Moduli of representations of one-point extensions

Arif Dönmez
<EMAIL_ADDRESS><EMAIL_ADDRESS>IUF – Leibniz Research Institute for Environmental Medicine, Düsseldorf, Germany

Markus Reineke
Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany

We study moduli spaces of (semi-)stable representations of one-point extensions of quivers by rigid representations. This class of moduli spaces unifies Grassmannians of subrepresentations of rigid representations and moduli spaces of representations of generalized Kronecker quivers. With homological methods, we find numerical criteria for non-emptiness and results on basic geometric properties, construct generating semi-invariants, expand the Gel'fand MacPherson correspondence, and derive a formula for the Poincaré polynomial in singular cohomology of these moduli spaces.

§ INTRODUCTION

In this paper we construct and study moduli spaces parametrizing isomorphism classes of representations of so-called one-point extensions of path algebras of quivers. This constitutes a class of algebras of global dimension two, for which many of the favourable properties of moduli spaces of representations of quivers still hold. Namely, we find numerical criteria for non-emptiness and results on basic geometric properties, construct generating semi-invariants, expand the Gel'fand MacPherson correspondence, and derive a formula for the Poincaré polynomial in singular cohomology of these moduli spaces. We explicitly apply the developed theory in several examples.

To do this, we fix a path algebra $A=kQ$ of a finite quiver, which we extend by a representation $T$ of $A$ to the one-point extension algebra $A[T]$.

We construct standard projective resolutions for representations of $A[T]$. One of the most important consequences of this is an explicit description of the space $\Ext^2$ of representations, which allows us to conclude its vanishing on so-called full representations (under the assumption of $T$ being rigid). See Theorems <ref>, <ref>, <ref>, <ref> for precise formulations. Moreover, the standard resolutions allow us to calculate the Euler form of $A[T]$ in Theorem <ref>, Corollary <ref>, <ref>.
After these preparations, we consider the representation varieties of $A[T]$, interpret the found homological properties in this geometric setting, and rediscover some results by Schofield and  Crawley-Boevey (with different methods) in Theorem <ref>, Corollary <ref>, <ref>. Moreover, in this way we can determine the  Zariski tangent space of the representation variety in each point (Theorem <ref>) and conclude that the open subset of full representations is smooth and irreducible (Theorem <ref>).

We follow the GIT approach of King in the construction of moduli spaces. For this, we choose a canonical stability condition, such that the resulting spaces unify quiver Grassmannians of subrepresentations of rigid representations (Theorem <ref>) and moduli spaces of representations of generalized Kronecker quivers. We find a numerical criterion for semistability in Theorem <ref>, which allows us to conclude that semi-stable representations are full representations (Corollary <ref>). In this way, we can apply the above geometric properties of representation varieties to prove that the resulting moduli spaces are smooth, irreducible and of expected dimension in Theorem <ref>.

After this, we prove a relative version of a recursive numerical criterion ([9]) for non-emptiness of the semi-stable locus in Theorems <ref>, <ref>. A set of generators for the ring of semi-invariants, closely following  Schofield and  Van den Bergh ([13]) is given in section $6$.

Moreover, we find a form of Gel'fand MacPherson correspondence in terms of these moduli spaces in  Theorem <ref>. We finish the paper by deriving a recursive formula to determine the  Poincaré polynomial of the moduli spaces  (Theorem <ref>).

§.§ Acknowledgements

The authors are supported by the DFG SFB / Transregio 191 `Symplektische Strukturen in Geometrie, Algebra und Dynamik'.

§ ONE-POINT EXTENSIONS AND THEIR REPRESENTATIONS
$ $

We fix an algebraically closed field $k$. Let $A=kQ$ be the path algebra of a finite quiver $Q$, and let $T$ be a  finite dimensional $A$-module. We consider the one-point extension of $A$ by $T$

\[
A[T] ~:=~\triAlg{A}{k}{T}
\left\lbrace
\begin{pmatrix}
a & t \\
0 & \lambda
\end{pmatrix}\;:\; a\in A,\,t\in T,\, \lambda\in k\right\rbrace.
\]

Recall that the multiplication in (the $k$-algebra) $A[T]$ is given by the formal matrix multiplication

\[
\begin{pmatrix}
a & t \\
0 & \lambda
\end{pmatrix} \cdot
\begin{pmatrix}
a' & t' \\
0 & \lambda'
\end{pmatrix}
\begin{pmatrix}
aa' & at'+t\lambda' \\
0 & \lambda \lambda'
\end{pmatrix}
\]

for $a,a'\in A,\, \lambda,\lambda'\in k,\,t,t'\in T$ and componentwise addition.

We define a category $\Rep_A(T\otimes_k ?_2 \to ?_1)$ as follows. Its objects $M$ are tuples $(M_1,M_2)$, where $M_1$ is a left $A$-module and $M_2$ is a $k$-vector space, together with a map of $A$-modules $f_M:T \otimes_k M_2 \to M_1$. A morphism $\varphi: M\to N$ is a tuple $(\varphi_1,\varphi_2)$, where $\varphi_1:M_1\to N_1$ is a map of $A$-modules and $\varphi_2:M_2\to N_2$ a $k$-linear map, such that the diagram

\[
\begin{tikzcd}
T\otimes_k M_2 \arrow[r, "f_M"] \arrow[d, "\varphi_1"]  & M_1 \arrow[d, "\varphi_2"] \\
T\otimes_k N_2 \arrow[r, "f_N"] & N_1
\end{tikzcd}
\]

commutes. Composition of morphisms is defined componentwise.

The category $\Rep_A(T\otimes_k ?_2 \to ?_1)$ is equivalent to the category of left $A[T]$-modules.

See for example [11].

We can easily describe a quiver $\hat{Q}$ and relations $\Rel$ in this situation such that $A[T]\simeq k\hat{Q}/\langle \Rel \rangle$. By extending $Q$ the following way

\begin{eqnarray*}
\hat{Q}_0 & = &  Q_0 \sqcup \{\infty \},  \\
\hat{Q}_1  & =  & Q_1 ~ \sqcup ~ \bigg\{ \rho_{l, (i)}:\infty \to i \;:\; i\in Q_0,\,l=1,\ldots, \dim T_i \bigg\},
\end{eqnarray*}

we obtain the quiver $\hat{Q}$. We obtain the relations $\Rel$ by using the transformation matrices induced by the fixed representation $T$: For each vertex $i\in Q_0$ we choose a $k$-basis $\rho_{1,(i)}, \ldots, \rho_{\dim T_i,(i)}$ of $T_i$ and write

T(\alpha)\rho_{l,(i)} = \sum_{s=1}^{\dim T_j}\lambda_{s,l}^{(\alpha)}\rho_{s,(j)}

for all $l=1,\ldots, \dim T_i$ and each arrow $(\alpha:i\to j)\in Q_1$.
The coefficients of these linear combinations induce the relations $\Rel$

\begin{equation} \label{eq:relationGabriel}
\alpha \cdot \rho_{l,(i)} ~=~ \sum_{s=1}^{\dim T_j}\lambda_{s,l}^{(\alpha)}\rho_{s,(j)}.
\end{equation}

Obviously this construction of relations does not depend (up to isomorphism of $k$-algebras) on the basis we choose in $T_i$ for each vertex $i\in Q_0$.

Let $Q=\left(1\to 2\right)$.

By extending the path algebra of $Q$ with $k^3 \xto{ [1~0~0] }k$ we get

\begin{tikzcd}
1	 \arrow{r}{m} &2  \\
0	 \arrow[bend left=50]{u}{\beta} \arrow{u}{\alpha} \arrow[swap,bend right=50]{u}{\gamma}   &
\end{tikzcd} \text{~~with $m \beta =  0,\;m \gamma =  0$.}

§ HOMOLOGICAL PROPERTIES

Since $A$ is the path algebra of a quiver $Q$, we can and will identify $A$ with the tensor algebra  $\mathsf{T}_RX$, where $R$ is the semisimple $k$-algebra generated by the vertices of $Q$ and $X$ is the $R$-$R$-bimodule generated (as a $k$-vector space) by the arrows of $Q$.

