v(3)|v(1)]-[v(2)|v(0)|v(2)] \\
+[v(0)|v(0)|v(4)]+[v(2)|v(0)|v(2)].
\end{multline*}
Then 
\[
\partial_3(g_3[v(2)|v(1)|v(1)]) = -2[v(1)|v(1)|v(1)]-[v(2)|v(1)|v(0)] + \dots,
\]
where all other summands are not Anick chains (contain $v(0)$ not at the last position).
Therefore, 
\[
(D^3_{\mathrm A}\psi )[v(2)|v(1)|v(1)] = \partial \psi(2,1,1) + 2\psi(1,1,1)+\psi(2,1,0).
\]

Proposition \ref{prop:MainTool} and homotopy equivalence of 
$\B_\bullet$ and $\A_\bullet$
immediately imply the following statement.

\begin{proposition}
  For a conformal algebra $C$ and a conformal $\C-$module $M$, 
  the reduced Hochschild complex 
  $\C^\bullet (\C,M)$ is homotopy equivalent 
  to $\C_{\A}^\bullet / D_{\A}^\bullet \C_{\A}^\bullet $.
\end{proposition}

The complex
$\mathrm C_{\mathrm A}^\bullet /D_{\mathrm A}^\bullet \mathrm C_{\mathrm A}^\bullet $ constructed 
on the algebra $\mathcal A_+ = \mathcal A_+(C)$
is called the {\em reduced Anick complex} of~$\mathcal A_+$.

\section{Cohomologies of the reduced Anick complex of $\mathcal A_+(U(2))$}

Given a conformal module $M$ over the 
associative conformal algebra $U(2)$, we denote 
by $\C^\bullet $
 the reduced Anick complex for 
$\mathcal A_+ = \mathcal A_+(U(2))$, and let
 $\tilde \C^\bullet$ stand for the non-reduced 
complex $\C_{\mathrm A}^\bullet $ with values in~$M$.
By $D^n$ we will denote the operation $D^n_{\mathrm A}$
on $\tilde C^n$.
In order to simplify notations, we will write 
$[i_1|i_2|\dots |i_n]$ for $[v(i_1)|v(i_2)| \ldots | v(i_n)]\in \mathrm B_n$.

In the case of non-scalar module $M$, like $M_{(\alpha,\Delta)}$, 
it is technically easier to work with the reduced complex 
$\C^\bullet $ rather than 
with the non-reduced one $\tilde{\C}^\bullet $.

\begin{lemma}\label{lem:Reduction}
The elements of $\C^n$ are in one-to-one 
correspondence with 
scalar sequences $\alpha_{(i_1,i_2,\ldots,i_n)}$, 
$[i_1|i_2|\ldots |i_n]\in \mathrm A_n$.
\end{lemma}

\begin{proof}
Let us identify elements of type $f(\partial )u \in M_{(\alpha,\Delta )}$ with polynomials $f(\partial )\in \Bbbk [\partial ]$.

Introduce the lexicographic order $\le_{\text{lex}}$ 
on the Anick chains 
$[i_1|i_2|\ldots |i_{n-1}|i_n]\in \mathrm A_n$ 
as on the sequences of non-negative integers.
This is a well order with the smallest element
$[1|1|\ldots |1|0]$. Let us prove by induction 
that 
for every $f\in \tilde \C^n$
there exists unique sequence of polynomials 
\[
h_{(j_1,j_2,\ldots ,j_{n-1},j_n)} \in \Bbbk [\partial ], 
\quad 
[j_1|j_2|\ldots |j_{n-1}|j_n] \in \mathrm A_n,
\]
such that 
\begin{equation}\label{eq:DerivationCoChain}
f[i_1|i_2|\ldots |i_{n-1}|i_n] - 
\partial h_{(i_1,i_2,\ldots ,i_{n-1},i_n)}
-\sum\limits_{k=1}^n i_k h_{(i_1,i_2,\ldots , i_k-1 , \ldots,i_{n-1},i_n)} = \alpha_{(i_1,i_2,\ldots,i_{n-1},i_n)}
\in \Bbbk 
\end{equation}
for every 
$[i_1|i_2|\ldots |i_{n-1}|i_n]\in \mathrm A_n$.
Here we assume that 
$h_{(j_1,j_2,\ldots ,j_{n-1},j_n)}=0$
if 
$[j_1|j_2|\ldots |j_{n-1}|j_n] \notin \mathrm A_n$.
If \eqref{eq:DerivationCoChain} holds then 
$f-D^nh$ takes scalar values at $\mathrm A_n$
for 
$h\in \tilde \C^n$ given by 
\[
h[i_1|i_2|\ldots |i_{n-1}|i_n] = h_{(i_1,i_2,\ldots ,i_{n-1},i_n)}.
\]

It is easy to find $h_{(1,1,\ldots, 1,0)}$ as 
$(b(\partial )- b(0))/\partial $
for $b = f[1|1|\ldots |1|0]$.