§.§ The standard resolution

Let $\{e_1,\ldots,e_n\}$ be the complete set of primitive orthogonal idempotents given by the length $0$ paths in $A$. Then

\[
\begin{pmatrix} e_1 & 0 \\ 0 & 0 \end{pmatrix},
\ldots,
\begin{pmatrix} e_n & 0 \\ 0& 0 \end{pmatrix},
\begin{pmatrix} 0 & 0 \\ 0& 1 \end{pmatrix}
\]

is a complete set of primitive orthogonal idempotents of $A[T]$.
Let $\tilde{R}$ be the $k$-subalgebra of $A[T]$ given by

\[
\tilde{R} ~~ := ~~
\bop_{i=1}^nk\begin{pmatrix} e_i & 0 \\ 0 & 0 \end{pmatrix}\;\; \op \;\;
k\begin{pmatrix} 0 & 0 \\ 0& 1 \end{pmatrix}.
\]

The multiplication of $A$ resp. $A[T]$ induces the Eilenberg sequence <cit.>

\[
\mathfrak{E}(A): 0\to \Omega(A) \xto{\kappa_A} A\otimes_R A \xto{\mu_A} A \to 0,
\]

\[
\mathfrak{E}(A[T]): 0\to \Omega(A[T]) \to A[T]\otimes_{\tilde{R}} A[T]\xto{\mu_{A[T]}} A[T] \to 0,
\]

with $\Omega(A):=\ker{\mu_A}$ resp. $\Omega(A[T]):=\ker{\mu_{A[T]}}$.

This sequence of

$A$-bimodules resp. $A[T]$-bimodules splits in the category of right $A$-modules resp. $A[T]$-modules.

Let $\tilde{M}=(M,V,f:T\otimes_k V\to M)$ be an $A[T]$-module. Then there is  a short exact sequence

\[
0\to \Omega(A[T])\otimes_{A[T]}\tilde{M}\to A[T]\otimes_{\tilde{R}}\tilde{M}\to\tilde{M}\to 0,
\]

\begin{eqnarray*}
A[T]\otimes_{\tilde{R}}\tilde{M} & = &
\bigg(T\otimes_k V,\,V,\;T\otimes_k V \xto\id T\otimes_k V\bigg)\;\op\\ &
&\quad \big(A\otimes_R M,0, {0}\big),   \\& &\\
\Omega(A[T])\otimes_{A[T]}\tilde{M} & = &
\bigg( T\otimes_k V,\,0,\,{0} \bigg) \op \bigg( \Omega(A)\otimes_A M,\,0,\,{0} \bigg).
\end{eqnarray*}

We obtain the short exact sequence by tensoring the split short exact sequence of right  $A[T]$-modules  $\mathfrak{E}(A[T])$ over $A[T]$ with $\tilde{M}$.

The first equation follows immediately using the definition of $\tilde{R}$. Namely, for $M_0:=V$ we have

\[
A[T]\tilde{e}_0   =  \begin{pmatrix} 0 & T\\ 0 & k \end{pmatrix}
\]

and $A[T]\otimes_{\tilde{R}}\tilde{M} =\bop_{i=0}^n A[T]\tilde{e}_i\otimes_k M_i$.

The second equation follows from the following commutative diagram with exact rows:

\[
\begin{tikzcd}[scale cd=.98, column sep=small]
0 \arrow[r] & 0 \arrow[r]\arrow[d] & T\otimes_k V \arrow[r, "\id"]\arrow[d, "(\id\;\; 0)^\top"] & T\otimes_k V \arrow[d, "f"]  \arrow[r] & 0 \\
0 \arrow[r] & (T\otimes_k V) \op \big(\Omega(A)\otimes_A M\big) \arrow[r, "g"] & (T\otimes_k V) \arrow[r, "(f\;\;\mu_M)" ]\op (A\otimes_R M) & M\arrow[r] & 0,
\end{tikzcd}
\]

\[
\mathfrak{E}(A)\otimes_A M: 0\to \Omega(A)\otimes_A M \xto{\kappa_M} A\otimes_R M\xto{\mu_M} M \to 0,
\]

and $g$ is defined as

\[
g\bigg((t\otimes v) + (\omega\otimes m) \bigg)~~:=~~
t\otimes v ~~- 1\otimes f(t\otimes v) + \kappa_M(\omega\otimes m)
\]

for $t\otimes v\in T\otimes_k V,\;\,\omega\otimes m\in \Omega(A)\otimes_A M$.

To simplify the notation, set $\Omega_A=\Omega(A)$ and $N=\Omega_A\otimes_A M$. Combining Theorem <ref> with the standard resolution

$$0\rightarrow \Omega_A\otimes_A X\rightarrow A\otimes X\rightarrow X\rightarrow 0$$

of $A$-modules, we obtain the following long exact sequence:

\[
\begin{tikzcd}[column sep=tiny, row sep=small]
0 \arrow[r] & \Omega_A\otimes_A (T\otimes_k V) \arrow[r] &\big(A\otimes_R (T\otimes_k V)\big)\op N \arrow[dr]\arrow[r]  &  A[T]\otimes_{\tilde{R}}\tilde{M} \arrow[r] & \tilde{M}\arrow[r] & 0. \\
& & & \big(T\otimes_k V\big)\;\op\;N\arrow[u] \arrow[dr]	&                                            &  \\
& &	&   0\arrow[u] &0 & \\
\end{tikzcd}
\]

Using this construction, we obtain:
For $A[T]$-modules $\tilde{M}=(M,V,f:T\otimes_k V \to M)$ we have the standard projective resolution:

\[
\begin{tikzcd}
0 \arrow[d]   & 0 \arrow[d] &  0 \arrow[d]  \\
P_2(\tilde{M})\arrow[d] :=  & 0 \arrow[d]\arrow[r] & (A\otimes_R X\otimes_R A)\otimes_A(T\otimes V) \arrow[d, "h"]\\
P_1(\tilde{M})\arrow[d] :=  & 0 \arrow[d]\arrow[r] & \big(A\otimes_R (T\otimes V)\big)\op \big( (A\otimes_RX\otimes A)\otimes_A M\big)\arrow[d, "g"] \\
P_0(\tilde{M})\arrow[d] :=  & T\otimes V \arrow[r, "(\id~~ 0)^\top"]\arrow[d, "\id"]  & (T\otimes V) \arrow[d, "(f\;\;\,\mu_M)" ]\op (A\otimes_R M) \\
~~~\tilde{M}\arrow[d]= & T\otimes V \arrow[r, "f"] \arrow[d]& M\arrow[d] \\
0 & 0  & 0.
\end{tikzcd}
\]

Here $g$ and $h$ are defined as

\begin{eqnarray*}
g\left(a\otimes (t\otimes v)+  a\otimes x \otimes  m \right) & = & at\otimes v - a\otimes f(t\otimes v)+ ax\otimes m -a\otimes xm,\\
h\left( a\otimes x \otimes (t\otimes v) \right) & = & \left( ax\otimes (t\otimes v)- a\otimes (xt\otimes v) \right)+a\otimes x \otimes f(t\otimes v),
\end{eqnarray*}
$ $

for $a\otimes (t\otimes v) \in A\otimes_R(T\otimes_k V), \,a\otimes x \otimes  m\in  (A\otimes_R X\otimes_R A)\otimes_A M$ and
$a\otimes x \otimes (t\otimes v) \in (A\otimes_R X\otimes_R A)\otimes_A(T\otimes_k V)$.

In particular, we have ${\rm gldim}\; A[T]\leq 2$.

§.§ Characterizing Ext2
$ $

Let $\tilde{M}=\big(M,V,f:T\otimes_kV\to M\big), \tilde{N}=\big(N,W,g:T\otimes_kW\to N\big)$ be $A[T]$-modules.

By using the Eilenberg sequence we obtain the following commutative diagram with exact rows and columns

\[
\begin{tikzcd}[row sep=large]
& 0 \arrow[d, dotted] &  0 \arrow[d, dotted] & 0 \arrow[d, dotted] & \\
0 \arrow[r]& \Omega(A)\otimes_A \ker{f}  \arrow[r, "\kappa_{\ker{f}}"] \arrow[d, dotted] & A\otimes_R \ker{f} \arrow[r, "\mu_{\ker{f}}"] \arrow[d, dotted] & \ker{f} \arrow[r] \arrow[d, dotted]& 0, \\
0 \arrow[r]& \Omega(A)\otimes_A (T\otimes_k V)  \arrow[r, "\kappa_{T\otimes_k V}"] \arrow[d, dotted]& A\otimes_R (T\otimes_k V) \arrow[r, "\mu_{ (T\otimes_k V)  }"] \arrow[d, dotted] & T\otimes_k V \arrow[r]\arrow[d, dotted, "f'"]& 0, \\
0 \arrow[r]& \Omega(A)\otimes_A \im{f}  \arrow[r, "\kappa_{\im{f}}"] \arrow[d, dotted] & A\otimes_R \im{f} \arrow[r, "\mu_{\im{f}}"] \arrow[d, dotted] & \im{f} \arrow[r] \arrow[d, dotted] & 0.\\
& 0 & 0  & 0 &
\end{tikzcd}
\]

Note that, since $R$ is a semisimple $k$-algebra, $\Omega(A)$ and $\Omega_A\otimes_A L$ are projective modules for all $A$-modules $L$.

Applying ${\rm Hom}_A(?,N)$, we obtain the following commutative diagram

\[
\begin{tikzcd}[row sep=large, column sep=small]
\Hom_R(\im{f},N) \arrow[r]\arrow[d] &  \Hom_A(\Omega_A\otimes_A \im{f}, N) \arrow[d,"(\Omega_A\otimes f')^*"]\arrow[r] & \Ext_A(\im{f},N) \arrow[d]\arrow[r] & 0   \\
\Hom_R(T\otimes_k V, N) \arrow[d]\arrow[r,"\kappa_{T\otimes_k V}^*"] & \Hom_A(\Omega_A\otimes_A (T\otimes_k V), N) \arrow[d]\arrow[r] & \Ext_A(T\otimes_k V, N) \arrow[d]\arrow[r] & 0\\
\Hom_R(\ker{f},N) \arrow[r, "\overline{\kappa_{T\otimes_k V}^*}"]\arrow[d] &  \Hom_A(\Omega_A\otimes_A \ker{f}, N) \arrow[r]\arrow[d] & \Ext_A(\ker{f},N)\arrow[r]& 0. \\
0                         & 0                                                               &
\end{tikzcd}
\]

Now we consider the standard projective resolution of $\tilde{M}$ of Theorem <ref>

\[
P_\bullet(\tilde{M}) \to \tilde{M} \to 0
\]

and the induced cochain complex $\mathsf{C}:=\Hom_B\big(P_\bullet(\tilde{M}),\tilde{N}\big)$:

\[
\mathsf{C}:\ldots \to 0 \to \Hom_{\tilde{R}}(\tilde{M},\tilde{N}) \to \Hom_A\big(\Omega_A\otimes_A M,N\big)\oplus \Hom_R\big(T\otimes_k V,N\big)
\]

\[
\xto{h^*}\Hom_A\big(\Omega_A\otimes_A (T\otimes_k V),N\big)\to 0 \to ...
\]

Then $H^i(\mathsf{C})=\Ext_{A[T]}^i(\tilde{M},\tilde{N})$ for $i=0,1,2$.

We have

\begin{eqnarray*}
h^* & = &  \Hom_{A[T]}(h,\tilde{N})\\
& = & \Hom_{A[T]}\left( [\kappa_{T\otimes_k V}\;\;\;\; \Omega_A\otimes f ]^\top ,\tilde{N}\right)\\
& = & \big[\kappa_{T\otimes V}^*\;\;\;\; (\Omega_A\otimes f)^* \big].
\end{eqnarray*}

Moreover, we have the following commutative diagrams:

\[
\begin{tikzcd}[row sep=large]
T\otimes_k V\arrow[r, "f"]\arrow[d, two heads, "f'"'] & M    \\
\im{f} \arrow[ur, hook , "\iota"']
\end{tikzcd}   ~  \overset{\Omega_A\otimes_A ?}{\leadsto} ~
\begin{tikzcd}[row sep=large]
\Omega_A\otimes_A(T\otimes_k V)\arrow[d, two heads, "\Omega_A\otimes f'"']\arrow[r, "\Omega_A\otimes f"]\arrow[d] & \Omega_A\otimes_A M    \\
\Omega_A\otimes_A\im{f} \arrow[ur, hook, "\Omega_A\otimes\iota"']
\end{tikzcd}
\]

\[
\overset{\Hom_A(?,N)}{\leadsto}~
\begin{tikzcd}[row sep=large]
\Hom_A(\Omega_A\otimes_A(T\otimes_k V),N)  &\arrow[l, "(\Omega_A\otimes f)^*"'] \Hom_A(\Omega_A\otimes_A M,N) \arrow[dl,two heads, "(\Omega_A\otimes\iota)^*"]   \\
\Hom_A(\Omega_A\otimes_A\im{f},N). \arrow[u,hook, "(\Omega_A\otimes f')^*"]
\end{tikzcd}
\]

Thus we obtain:

\begin{equation} \label{eq:HilfsGleichungx}
\im{(\Omega_A\otimes f')^*} ~~=~~ \im{(\Omega_A\otimes f)^*}.
\end{equation}

We sum up and obtain finally:

\begin{eqnarray*}
\Ext_{A[T]}^2(\tilde{M},\tilde{N}) & = &\Hom_A\big(\Omega_A\otimes_A (T\otimes_k V),N\big) \big/ \im{h^*} \\
& = & \Hom_A\left(\Omega_A\otimes_A (T\otimes_k V),N\right) /  \im{[\kappa_{T\otimes V}^*\;\;\;\; (\Omega_A\otimes f)^* ]} \\
&\overset{(\ref{eq:HilfsGleichungx})}{=} & \left(\Hom_A\left(\Omega_A\otimes_A (T\otimes_k V),N\right) / \im{(\Omega_A\otimes f')^*} \right) / \im{\overline{\kappa_{T\otimes V}^*}} \\
& = & \Hom_A(\Omega_A\otimes_A \ker{f}, N)  /  \im{\overline{\kappa_{T\otimes V}^*}} \\
& = &\Ext_A(\ker{f},N).
\end{eqnarray*}

We thus arrive at the folllowing description of $\Ext_{A[T]}^2$:
$ $
For $A[T]$-modules $\tilde{M}=\big(M,V,f:T\otimes_kV\to M\big)$, $\tilde{N}=\big(N,W,g:T\otimes_kW\to N\big)$ there is an isomorphism:

\[
\Ext^2_{A[T]}(\tilde{M},\tilde{N}) ~\xto{\sim}~ \Ext_A(\ker f,N).
\]

In particular, if $T$ is projective, so is $\ker{f}$, thus ${\rm Ext}^2_{A[T]}$ vanishes identically. In other words, $A[T]$ is again hereditary in this case.

We call an $A[T]$-module $\tilde{M}=\big(M,V,f:T\otimes_kV\to M\big)$ full, if $f$ is a surjective map.

Assume $\Ext_A(T,T)=0$. Then, for full $A[T]$-modules $\tilde{M}, \tilde{N}$, the vanishing property

\[
\Ext_{A[T]}^2(\tilde{M},\tilde{N}) = 0
\]

Write $\tilde{M}=\big(M,V,f:T\otimes_kV\to M\big)$ and $\tilde{N}=\big(N,W,g:T\otimes_kW\to N\big)$.
We consider

\[
0\to \ker{g} \to T\otimes_k W \xto{g}  N \to 0
\]

and apply $\Hom_A(\ker{f},?)$  and obtain the long exact sequence

\[
0\to\Hom_A(\ker{f},\ker{g})\to\Hom_A(\ker{f},T\otimes_k W)\to \Hom_A(\ker{f},N)
\]

\[
\to \Ext_A(\ker{f},\ker{g})\to\Ext_A(\ker{f},T\otimes_k W)\to \Ext_A(\ker{f},N) \to 0.