Assume $h_{(j_1,j_2,\ldots ,j_{n-1},j_n)}$
are already constructed for all 
$[j_1|j_2|\ldots |j_{n-1}|j_n]\le_{\text{lex}}
[i_1|i_2|\ldots |i_{n-1}|i_n]$.
Then 
$h_{(i_1,i_2,\ldots ,i_{n-1},i_n)}$
is uniquely defined 
as  $(b(\partial )- b(0))/\partial $
for 
\[
b(\partial )=
f[i_1|i_2|\ldots |i_{n-1}|i_n] 
+\sum\limits_{k=1}^n i_k h_{(i_1,i_2,\ldots , i_k-1 , \ldots,i_{n-1},i_n)} .
\]
\end{proof}

\begin{theorem} For the conformal module $M_{(\alpha,1)}$ 
over $U(2)$, we have 
\[ 
\dim_\Bbbk\Homol^1(U(2), M_{(\alpha,1)})=
\begin{cases}
1 & \alpha=0 ,\\
0 & \alpha\ne0.
\end{cases}
\]
\end{theorem}

\begin{proof}
We are interested in the space
$\Homol^1(U(2),M) $ which is isomorphic 
to the space of non-coboundary cocycles in $\C^1 = \tilde \C^1/D^1\tilde \C^1$.
Suppose $\varphi \in \tilde \C^1 = \Hom_\Lambda (\A_1, M)$. 
By Lemma~\ref{lem:Reduction} we may assume 
$\varphi [n] = \alpha_n \in \Bbbk $
for $n\ge 0$.
Then the differential $\Delta^1\varphi$ takes the following 
values on the Anick 1-chains (see Figure~\ref{FIG:1}):
\[
(\Delta^1\varphi)[n|m] =\varphi(\delta_2[n|m]) = v(n)\alpha_{m}-v(0)\alpha_{n+m}-n\alpha_{n+m-1}.
\]
Hence by \eqref{eq:v(n)-Action} we have 
\[
\begin{gathered}
(\Delta^1\varphi)[1|m] =-(\partial+\alpha)\alpha_{1+m},\quad m\ge0, 
\\
(\Delta^1\varphi)[n|m]  =-(\partial+\alpha)\alpha_{n+m}-n\alpha_{n+m-1},\quad n\ge2,\ m\ge0.
\end{gathered}
\]
In order to find the constants that define $\Delta^1(\varphi +D^1\tilde \C^1) \in \C^2$
choose $\psi \in \tilde C^2$
such that 
\[
 \psi[n|m] =- \alpha_{n+m},\quad n\ge1,\ m\ge0.
\]
Then  
\[
(\Delta^1\varphi-D^2\psi)[n|m]=-\alpha\alpha_{n+m}+m\alpha_{n+m-1}
\]
for all $[n|m]\in \mathrm A_2$.
Therefore, $\varphi +D^1\tilde \C^1 $ is a 1-cocycle in $\C^1$
if and only if 
$\alpha \alpha_{n+m} = m \alpha_{n+m-1}$
for all $n\ge 1$, $m\ge 0$.
The latter is possible only in $\alpha_m=0$ for all $m\ge 1$.
Hence, 1-cocycles in $\C^1$ form a 1-dimensional space:
every 1-cocycle is determined by $\alpha _0$.

On the other hand, coboundary cocycles in $\C^1$ are given by 
$\Delta^0 (h+D^0\tilde C^0)$, where $h\in \Bbbk [\partial ]$. 
Modulo $D^0\tilde C^0$, we may assume $h(0) = \beta \in \Bbbk $, 
then 
\[
(\Delta^0h)[n] = v(n)\beta =
\begin{cases}
(\partial + \alpha)\beta , & n=0, \\
\beta , & n=1, \\
0, & n>1. \end{cases}
\]
Reduce $\partial $ modulo $D^1\tilde \C^1$: 
choose $\xi \in \tilde \C^1$ such that 
$\xi [n]=\delta_{n,0}\beta  $.
Then $(D^1\xi )[n] = \delta_{n,0}\partial \beta + \delta_{n,1}\beta $,
\[
(\Delta^0 h - D^1\xi )[n]=\delta_{n,0}\alpha\beta.
\]
As a result, the space of 1-coboundaries is 1-dimensional for 
$\alpha\ne 0$ and zero for $\alpha =0$.
Hence, 
\[ 
\dim_\Bbbk\Homol^1(U(2), M_{(\alpha,1)})=
\begin{cases}
1 & \alpha=0,\\
0 & \alpha\ne0.
\end{cases}
\]
\end{proof}

The following statement is a corollary of 
\cite{Kozlov2017} where it was proved that 
the second Hochschild cohomology group of $U(2)$
is always trivial. Let us still present its proof 
by means of our methods since the same approach 
would work later for an arbitrary $n\ge 2$.

\begin{theorem}\label{thm:Anick} The the second
cohomology group of $U(2)$ with the value in $M_{(\alpha,1)}$ is trivial, 
\[
\Homol^2(U(2),M_{(\alpha,1)})=0.
\]
\end{theorem}

\begin{proof}
 Suppose $\varphi \in \tilde \C^2$. 
The class $\varphi +D^2\tilde \C^2$ is completely 
defined by a sequence of scalars $\alpha_{(n,m)}$
by Lemma~\ref{lem:Reduction}.

The differential $\Delta^2\varphi$ takes the following 
values on the Anick 2-chains ($n,m\ge 1$, $p\ge 0$):
\begin{multline*}
(\Delta^2\varphi)[n|m|p] = \varphi (\delta_3[n|m|p])
= v(n)\alpha_{(m,p)} - n\alpha_{(n+m-1,p)}- v(0)\alpha_{(n+m,p)}\\
   + v(0)\alpha_{(n,m+p)} + n\alpha_{(n-1,m+p)}	+ m\alpha_{(n,m+p-1)}.
\end{multline*}
The summand $n\alpha_{(n-1,m+p)}$ does not appear if $n=1$ since 
$[0|m+p]$ is not an Anick chain. Hence, we have 
to consider two cases: $n=1$ and $n\ge 2$.

For all $n\ge2$, $m\ge1$, $p\ge0$, we have 
\[
    (\Delta^2\varphi)[n|m|p] 
    = - n\alpha_{(n+m-1,p)}- (\partial+\alpha)\alpha_{(n+m,p)}
   + (\partial+\alpha)\alpha_{(n,m+p)}
   + n\alpha_{(n-1,m+p)}	+ m\alpha_{(n,m+p-1)}
\]
by \eqref{eq:v(n)-Action}.
Similarly, for $n=1$, $m\ge1$, $p\ge0$ we obtain
\[
(\Delta^2\varphi)[1|m|p] =- (\partial+\alpha)\alpha_{(1+m,p)} + (\partial+\alpha)\alpha_{(1,m+p)}+ m\alpha_{(1,m+p-1)}. 
\]
Reduce the result by means of $D^3\psi $, where
$\psi \in \tilde C^3 $ is given by 
\[
\psi[n|m|p]=-\alpha_{(n+m,p)} + \alpha_{(n,m+p)}, 
 \quad n\ge1,\ m\ge1,\ p\ge0.
    \]
In both cases, we obtain the same expression for 
 $\Delta^2\varphi - D^3\psi$:
\[
  (\Delta^2\varphi-D^3\psi)[n|m|p] =
  - \alpha\alpha_{(n+m,p)} + \alpha\alpha_{(n,m+p)} 
  + m\alpha_{(n+m-1,p)} + p\alpha_{(n+m,p-1)}- p\alpha_{(n,m+p-1)}.
\]
Hence, $\varphi +D^2\tilde \C^2$ is a 2-cocycle in $\C^2$
if and only if 
\begin{equation}\label{eq:2-cocycleCond}
 - \alpha\alpha_{(n+m,p)} + \alpha\alpha_{(n,m+p)} 
  + m\alpha_{(n+m-1,p)} + p\alpha_{(n+m,p-1)}- p\alpha_{(n,m+p-1)} = 0
\end{equation}
for all $[n|m|p]\in \mathrm A_3$.