\]

$ $

Now we will show that $\Ext_A(\ker{f},N)=0$ holds by showing that

\[
\Ext_A(\ker{f},T\otimes_k W)=0
\]

holds. We also have:

\[
0\to \ker{f} \to T\otimes_k V \xto{f}  M \to 0.
\]

Applying $\Hom_A(?,T\otimes_k W)$ to this, we obtain the long exact sequence:

\[
0\to\Hom_A(M,T\otimes_k W)\to\Hom_A(T\otimes_k V,T\otimes_k W)\to \Hom_A(\ker{f},T\otimes_k W)
\]

\[
\to \Ext_A(M,T\otimes_k W)\to\Ext_A(T\otimes_k V,T\otimes_k W)\to\Ext_A(\ker{f},T\otimes_k W)\to 0.
\]

Since $\Ext_A(T,T)=0$, in particular $\Ext_A(T\otimes_k V,T\otimes_k W)=0$ holds.

So $\Ext_A(\ker{f},T\otimes_k W)=0$ and therefore $\Ext_A(\ker{f},N)=0$. Using Theorem <ref>, we conclude the proof.

§.§ Derivations and the Euler form of A[T]
$ $

Let $\tilde{M}=\big(M,V,f:T\otimes_kV\to M\big), \tilde{N}=\big(N,W,g:T\otimes_kW\to N\big)$ be $A[T]$-modules. To shorten notation, we define $B=A[T]$. In this section we determine the dimension of a space of derivations

\[
\dim \Der_{\tilde{R}}\big(B,\Hom_{\tilde{R}}(\tilde{N},\tilde{M})\big).
\]

To do this, we consider the canonical exact sequence

\[
0\to\Hom_B(\tilde{N},\tilde{M})\to \Hom_{\tilde{R}}(\tilde{N},\tilde{M}) \xto c \Der_{\tilde{R}}\big(B,\Hom_{\tilde{R}}(\tilde{N},\tilde{M})\big)
\]

\[\to\Ext_B(\tilde{N},\tilde{M})\to 0,
\]

where $c$ is defined as

\[
\big(\beta_i:\tilde{N}_i\to \tilde{M}_i\big)_{i\in Q_0}
\mapsto \sum_{\substack{\alpha\in Q_1\\ \alpha:i\to j}}
\tilde{M}_\alpha\beta_i - \beta_j\tilde{N}_\alpha.
\]

We obtain the equality:

$\dim \Der_{\tilde{R}}\big(B,\Hom_{\tilde{R}}(\tilde{N},\tilde{M})\big)$

$-\left( \dim \Hom_B(\tilde{N},\tilde{M}) - \dim \Ext_B(\tilde{N},\tilde{M}) \right) + \Hom_{\tilde{R}}(\tilde{N},\tilde{M})$.

$ $

On the other hand, we have the following description:

We consider the standard projective resolution of $\tilde{N}$ (Theorem <ref>)

\[
P_\bullet(\tilde{N})\to \tilde{N } \to 0
\]

and  the induced cochain complex $\mathsf{C}:=\Hom_B\big(P_\bullet(\tilde{N}),\tilde{M}\big)$:

\[
\mathsf{C}:\ldots \to 0 \to \Hom_{\tilde{R}}(\tilde{N},\tilde{M}) \to \Hom_A\big(\Omega_A\otimes_A N,M\big)\oplus \Hom_R\big(T\otimes_k W,M\big)
\]

\[
\xto{h^*}\Hom_A\big(\Omega_A\otimes_A (T\otimes_k W),M\big)\to 0 \to ...
\]

Then $H^i(\mathsf{C})=\Ext_{A[T]}^i(\tilde{N},\tilde{M})~~~(i=0,1,2)$.

This way we obtain the equality:

$\dim \Hom_B(\tilde{N},\tilde{M})-\dim\Ext_B(\tilde{N},\tilde{M}) + \dim
\Ext^2_B(\tilde{N},\tilde{M})$

$\dim\Hom_{\tilde{R}}(\tilde{N},\tilde{M}) - \dim\Hom_A\big(\Omega_A\otimes_A N,M\big)-\dim\Hom_R\big(T\otimes_k W,M\big) + \dim\Hom_A\big(\Omega_A\otimes_A (T\otimes_k W),M\big)$.

$ $

We easily determine:

\begin{eqnarray*}
\dim\Hom_A\big(\Omega_A\otimes_A N,M\big) & = & \sum_{\substack{\alpha\in Q_1\\ \alpha:i\to j}} \dim N_i\cdot\dim M_j, \\
\dim\Hom_R\big(T\otimes_k W,M\big) & = & \dim W\cdot \sum_{i\in Q_0} \dim T_i \cdot \dim M_i, \\
\dim\Hom_A\big(\Omega_A\otimes_A (T\otimes_k W),M\big) & = & \dim W \cdot \sum_{\substack{\alpha\in Q_1\\ \alpha:i\to j}} \dim T_i\cdot\dim M_j.
\end{eqnarray*}

By using the characterization of $\Ext_{A[T]}^2$ of Theorem <ref>, we end up in:
For finite-dimensional $A[T]$-modules $\tilde{M}=\big(M,V,f:T\otimes_kV\to M\big)$ and $\tilde{N}=\big(N,W,g:T\otimes_kW\to N\big)$ we have

\[
\dim \Der_{\tilde{R}}\big(B,\Hom_{\tilde{R}}(\tilde{N},\tilde{M})\big) ~~=~~ \dim \Der_{R}\big(A,\Hom_{R}(N,M)\big)
\]

\[
~+~\dim W \cdot \langle T, M\rangle_A ~~+~~ \dim \Ext_A(\ker{g},M),
\]

where $\langle .,.\rangle_A$ denotes the homological Euler-form of $A$.

$ $
If we assume $\Ext_A(T,T)=0$, then for  full $A[T]$-modules $\tilde{M}$ and $\tilde{N}$ we have

\[
\dim \Der_{\tilde{R}}\big(B,\Hom_{\tilde{R}}(\tilde{N},\tilde{M})\big) ~~=~~ \dim \Der_{R}\big(A,\Hom_{R}(N,M)\big) + \dim W \cdot \langle T, M\rangle_A.
\]

The claim follows immediately using Theorem <ref> and <ref>.

We notate a dimension vector $\vec{d}\in\cN^{\hat{Q}_0}$ as a tuple $(s,d)$, where $s=\vec{d}_{\infty}$ and $d\in\cN^{Q_0}$.
For dimension vectors $(s,d),(s',d')\in\cN^{\hat{Q}_0}$ we set

\[
\langle (s,d),(s',d')\rangle_{A[T]} := ss' - s\cdot\langle \dimv T, d'\rangle_Q
+\langle d,d' \rangle_Q.
\]

$ $

For the homological Euler-form of $A[T]$ the following identity holds:

\[
\langle \tilde{M},\tilde{N}\rangle_{A[T]} ~=~ \langle \dimv\tilde{M},\dimv\tilde{N}\rangle_{A[T]},
\]

where $\tilde{M},\tilde{N}$ are (finite-dimensional) $A[T]$-modules and $\langle .,.\rangle_Q$ denotes the Euler-form of $Q$.

The identity follows from the above discussion.

§ VARIETIES OF REPRESENTATIONS OF ONE-POINT EXTENSIONS

For all standard notions on varieties of representations of algebras, we refer to [5]. Let $(s,d)$ be a dimension vector of $\hat{Q}$. Using the isomorphism $A[T]\simeq k\hat{Q}/(\Rel)$, we can realize the
variety of representations of $A[T]$ with dimension $(s,d)$ as a (Zariski-)closed subvariety of the variety of representations of $\hat{Q}$ with dimension $(s,d)$ denoted by $\Rep_{(s,d)}(\hat{Q})$:

\[
\Rep_{(s,d)}(A[T]) \overset{\mathrm{clsd}}\subseteq \Rep_{(s,d)}(\hat{Q}) .
\]

§.§ The Zariski-tangent space
$ $

For $\tilde{M}\in\Rep_{(s,d)}(A[T])$ we have (see <cit.>):

\[
\mathsf{T}_{\tilde{M}}\Rep_{(s,d)}(A[T])~=~\Der_{\tilde{R}}\big(A[T],\End_{\tilde{R}}(\tilde{M})\big).
\]

We set $t:=\dimv \,T$ and get by using Theorem <ref>:

For $\tilde{M}=\big(M,V,f:T\otimes_kV\to M\big)\in\Rep_{(s,d)}(A[T])$ we have

\[
\dim \mathsf{T}_{\tilde{M}}\Rep_{(s,d)}\big(A[T]\big)~=~ \dim\Rep_d(Q) + s\cdot\langle t,d\rangle_Q + \dim \Ext_A(\ker{f},M).
\]

§.§ A well-behaved subvariety in the rigid case
$ $

We consider

\[
\Rep_{(s,d)}\voll\big(A[T]\big)~:=~ \bigg\lbrace (f,M)\;:\; M\in\Rep_d(Q) ,\; \, \rank{f}=\gamma_{T^s,d} \bigg\rbrace
\]

\[
\overset{\text{open}}{\subseteq} \, \Rep_{(s,d)}\big(A[T]\big),
\]

where $\gamma_{T^s,d}\in\cN^{Q_0}$ denotes the general rank of homomorphism from $T^s$ to a representation of dimension $d$ (see <cit.>).

If we assume $\gamma_{T^s,d}=d$ and $\Ext_A(T,T)=0$, then $\Rep_{(s,d)}\voll\big(A[T]\big)$
is smooth and every component has dimension

\[
\dim \Rep_d(Q) + s\cdot\langle \dimv\, T,d\rangle_Q,
\]

i.e. $\Rep_{(s,d)}\voll\big(A[T]\big)$ is a local complete intersection.

By counting the explicit defining polynomial equations

\[
\Rep_{(s,d)}(A[T]) \overset{\mathrm{clsd}}{\subseteq}\Rep_{(s,d)}(\hat{Q}),
\]

induced by (<ref>), we find that every (nonempty) component of $\Rep_{(s,d)}(A[T])$ has dimension $\geq \dim\,\Rep_d(Q)+s\cdot\langle \dimv \,T,d\rangle_Q$.