{\sc Case 1:} $\alpha \ne 0$. 
Put $p=0$ in \eqref{eq:2-cocycleCond} to obtain  
\[
 \begin{gathered}
-\alpha\alpha_{(1,m)}= - \alpha\alpha_{(1+m,0)}+m\alpha_{(m,0)}, \quad m\ge1, \\
-\alpha\alpha_{(n,m)}= - \alpha\alpha_{(n+m,0)}+m\alpha_{(n+m-1,0)}, 
\quad n\ge2,\ m\ge1.
  \end{gathered}
\]
Therefore, cocycles in $\C^2$ are determined by $\alpha_{(n,0)}$
for $n\ge 1$. 

Choose $\varphi_1\in \tilde \C^1$, $\psi_1 \in \tilde \C^2$ 
such that
    \[
    \begin{aligned}
    \varphi_1[n]&= 
    -\dfrac{\alpha_{(n,0)}}{\alpha},\quad n \ge 1.\\
    \psi_1[n|m]& = - \alpha_{n+m}, \quad n\ge1,\  m\ge0.
    \end{aligned}
    \]
Then $\Delta^1\varphi_1 $ is a coboundary in $\tilde \C^2$, and
\[
\begin{gathered}
(\Delta^1\varphi_1-D^2\psi_1)[n|0]=\alpha_{(n,0)}=\varphi[n|0], \quad n\ge1.
   \end{gathered}
\]
Hence, 
$\varphi -\Delta^1\varphi_1 \in D^2\tilde \C^2$, so 
every cocycle in $\C^2$ is a coboundary.

{\sc Case 2:} $\alpha =0$. 
 In this case, we have
\begin{equation}\label{eq:2-cocycleCond-II}
 m\alpha_{(n+m-1,p)} + p\alpha_{(n+m,p-1)}- p\alpha_{(n,m+p-1)} = 0
\end{equation}
for all $[n|m|p]\in \mathrm A_3$.

Put $p=1$ in \eqref{eq:2-cocycleCond-II} to get
   \[
   0=m\alpha_{(n+m-1,1)}+\alpha_{(n+m,0)}- \alpha_{(n,m)},
   \quad n\ge 1,\ m\ge1.
   \]
For $n=1$ and $p=0$ we simply obtain from \eqref{eq:2-cocycleCond-II}
that $\alpha_{(m,0)}=0$  for all $m\ge 1$.
So 
   \[
    \alpha_{(n,m)}=m\alpha_{(n+m-1,1)},\quad n\ge1,\ m\ge1,
    \]
i.e., cocycles in $C^2$ are determined by 
 $\alpha_{(n,1)}$, $n\ge1$.

Choose 
$\varphi_1\in\tilde \C^1$ and $\psi_1\in\tilde \C^2$ such that
    \[
    \begin{aligned}
    \varphi_1[n]&=\alpha_{(n,1)},\quad n\ge1, \\
  \psi_1[n|m]& =- \alpha_{n+m},\quad n\ge1, \ m\ge0.
    \end{aligned}
    \]
Then $\Delta^1\varphi_1 $ is a coboundary in $\tilde \C^2$, 
and
\[
(\Delta^1\varphi_1-D^2\psi_1)[n|1]=\alpha_{(n,1)},
\quad n\ge1.
\]
Hence, every 2-cocycle is a coboundary.
\end{proof} 

 
\begin{theorem}\label{the5.5}
 For a conformal module $M_{(\alpha,1)}$ over the 
associative conformal algebra $U(2)$, and for all $n\ge3$  we have
 \[
 \Homol^{n-1}(U(2),M_{(\alpha,1)})=0 .
 \]
\end{theorem}
 
 \begin{proof}
Suppose $\varphi \in \tilde \C^{n-1} = \Hom_\Lambda (\A_{n-1}, M)$, $M=M_{(\alpha, 1)}$. 
As above, let us identify $f(\partial )u\in M$
and $f\in \Bbbk [\partial ]$.
Suppose
$\varphi[i_1|i_2|\ldots |i_{n-1}] = f_{(i_1,i_2,\ldots,i_{n-1})}(\partial) \in \Bbbk [\partial]$ for $[i_1|i_2|\ldots |i_{n-1}]\in \mathrm A_{n-1}$.
It is convenient to assume that 
$f_{(i_1,i_2,\ldots,i_{n-1})}(\partial)=0$
if 
$[i_1|i_2|\ldots |i_{n-1}]\notin \mathrm A_{n-1}$.

Recall that the operation $D^{n-1}$
acts on $\tilde \C^{n-1}$
as 
\[
(D^{n-1}\varphi )[i_1|i_2|\ldots|i_{n-1}] = \partial f_{(i_1,i_2,\ldots,i_{n-1})} + i_1f_{(i_1-1,i_2,\ldots,i_{n-1})}+\ldots+i_{n-1}f_{(i_1,i_2,\ldots,i_{n-1}-1)}. 
\]
By Lemma~\ref{lem:Reduction}, the elements of $\C^{n-1}=\tilde \C^{n-1}/D^{n-1}\tilde \C^{n-1}$ 
are defined by 
scalar sequences 
$\alpha_{(i_1,i_2,\ldots,i_{n-1})}$.

The differential $\Delta^{n-1}$ takes the following
values on the Anick $(n-1)$-chains:
\[
\begin{aligned}
	&(\Delta^{n-1}\varphi)[1|i_2|\ldots|i_{n-1}|i_n]=\alpha_{(i_2,i_3,\ldots,i_{n-1})}\\
	&+\sum\limits_{j=1}^{n-1}(-1)^ji_j\alpha_{(1,i_2,\ldots,i_j+i_{j+1}-1,\ldots,i_{n})} +\\ &+\sum\limits_{j=1}^{n-1}(-1)^j(\partial+\alpha)\alpha_{(1,i_2,\ldots,i_j+i_{j+1},\ldots,i_n)} + \\
	&+\sum\limits_{j=2}^{n-1}\sum\limits_{k=1}^{j-1}(-1)^ji_k\alpha_{(1,\ldots,i_k-1,\ldots,i_j+i_{j+1},\ldots,i_n)},
 \quad i_2,\ldots,i_{n-1}\ge1, \ i_n\ge0,
	\end{aligned}
\]
\[
\begin{aligned}
	&(\Delta^{n-1}\varphi)[i_1|i_2|\ldots|i_{n-1}|i_n]=
	\sum\limits_{j=1}^{n-1}(-1)^ji_j\alpha_{(i_1,i_2,\ldots,i_j+i_{j+1}-1,\ldots,i_{n})} \\ &+\sum\limits_{j=1}^{n-1}(-1)^j(\partial+\alpha)\alpha_{(i_1,i_2,\ldots,i_j+i_{j+1},\ldots,i_n)}\\
	&+\sum\limits_{j=2}^{n-1}\sum\limits_{k=1}^{j-1}(-1)^ji_k\alpha_{(i_1,\ldots,i_k-1,\ldots,i_j+i_{j+1},\ldots,i_n)},
 \quad i_1\ge2,\ i_2,\ldots,i_{n-1}\ge1,\ i_n\ge0.
	\end{aligned}
\]
 Let us reduce 
 $\Delta^{n-1}\varphi $
 by means of $D^n\psi $ for 
 $\psi \in \tilde C^{n} $
 given by
\[
\psi[i_1|i_2|\ldots|i_{n}]
=\sum\limits_{j=1}^{n-1}(-1)^j\alpha_{(i_1,i_2,\ldots,i_j+i_{j+1},\ldots,i_n)}.
\]
Namely, 
\begin{align}
\nonumber
  (\Delta^{n-1}\varphi-D^n\psi)[i_1|i_2|\ldots|i_{n}]&
  {} =\sum\limits_{j=1}^{n-1}(-1)^{j}\alpha\alpha_{(i_1,i_2,\ldots,i_j+i_{j+1},\ldots,i_n)}\\
 &
\nonumber
 +\sum_{j=3}^{n}\sum_{k=1}^{j-2}(-1)^{k}i_j\alpha_{(i_1,\ldots,i_k+i_{k+1},\ldots,i_j-1,\ldots,i_n)}\\
 &
+\sum^{n-1}_{j=1}(-1)^{j+1}i_{j+1}\alpha_{(i_1,\ldots, i_j+i_{j+1}-1,\ldots ,i_n)} .
\label{eq:Cocycle-n-1}
\end{align}
Suppose $\varphi + D^{n-1}\tilde \C^{n-1} $
is a cocycle in $\C^{n-1}$. 
Then the right-hand side of 
\eqref{eq:Cocycle-n-1} is zero for every 
$[i_1|i_2|\ldots |i_n]\in \mathrm A_n$.