On the other hand, for each $\tilde{M}\in\Rep\voll_{(s,d)}(A[T])$ we have

\[
\dim_{\tilde{M}}\Rep\voll_{(s,d)}(A[T]) \leq \dim \mathsf{T}_{\tilde{M}}\Rep_{(s,d)}(A[T]).
\]

The claim follows immediately by using the dimension formula in Theorem <ref>.

§.§ On homomorphisms from a fixed representation
$ $

We denote the open dense subset

\[
\bigg\{ M\in \Rep_d(Q)\;:\; \dim\Hom_A(T,M)=\hom(T,d) ~~\text{und}~~  \gamma_{T,M}=\gamma_{T,d} \bigg\}
\]

of $\Rep_d(Q)$ by $\mathcal{V}_{T,d}$. Here, $\hom(T,d)$ is the dimension of the space of homomorphisms from the fixed representation $T$ to a general representation of dimension vector $d$ (see [3]). Moreover, $\gamma_{T,M}$ is the unique maximal rank of homomorphisms from $T$ to $M$.

We consider the regular map

\[
\tilde{\pi}:\Rep_{(1,d)}\voll\big(A[T]\big) \to \Rep_d(Q),\,(f,M)\mapsto M,
\]

which induces a regular map

\[
\pi:\mathcal{V} \to \mathcal{V}_{T,d},
\]

where $\mathcal{V}$ is the open preimage $\tilde{\pi}\inv(\mathcal{V}_{T,d})\subseteq \Rep_{(1,d)}\voll\big(A[T]\big)$.

Obviously $\mathcal{V}_{T,d}$ is irreducible and the fibers of $\pi$ are irreducible of dimension $\mathrm{hom}(T,d)$. So there is a unique irreducible component $\mathcal{V}_0$ of $\mathcal{V}$ of maximal dimension which dominates $\pi$, that is, we have $\overline{\pi(\mathcal{V}_0)} = \mathcal{V}_{T,d}$ (Proposition <ref>)and

\[
\dim \mathcal{V} = \dim \mathcal{V}_0=\dim \Rep_d(Q) + \mathrm{hom}(T,d).
\]

Since the regular points in $\mathcal{V}$ form an open dense subset, there is regular point $\tilde{M}$ in the irreducible component $\mathcal{V}_0$, and we have:

\begin{eqnarray*}
\dim\mathcal{V}_0  & =  & \dim_{\tilde{M}}\mathcal{V} \\
& =  & \dim \mathsf{T}_{\tilde{M}}\mathcal{V} ~=~\dim \mathsf{T}_{\tilde{M}}\Rep_{(1,d)}\voll\big(A[T]\big).
\end{eqnarray*}

Using the description of the tangent space by derivations, we thus find:
We have

\[
\hom(T,d)~=~\langle t,d\rangle_Q ~+~ \dim \Ext_A(\ker{f},M)
\]

for a general $A$-module homomorphism $f:T\to M$.

As a side remark, we rediscover the following result of Schofield [12] and Crawley-Boevey [3] :

We have

\[
\hom(T,d)~=~\langle \gamma_{T,d},d\rangle_Q ~+~ \dim \Hom_A(\ker{f},M)
\]

for a general $A$-module homomorphism $f:T\to M$.

Use the the Euler form of $Q$ and the identity

\[
\dim\Hom_A(\ker{f},M) ~- ~\dim \Ext_A(\ker{f},M) ~~=~~ \langle \dimv\,\ker{f}, d\rangle .
\]

$ $

If we assume $\gamma_{T,d}=d$ and $\Ext_A(T,T)=0$, then $\hom(T,d)~=~\langle t,d\rangle_Q$.

§.§ An irreducible component in the rigid case
$ $

We need the following facts from algebraic geometry:

Let $\pi:\mathcal{X}\to\mathcal{Y}$ be a regular map of affine varieties. Then there is an open dense subset $\mathcal{U}\subseteq \mathcal{X}$ such that for all $x\in\mathcal{U}$

\[
\mathsf{T}_x\pi\inv\big(\pi(x)\big) = \ker{\mathsf{d}\pi_{x}}.
\]

See for example [4].

Let $f:\X\to\Y$ be a regular map of quasi-projective varieties.

If $\Y$ is irreducible and all fibers of $f$ are irreducible and of same dimension $d$ (in particular $f$ is surjective), then:

a) There is a unique irreducible component $\X_0$ of $\X$ that dominates $\Y$, i.e. $\overline{f(\X_0)}=\Y$.

b)Each irreducible component $\X_i$ of $\X$ is a union of fibers of $f$. Its dimension is equal to $\dim\overline{f(\X_i)}+d$.

In particular, we can conclude $\X$ is irreducible if either of the following holds:

i) $\X$ is equidimensional.

ii) $f$ is closed.

See [6].

$ $
If we assume $\gamma_{T^s,d}=d$ and $\Ext_A(T,T)=0$, then $\Rep_{(s,d)}\voll\big(A[T]\big)$ is irreducible and smooth of dimension

\[
\dim \Rep_{(s,d)}\voll\big(A[T]\big)~=~ \dim \Rep_d(Q) + s\cdot\langle t,d\rangle_Q .
\]

By Theorem <ref>, it remains to prove that $\Rep_{(s,d)}\voll\big(A[T]\big)$ is irreducible. We consider the dominant map

\[
\tilde{\pi}: \Rep_{(s,d)}\voll\big(A[T]\big) \to \Rep_{d}(Q),\tilde{M}=(M,k^s,f)\mapsto M.
\]

For each $\tilde{M}=(M,k^s,f)\in\Rep_{(s,d)}\voll\big(A[T]\big)$ the regular map

\[
\Rep_{d}(Q) \to \Rep_{(s,d)}\big(A[T]\big),\,N\mapsto (N,k^s,0)
\]

induces a split mono for the differential $\tilde{\pi}$ in $\tilde{M}$

\[
\mathsf{d}\tilde{\pi}_{\tilde{M}}:\mathsf{T}_{\tilde{M}}\Rep_{(s,d)}\voll\big(A[T]\big) \to \mathsf{T}_{M}\Rep_{d}(Q),
\]

i.e. $\mathsf{d}\tilde{\pi}_{\tilde{M}}$ is surjective.
For each $\tilde{M}=(M,k^s,f)\in\Rep_{(s,d)}\voll\big(A[T]\big)$ we have

\[
\Hom_A(T^s,M) ~~ \simeq ~~ \mathsf{T}_{\tilde{M}}\tilde{\pi}\inv\big(\tilde{\pi}(\tilde{M})\big).
\]

There is an open dense subset $\mathcal{U}\subseteq \Rep_{(s,d)}\voll\big(A[T]\big)$ such that for each $\tilde{M}\in\mathcal{U}$ we have

\[
\Hom_A(T^s,M)~\simeq ~\ker{\mathsf{d}\tilde{\pi}_{\tilde{M}}}
\]

(Theorem <ref>). Since $\mathsf{d}\tilde{\pi}_{\tilde{M}}$ is surjective, by using Theorem <ref> we obtain:

\[
\dim\Hom_A(T^s,M)~=~\hom(T^s,d)
\]

for each $\tilde{M}\in\mathcal{U}$.

\[
\mathcal{U}~~\subseteq ~~\bigg\{ (M,k^s,f)\in\Rep_{(s,d)}\voll(A[T])\;:\; \hom(T^s,M)=\dim\Hom_A(T^s,M) \bigg\}=:\mathcal{V}.
\]

Since $\mathcal{U}\subseteq\Rep_{(s,d)}\voll(A[T])$ is dense,
$\mathcal{V}\subseteq\Rep_{(s,d)}\voll(A[T])$ is dense, too.

The map $\tilde{\pi}$ induces a surjective regular map

\[
\pi:\mathcal{V}\to\mathcal{V}_{T^s,d},(M,k^s,f)\mapsto M,
\]

where $\mathcal{V}_{T^s,d}\subseteq\Rep_d(Q)$ is dense (and thus irreducible). All fibers of $\pi$ are irreducible of equal dimension and $\mathcal{V}$ is equidimensional, thus $\mathcal{V}$ has to be irreducible by Theorem <ref>.
Using $\overline{\mathcal{V}}=\Rep_{(s,d)}\voll(A[T])$, we can conclude the claim.

§ SEMISTABILITY

For all notions concerning stability and moduli spaces of representations we refer to [9]. For an $A[T]$-module $\tilde{M}=(M,V,f:T\otimes_k V\to M)$ we define its slope

\[
\mu\big(\tilde{M}\big)~=~\mu\big(M,V,f\big):=~ \frac{\dim V }{\dim V + \dim M}.
\]

An  $A[T]$-module $\tilde{M}=(M,V,f:T\otimes_k V\to M)$ is called semi-stable (resp. stable) if
$\mu(U)\leq \mu(\tilde{M})$ (resp. $\mu(U)<\mu(\tilde{M})$) for all proper subrepresentation $0\neq U\subset\tilde{M}$.

§.§ A first criterion

Let $\tilde{M}=(M,V,f:T\otimes_k V\to M)$ be an $A[T]$-module with $\tilde{M}_\infty\neq 0$. Then $\tilde{M}$ is (semi-)stable iff for all subspaces $W\subseteq \tilde{M}_\infty$ the following inequality is fulfilled:

\[
\dim M \cdot \frac{\dim W}{\dim V} \underset{(=)}{<}
\sum_{i\in Q_0} \dim \left(\sum_{l=1}^{t_i}\tilde{M}_{\rho_{l, (i)}}(W)\right).
\]

For all subobjects $\tilde{U}=(U,W,g)\subsetneq \tilde{M}$, we have:

\[ \mu(\tilde{U})  \underset{(=)}{<}  \mu(\tilde{M}) \Leftrightarrow \dim M \cdot \frac{\dim W}{\dim V} \underset{(=)}{<}  \dim U.   \]

We can easily determine the total space of the smallest subobject $\overline{W}$ of $\tilde{M}$ containing a given $W\subset V$. Namely, we have

\[
\overline{W}_i~=~\sum_{l=1}^{t_i}\tilde{M}_{\rho_{l,(i)}}(W)
\]

for each $i\in Q_0$.

§.§ An observation on the Harder-Narasimhan filtration
$ $

At first we recall the notion of Harder-Narasimhan filtration.
$ $

A dimension vector $(s,d)\in \cN\times\cN Q_{0}$ is called (semi-)stable if there is a
(semi-)stable representation of $A[T]$ with dimension vector $(s,d)$.

b) A tuple

\[
(s,d)^*~=~\big( (s^1,d^1),\ldots, (s^r,d^r) \big)~~\in \big(\cN\times \cN Q_{0}\big)^r
\]

of dimension vectors is of HN-type if each $(s^l,d^l)$ ($l=1,\ldots,r$) is semi-stable and

\[
\mu(s^1,d^1) > \ldots > \mu(s^r,d^r).
\]

c) A filtration

\[
0 = \tilde{M}_0 \subset \tilde{M}_1 \subset \ldots \subset \tilde{M}_r =\tilde{M}
\]

of a representation $\tilde{M}$ (of $A[T]$) is called Harder-Narasimhan (HN) if each quotient $\tilde{M}_{l}/\tilde{M}_{l-1}$ ($l=1,\ldots,r$) is  semi-stable and

\[
\mu\big(\tilde{M}_{1}/\tilde{M}_{0}\big) > \mu\big(\tilde{M}_{2}/\tilde{M}_{1}\big) > \ldots >  \mu\big(\tilde{M}_{r}/\tilde{M}_{r-1}\big).
\]

$ $

Every representation $\tilde{M}$ of $A[T]$ admits a unique Harder-Narasimhan filtration.

See [9].

For a HN type $(s,d)^*$ we denote by

\[
\Rep_{(s,d)^*}^{HN}(A[T])\subseteq \Rep_{(s,d)}(A[T])
\]

the subset of representations whose HN filtration is of type $(s,d)^*$. $\Rep_{(s,d)*}^{HN}(A[T])$ is called HN stratrum for the HN type $(s,d)^*$.
More generally,  we denote by

\[
\Rep_{(s,d)}^{(s,d)^*}(A[T]) \subseteq \Rep_{(s,d)}(A[T])
\]

the subset of representations $\tilde{M}$ possessing a filtration of type $(s,d)^*$, i.e.  there is a chain of subrepresentations
0=\tilde{M}_0\subset \tilde{M}_1\subset \ldots \subset \tilde{M}_r=\tilde{M}
with $\dimv\,\tilde{M}_l/\tilde{M}_{l-1} = (s^l,d^l)$ for $l=1,\ldots,r$.

Let $\tilde{M}=(M,V,f:T\otimes_{k} V\to M)$ be a representation of $A[T]$ with $V\neq 0$. Then the following are equivalent:

a) $f:T\otimes_{k} V \to M$ is surjective.

b) For the HN filtration of $\tilde{M}$

\[
0=\tilde{M}_0\subset \tilde{M}_1\subset \ldots \subset \tilde{M}_r = \tilde{M}
\]

we have

\[
\mu(\tilde{M}_r/\tilde{M}_{r-1}) ~\neq ~ 0.
\]

a) $\Rightarrow$ b):
We denote $\tilde{U}=\tilde{M}_{r-1}$, which we write as

\[
\tilde{U}=(U,W,g:T\otimes_{k} W\to U).
\]

Since $\tilde{U}\subsetneq \tilde{M}$ is a subrepresentation, we have the following commutative diagram:

\[
\begin{tikzcd}
T\otimes_{k} V \ar[r, "f"]  & M  \\
T\otimes_{k} W \ar[r, "g"] \arrow[u, hook] & U \arrow[u, hook]  \\
\end{tikzcd} .
\]

Assume $\mu(\tilde{M}/\tilde{U})=0$, i.e. $\dim V - \dim W =0$. So we obtain $W=V$, and since $f$ is surjective we can conclude from the above commutative diagram that $U$ already equals $M$. In other words, $\tilde{U}=\tilde{M}$,  contradicting our assumption.

b) $\Rightarrow$ a):
Assume the structure morphism $f:T\otimes_{k} V\to M$ is not surjective.
Then we can consider the proper subobject $\tilde{U}\subsetneq \tilde{M}$ induced by the commutative diagram

\[
\begin{tikzcd}
\tilde{M}: &  T\otimes_{k} V \ar[r, "f"]  & M  \\
\tilde{U}:\arrow[u, hook] & T\otimes_{k} V \ar[r, "f'"] \arrow[u, hook] & \im{f}. \arrow[u, hook]  \\
\end{tikzcd}
\]

Obviously then we have $\mu(\tilde{M}/\tilde{U})=0$, and since $f$ is not surjective, we neither have $\dimv\,\tilde{M}/\tilde{U} = 0$.
So $\tilde{M}/\tilde{U}$ is a semi-stable representation.

Now look at  the HN filtration of $\tilde{U}$. Since the structure morphism of $\tilde{U}$ is surjective we can conclude from the first part of this proof that for the HN filtration of $\tilde{U}=Y_l\supset Y_{l-1}\supset \ldots \supset Y_1\supset Y_0=0$ we have

\[
\mu(Y_l/Y_{l-1})\neq 0.
\]

Since by definition the slope is always $\geq 0$ we can conclude

\[
\mu(Y_l/Y_{l-1}) > \mu(\tilde{M}/\tilde{U}).
\]

Using the uniqueness of the HN filtration we finally obtain that

\[
0=Y_0\subset Y_1\subset \ldots Y_l = \tilde{U} \subset \tilde{M}
\]

must be the HN filtration of $\tilde{M}$. But this contradicts our assumption about the HN filtration of $\tilde{M}$. So $f$ has to be surjective.

Consequences of Theorem <ref> are:

For $(s,d)\in \cN\times \cN^{Q_{0}}$ we have the following connection between the semi-stable representations and full representations:

\[
\Rep_{(s,d)}^\sst(A[T]) ~ \subseteq ~ \Rep_{(s,d)}\voll(A[T]).
\]

§.§ Geometric consequences for the moduli space
$ $

The linear algebraic group

\[
\G_{(s,d)}:=\Gl_{s}(k)\times\prod_{i\in Q_0}\Gl_{d_i}(k)
\]

acts on $\Rep_{(s,d)}( \hat{Q})$ via the base change action

\[
(g_i)_i\cdot (\tilde{M}_\alpha)_{\alpha\in\hat{Q}_1}
\big( g_j \tilde{M}_\alpha g_i\inv \big)_{\alpha:i\to j}.
\]

$\Rep_{(s,d)}(A[T])$ is stable under this $\G_{(s,d)}$-action. By definition, the $\G_{(s,d)}$-orbits in $\Rep_{(s,d)}(A[T])$ correspond bijectively to isomorphism classes $[\tilde{M}]$ of representations of $A[T]$ of dimension vector $(s,d)$. We consider the stability function $\tilde{\Theta}$ for $\hat{Q}$ given by $\tilde{\Theta}_{\infty}=1$ and $\tilde{\Theta}_i=0$ for $i\in Q_0$. The associated slope function on representations of $\hat{Q}$ coincides with the slope function $\mu$ on $A[T]$-modules. This allows us to define moduli spaces $\M_{(s,d)}^\sst\big(A[T]\big)$ resp. $\M_{(s,d)}^\st\big(A[T]\big)$ as the algebraic quotient of $\Rep_{(s,d)}^\sst(A[T])$, resp. the geometric quotient of $\Rep_{(s,d)}^\st(A[T])$, by $\G_{(s,d)}$.
Assume $\Ext_A(T,T)=0$ and $\gamma_{T^s,d}=d$.

If $\Rep_{(s,d)}^\st\big(A[T]\big)\neq\emptyset$, then both $\M_{(s,d)}^\sst\big(A[T]\big)$ and $\M_{(s,d)}^\st\big(A[T]\big)$ are irreducible and smooth of dimension

\[
\dim\,\M_{(s,d)}^\sst\big(A[T]\big)~=~ 1 - \langle (s,d),(s,d)\rangle_{A[T]}.
\]

We calculate fibre dimensions for the geometric quotient

\[
\pi:\Rep_{(s,d)}^\st\big(A[T]\big) \to \M_{(s,d)}^\st\big(A[T]\big),
\]

and use Corollary <ref>, Theorem <ref> and the fact that the endomorphism rings of stable representations are trivial.

Next, we introduce the Harder-Narasimhan stratification. Note that the term stratification is used in a weak sense, meaning a finite decomposition of a variety into locally closed subsets.

§.§ Harder-Narasimhan stratification
$ $

In this section we write $\Rep_{(s,d)}$ for $\Rep_{(s,d)}\voll(A[T])$ to simplify notation.
Let assume $\Ext_A(T,T)=0$ and $\gamma_{T^s,d}=d$.
The HN-strata for the HN-types