{\sc Case 1:} $\alpha\ne 0$.
Then 
for $i_1,i_2,\ldots,i_{n-1}\ge1$, $i_n=0$
relation \eqref{eq:Cocycle-n-1} implies
\begin{multline}\label{eq:5.4}
 (-1)^{n}\alpha\alpha_{(i_1,i_2,,\ldots,i_{n-1})} =\sum\limits_{j=1}^{n-2}(-1)^{j}\alpha\alpha_{(i_1,i_2,\ldots,i_j+i_{j+1},\ldots,i_{n-1},0)}\\
 +\sum_{j=3}^{n-1}\sum_{k=1}^{j-1}(-1)^{k+1}i_j\alpha_{(i_1,\ldots,i_k+i_{k+1},\ldots,i_j-1,\ldots,i_{n-1},0)}\\
  +\sum^{n-2}_{j=1}(-1)^{j+1}i_{j+1}\alpha_{(i_1,\ldots,
  i_j+i_{j+1}-1,\ldots ,i_{n-1},0)}
\end{multline}
Hence, the entire sequence 
$\alpha_{(i_1,i_2,\ldots,i_{n-1})}$
is determined by $\alpha_{(i_1,i_2,,\ldots,i_{n-2},0)}$ for $i_1,i_2,\ldots,i_{n-1}\ge 1$.

Choose $\varphi_1\in \tilde C^{n-2}$, 
$\psi_1\in \tilde \C^{n-1}$ 
such that
\[
\varphi_1[i_1|i_2|\ldots |i_{n-2}]
 =\beta_{(i_1,i_2,\ldots,i_{n-2})},
 \quad 
 \psi_1[i_1|i_2|\ldots|i_{n-1}] 
 =\sum\limits_{j=1}^{n-2}(-1)^{j}
  \beta_{(i_1,i_2,\ldots,i_j+i_{j+1},\ldots,i_{n-1})},
\]
where 
\[
\beta_{(i_1,i_2,\ldots,i_{n-2})}  
=
\begin{cases}
(-1)^{n}\dfrac{\alpha_{(i_1,i_2,\ldots,i_{n-2},0)}}{\alpha}, & i_1,i_2,\ldots,i_{n-1}\ge1, \\
0,& \text{otherwise}.
\end{cases}
\]
Then $\Delta^{n-2}\varphi_1-D^{n-1}\psi_1$
represents a coboundary in $C^{n-1}$, and
\[
\begin{aligned}
  (\Delta^{n-2}\varphi_1-D^{n-1}\psi_1)
   [i_1|i_2|\ldots |i_{n-1}] 
   & 
   =\sum\limits_{j=1}^{n-2}(-1)^{j}\alpha
      \beta_{(i_1,i_2,\ldots,i_j+i_{j+1},\ldots,i_{n-1})}
      \\
 &+\sum_{j=3}^{n-1}\sum_{k=1}^{j-2}(-1)^{k}i_j
  \beta_{(i_1,\ldots,i_k+i_{k+1},\ldots,i_j-1,\ldots,i_{n-1})}
  \\
     &+\sum^{n-2}_{j=1}(-1)^{j+1}i_{j+1}
 \beta_{(i_1,\ldots ,i_j+i_{j+1}-1,\ldots ,i_{n-1})}.
\end{aligned}
\]
For all $i_1,i_2,\ldots,i_{n-1}\ge1$, we get
\[
(\Delta^{n-2}\varphi_1-D^{n-1}\psi_1)
[i_1|i_2|\ldots |i_{n-2}|0]
 =\alpha_{(i_1,i_2,\ldots,i_{n-2},0)}.
\]
Hence, the element 
$\varphi +D^{n-1}\tilde \C^{n-1}$
is a coboundary in $\C^{n-1}$.

{\sc Case 2:} $\alpha =0$. Then \eqref{eq:Cocycle-n-1} implies
\begin{equation}\label{eq:5.3-prime}
\sum_{j=3}^{n}\sum_{k=1}^{j-2}(-1)^{k}i_j\alpha_{(i_1,\ldots,i_k+i_{k+1},\ldots,i_j-1,\ldots,i_n)}
+
\sum^{n-1}_{j=1}(-1)^{j+1}i_{j+1}\alpha_{(i_1,\ldots ,i_j+i_{j+1}-1,\ldots, i_n)}  = 0.
\end{equation}
For $i_n=1$, we obtain
\begin{align}
\nonumber
(-1)^n\alpha_{(i_1,i_2\ldots,i_{n-1})}&=\sum_{j=3}^{n-1}\sum_{k=1}^{j-2}(-1)^{k}i_j\alpha_{(i_1,\ldots,i_k+i_{k+1},\ldots,i_j-1,\ldots,i_{n-1},1)}\\
\nonumber
&+\sum_{k=1}^{n-2}(-1)^{k}i_j\alpha_{(i_1,\ldots,i_k+i_{k+1},\ldots,i_j-1,\ldots,i_{n-1},0)}\\
\label{eq:Cocycle-all-0,1}
 &+\sum^{n-2}_{j=1}(-1)^{j+1}i_{j+1}\alpha_{(i_1,\ldots ,i_j+i_{j+1}-1,\ldots ,i_{n-1},1)},
 \end{align}
and for $i_n=0$, $i_{n-1}=1$ it follows from \eqref{eq:5.3-prime} that
\begin{align}
\nonumber
  (-1)^{n-1}\alpha_{(i_1,i_2,\ldots,i_{n-2},0)}&=\sum_{j=3}^{n-1}\sum_{k=1}^{j-2}(-1)^{k}i_j\alpha_{(i_1,\ldots,i_k+i_{k+1},\ldots,i_j-1,\ldots,i_{n-2},1,0)}\\
  \label{eq:Cocycle-all-0}
     &+\sum^{n-2}_{j=1}(-1)^{j+1}i_{j+1}\alpha_{(i_1,\ldots ,i_j+i_{j+1}-1,\ldots ,i_{n-2},1,0)}.
\end{align}
Choose 
$\varphi_1\in \tilde \C^{n-2}$, 
$\psi_1\in \tilde \C^{n-1}$ in such a way that
\begin{equation}\label{eq:RedConstr}
\varphi_1[i_1|i_2|\ldots|i_{n-2}]=\beta_{(i_1,i_2,\ldots,i_{n-2})},
\quad 
\psi_1[i_1|i_2|\ldots|i_{n-1}]
=\sum\limits_{j=1}^{n-2}(-1)^{j}\beta_{(i_1,i_2,\ldots,i_j+i_{j+1},\ldots,i_{n-1})} ,
\end{equation}
where 
\[
\beta_{(i_1,i_2,\ldots,i_{n-3},0)}=
\begin{cases}
(-1)^{n-1}\alpha_{(i_1,i_2,\ldots,i_{n-3},1,0)},
   & i_1,i_2,\ldots,i_{n-3}\ge1,\ i_{n-2}=0, \\
0, & \text{otherwise}.
\end{cases}
\]
Then $\Delta^{n-2}\varphi_1-D^{n-1}\psi_1$ represents a coboundary in $C^{n-1}$, 
and
\[
\begin{aligned}
  (\Delta^{n-2}\varphi_1-D^{n-1}\psi_1)[i_1|i_2|\ldots |i_{n-1}]
  =
 &\sum_{j=3}^{n-1}\sum_{k=1}^{j-2}(-1)^{k}i_j\beta_{(i_1,\ldots,i_k+i_{k+1},\ldots,i_j-1,\ldots,i_{n-1})}\\
     &+\sum^{n-2}_{j=1}(-1)^{j+1}i_{j+1}\beta_{(i_1,\ldots ,i_j+i_{j+1}-1,\ldots ,i_{n-1})} .
\end{aligned}
\]
Hence,
\[
  (\Delta^{n-2}\varphi_1-D^{n-1}\psi_1)[i_1|i_2|\ldots|i_{n-3}|1|0]=\alpha_{(i_1,i_2,\ldots,i_{n-3},1,0)}=\varphi[i_1|i_2|\ldots|i_{n-3}|1|0].
\]
Therefore, we may assume that 
all $\alpha_{(i_1,i_2,\ldots,i_{n-3},1,0)}=0$. 
Relation \eqref{eq:Cocycle-all-0} implies 
$\alpha_{(i_1,i_2,\ldots,i_{n-2},0)}=0$  
for all $i_1,i_2,\ldots, i_{n-2}\ge 1$.

Let us repeat the construction \eqref{eq:RedConstr} 
with new values of $\beta$'s to get  
$\varphi'_1$ and $\psi'_1$ for 
\[
\beta{(i_1,i_2,\ldots,i_{n-2})}=
\begin{cases}
(-1)^n\alpha_{(i_1,i_2,\ldots,i_{n-2},1)},& i_1,i_2,\ldots,i_{n-2}\ge1, \\
0,& \text{otherwise}.
\end{cases}
\]
Then 
for all $i_1,\ldots,i_{n-2}\ge 1$ we get
\[
 (\Delta^{n-2}\varphi'_1-D^{n-1}\psi'_1)
   [i_1|i_2|\ldots|i_{n-3}|1|0]
   =0
\] 
and 
\[
 (\Delta^{n-2}\varphi'_1-D^{n-1}\psi'_1)
   [i_1|i_2|\ldots|i_{n-2}|1]
   =\alpha_{i_1,i_2,\ldots,i_{n-2},1}.
\]
Therefore, the cocycle $\varphi + D^{n-1}\tilde \C^{n-1}$
is a coboundary.
\end{proof}