\[
(s,d)^* ~=~\big( (s^1,d^1),\ldots, (s^r,d^r) \big)
\]

with weight $(s,d)$, i.e. $(s,d)=\sum_{i=1}^r(s^i,d^i)$, and $s^r\neq 0$ define a stratification of $\Rep_{(s,d)}$.

The codimension of $\Rep_{(s,d)^*}^{HN}$ in $\Rep_{(s,d)}$ is given by:

\[
- \sum_{k<l}\langle (s^k,d^k),(s^l,d^l)\rangle_{A[T]}.
\]

Let $F^*:0=F^0\subset F^1\subset \ldots \subset F^r$ be a flag of type $(s,d)^*$ in the $\hat{Q}_0$-graded vector space $k^d\times k^s =\oplus_{i\in Q_0}k^{d_i}~\times~k^{s}$, i.e. $F^{l}/F^{l-1}\simeq k^{d^l}\times k^{s^l}$ for $l=1,\ldots, r$, and denote by $F_i^l$ the $i$-component of $F^l$.

Denote by $\tilde{\mathcal{Z}}_{(s,d)^*}$ the closed subvariety of $\Rep_{(s,d)}(A[T])$ of representations $\tilde{M}$ which are compatible with $F^*$, i.e.
$\tilde{M}_\gamma(F_i^l)\subset F_j^l$ for $l=1,\ldots, r$ and for all arrows $(\gamma:i\to j)$ in $\hat{Q}$.
We have the regular map

\[
\tilde{p}_{(s,d)^*}:\tilde{\mathcal{Z}}_{(s,d)^*} \to \Rep_{(s^1,d^1)}(A[T]) \times \ldots \times \Rep_{(s^r,d^r)}(A[T])
\]

given by the projection $\tilde{p}_{(s,d)^*}$ mapping $\tilde{M}\in \tilde{\mathcal{Z}}_{(s,d)^*}$ to the sequence of subquotients with respect to $F^*$.
The map $\tilde{p}_{(s,d)^*}$ induces a regular map

\[
p_{(s,d)^*}:\mathcal{Z}_{(s,d)^*}\to \Rep_{(s^1,d^1)}\times \ldots \times\Rep_{(s^r,d^r)},
\]

\[
\mathcal{Z}_{(s,d)^*}:=\tilde{p}_{(s,d)^*}\inv\big(\Rep_{(s^1,d^1)}\times \ldots \times \Rep_{(s^r,d^r)}\big) \subseteq \tilde{\mathcal{Z}}_{(s,d)^*}~~~\text{open.}
\]

A minute reflection shows that

\[
\mathcal{Z}_{(s,d)^*}\subset \Rep_{(s,d)},
\]

and it is a locally closed subset. By Theorem <ref>,

\[
\Rep_{(s,d)},\Rep_{(s^1,d^1)},\ldots,\Rep_{(s^r,d^r)}~~~\text{are irreducible.}
\]

We set

\[
e^*_2 ~:=~\big( (s^1,d^1), (s^2,d^2) \big).
\]

We obtain the regular map

\[
p_{e^*_2}: \mathcal{Z}_{e^*_2}  \to \Rep_{(s^1,d^1)}\times \Rep_{(s^2,d^2)},
\]

with fibers at $(\tilde{M}_1,\tilde{M}_2)\in\Rep_{(s^1,d^1)}\times \Rep_{(s^2,d^2)}$ given by

\[
p_{e^*_2}\inv(\tilde{M}_1,\tilde{M}_2)~=~ \Der_{\tilde{R}}\big(B,\Hom_{\tilde{R}}(\tilde{M}_2,\tilde{M}_1)\big).
\]

Thus all fibers are irreducible and of equal dimension (Corollary <ref> and <ref>). By  Theorem <ref> we have

\[
\dim\mathcal{Z}_{e^*_2} ~=~\dim p_{e^*_2}\inv(\tilde{M}_1,\tilde{M}_2) + \dim \left( \Rep_{(s^1,d^1)}\times \Rep_{(s^2,d^2)} \right).
\]

Now we set

\begin{eqnarray*}
\overline{e}^*_3  &:= &\big( (s^1,d^1) + (s^2,d^2), (s^3,d^3) \big),\\
e^*_3  &:= &\big( (s^1,d^1), (s^2,d^2), (s^3,d^3) \big).
\end{eqnarray*}

Therefore we obtain the regular map

\[
p_{\overline{e}^*_3}: \mathcal{Z}_{\overline{e}^*_3}  \to \Rep_{(s^1,d^1) + (s^2,d^2)}\times \Rep_{(s^3,d^3)}.
\]

\[
Z_{e^*_3}=p_{\overline{e}^*_3}\inv( \mathcal{Z}_{e^*_2} \times \Rep_{(s^3,d^3)} ),
\]

the map $p_{\overline{e}^*_3}$ induces a regular map

\[
\mathcal{Z}_{e^*_3} \to \mathcal{Z}_{e^*_2} \times \Rep_{(s^3,d^3)}.
\]

As above, we see that this regular map has irreducible fibers of equal dimension, that is, for  $(\tilde{M}_1,\tilde{M}_3)\in\mathcal{Z}_{e^*_2} \times \Rep_{(s^3,d^3)}$, we have:

\[
\dim \mathcal{Z}_{e^*_3} ~=~ \dim\Der_{\tilde{R}}\big(B,\Hom_{\tilde{R}}(\tilde{M}_3,\tilde{M}_1)\big) + \dim\mathcal{Z}_{e^*_2} + \dim\Rep_{(s^3,d^3)}.
\]

Inductively, we obtain in this way a regular map

\[
\mathcal{Z}_{e^*_r} \to \mathcal{Z}_{e^*_{r-1}} \times \Rep_{(s^r,d^r)}
\]

with irreducible fibers of equal dimension, where

\begin{eqnarray*}
e^*_{r-1}  & = & \big( (s^1,d^1), \ldots, (s^{r-1},d^{r-1}) \big), \\
e^*_{r}  & = & (s,d)^*.
\end{eqnarray*}

Summing up, for $(\tilde{M}_1,\tilde{M}_r)\in\mathcal{Z}_{e^*_{r-1}} \times \Rep_{(s^r,d^r)}$ this yields:

\[
\dim\mathcal{Z}_{(s,d)^*} ~=~
\dim\Der_{\tilde{R}}\big(B,\Hom_{\tilde{R}}(\tilde{M}_r,\tilde{M}_1)\big) + \dim\mathcal{Z}_{e^*_{r-1}} + \dim\Rep_{(s^r,d^r)}.
\]

Together with the formula in Corollary <ref>, we find:

\[
\dim \mathcal{Z}_{(s,d)^*} ~=~ \sum_{n<l} \sum_{\substack{\alpha\in Q_1\\ \alpha: i\to j}} d_i^ld_j^n + s^l \langle t, d^n\rangle_Q ~~+~~\sum_{i=1}^r\dim \Rep_{(s^i,d^i)}.
\]

To simplify the notation in the following, we write $\mathcal{Z}$ (resp. $p$) for $\mathcal{Z}_{(s,d)^*}$ (resp. $p_{(s,d)^*}$).
The preimage of

\[
\Rep_{(s^1,d^1)}^\sst\times \ldots \times \Rep_{(s^r,d^r)}^\sst
\]

under $p$ gives us an open subvariety $\mathcal{Z}_0$ of $\mathcal{Z}$ and $\tilde{\mathcal{Z}}$. Since the varieties

\[
\Rep_{(s^1,d^1)},\ldots, \Rep_{(s^r,d^r)}
\]

are irreducible, we see in a similar manner to $\mathcal{Z}$ that $\dim \mathcal{Z}_0=\dim \mathcal{Z}$ holds.

The action of $\mathcal{G}_{(s,d)}$ on $\Rep_{(s,d)}(A[T])$ induces actions of the parabolic subgroup $\mathcal{P}_{(s,d)^*}$ of $\mathcal{G}_{(s,d)}$, consisting of elements fixing the flag $F^*$, on $\mathcal{Z}_0$ and $\mathcal{Z}$. The image of the associated fiber bundle $\mathcal{G}_{(s,d)}\times^{\mathcal{P}_{(s,d)^*}} \tilde{\mathcal{Z}}$ under the action morphism $m$ equals $ \Rep^{(s,d)^*}_{(s,d)}(A[T])$, which is thus a closed subvariety of $\Rep_{(s,d)}(A[T])$. The image of $\mathcal{G}_{(s,d)}\times^{\mathcal{P}_{(s,d)^*}} \mathcal{Z}_0$ under $m$ equals $\Rep_{(s,d)^*}^{HN}$, and  $\mathcal{G}_{(s,d)}\times^{\mathcal{P}_{(s,d)^*}} \mathcal{Z}_0$ is the full preimage. By the uniqueness of the HN filtration, the morphism $m$ is bijective over $\Rep_{(s,d)^*}^{HN}$,  which therefore is a locally closed subvariety of $\Rep_{(s,d)}(A[T])$.

The canonical map $\mathcal{G}_{(s,d)}\times^{\mathcal{P}_{(s,d)^*}} \mathcal{Z}_0 \to \mathcal{G}_{(s,d)}\big/ \mathcal{P}_{(s,d)^*}$ is (Zariski) locally trivial. Therefore

\[
\dim \mathcal{G}_{(s,d)}\times^{\mathcal{P}_{(s,d)^*}} \mathcal{Z}_0~~~=~~
(\dim \mathcal{G}_{(s,d)} ~-~ \mathcal{P}_{(s,d)^*})~~+~~\dim \mathcal{Z}_0  .
\]

The codimension of $\Rep_{(s,d)^*}^{HN}$ in $\Rep_{(s,d)}$ is now easily computed as

\[
- \sum_{n<l}\langle (s^n,d^n),(s^l,d^l)\rangle_{A[T]},
\]

using the identity $\langle d,d \rangle_Q  =  \dim \mathcal{G}_d - \dim \Rep_d(Q)$ and the above description of $\mathcal{Z}_0$.

From this description, we can derive a recursive criterion for the existence of semi-stable representations:
Let us assume $\Ext_A(T,T)=0$. $ $

A dimension type $(s,d)\in \cN\times \cN Q_{0}$ is semi-stable if and only if $\gamma_{T^s,d}=d$ and there exists no HN type

\[
(s,d)^* ~=~\big( (s^1,d^1),\ldots, (s^r,d^r) \big)
\]

with weight $(s,d)$ and $s^r\neq 0$ such that

\[
~~~~~~~~~~~~~~~~~~~~~~~~ \langle (s^n,d^n),(s^l,d^l)\rangle_{A[T]} ~=~0 \;\;\;\;\;\text{(for all $1\leq n<l\leq r$)}.
\]