\begin{corollary}\label{cor:AllFinite}
Let $M$ be a finite module over Weyl associative conformal algebra $U(2)$. Then $\Homol^k(U(2),M)=0$ for all $k\ge 2$.
\end{corollary}

\begin{proof}
Let $M$ be a finite conformal module over $U(2)$.
Then, in particular, $M$ is a finite module over the Virasoro Lie conformal algebra $\mathop {\fam 0 Vir} \nolimits $. 
Hence there exists a chain of $\mathop {\fam 0 Vir} \nolimits$-submodules
 (see, e.g., \cite[Lemma 3.3]{kolesnikov})
 \[
 0= M_{-1} \subset M_0 \subset \ldots \subset M_n=M,
 \]
where $M_i/M_{i-1}$,  $i=0,\ldots,n$, is 
either isomorphic to a $\mathop {\fam 0 Vir} \nolimits $-module $M_{(\alpha,\Delta)}$,
or trivial torsion-free module $\Bbbk [\partial ]u$ with 
$(v\oo\lambda u)=0$, 
or coincides with its torsion (hence, trivial).
Note that a $\mathop {\fam 0 Vir} \nolimits$-submodule of an $U(2)$-module $M$ is itself a 
$U(2)$-module, therefore, all $M_i$ are $U(2)$-modules and 
so are $M_i/M_{i-1}$. Hence, all irreducible quotients 
are of type $M_{(\alpha, 1)}$.

The case of torsion module was considered \cite{AlKol-JMP}. 
There is no difference between the scalar $U(2)$-module 
$\Bbbk $ 
and the trivial torsion-free module $M=\Bbbk [\partial ]u$. 
One may also apply the technique 
of Theorem~\ref{the5.5} to the case of trivial module $M$ as above 
to prove that $\Homol^k(U(2),M)=0$ for all $k\ge 2$.
Finally, Theorem~\ref{the5.5} implies
that
 \[
\Homol^k(U(2), M_i/M_{i-1} )=0, \quad i=0,\dots, n, 
\]
for all $k\ge 2$.
The short exact sequence of modules 
\[
0\to M_{i-1} \to M_i \to 
M_i/M_{i-1} \to 0
\]
leads to the long exact sequence of cohomology groups
\[
\begin{aligned}
\dots & \to \Homol^k(U(2),M_{i-1})
        \to \Homol^k(U(2),M_i)
        \to \Homol^k (U(2), {M_i/M_{i-1}}) \\
& \to \Homol^{k+1}(U(2),M_{i-1}) 
  \to \Homol^{k+1}(U(2),M_i)
        \to \Homol^{k+1} (U(2), {M_i/M_{i-1}}) \\
&        \to \dots .
\end{aligned}
\]
for every $i=1,\ldots,n$. 
Since  
\[
\Homol^k(U(2),M_0)=0,\quad \Homol^k(U(2),{M_1/M_0})=0
\]
for all  $k\ge2$, we obtain $\Homol^k(U(2), M_1)=0$.
Proceed by induction on $i=1,\dots, n$ to obtain
$\Homol^k(U(2),M_n)=0$, for all $k\ge2$.
\end{proof}

The series of modules $M_{(\alpha, \Delta)}$ also includes 
non-irreducible modules with $\Delta = 0$. 
For example, the module $M_{(0,0)}=\Bbbk [\partial ]u$
is defined by $(v\oo\lambda u) = \partial u$, 
this representation corresponds to the embedding 
of $\mathop {\fam 0 Vir} \nolimits $ into $\mathop {\fam 0 Cend} \nolimits_{x,1}$ \cite{BKL}.

\begin{corollary}
For all $n\ge 2$ we have     
$\Homol^n (U(2),M_{(\alpha, 0)}) = 0$.
\end{corollary}

Indeed, $M_{(\alpha,0)}$ is a finite module. The chain of submodules 
mentioned in the proof of Corollary~\ref{cor:AllFinite}
is given by 
\[
0=M_{-1} \subset (\partial+\alpha)M=M_0 \subset M=M_1, 
\]
where 
$M_0/M_{-1}\simeq M_{(\alpha, 1)}$, $M_1/M_0\simeq \Bbbk $.