\[
H:=\left\{ (s,d)^* \,:\, \exists\, r\in\cN_{>1},\,\text{$(s,d)^* = \big( (s^1,d^1),\ldots, (s^r,d^r) \big)$ is of HN-type} \right.
\]

\[
\left. \text{with weight $(s,d)$ and $s^r\neq0$}  \right\}.
\]

Obviously $H$ is a finite set. Using Theorem <ref> we get

\[
\Rep_{(s,d)}~~=~~ \Rep_{(s,d)}^\sst ~\sqcup \bigsqcup_{(s,d)^*\in H} \Rep^{HN}_{(s,d)^*}.
\]

If $\Rep_{(s,d)}^\sst\neq \emptyset$, $\Rep_{(s,d)}^\sst$ is of equal dimension like $\Rep_{(s,d)}$.  So

\[
\mathrm{codim}_{\Rep_{(s,d)}}\Rep^{HN}_{(s,d)^*} > 0
\]

for $(s,d)^*\in H$ implies $\Rep_{(s,d)}^\sst\neq\emptyset$.

Let $n<l$. Since $(s^n,d^n)$ resp. $(s^l,d^l)$ are semi-stable, we find semi-stable representations $\tilde{N}$ resp. $\tilde{M}$
of $A[T]$ with $\dimv\,\tilde{N}=(s^n,d^n)$ resp. $\dimv\,\tilde{M}=(s^l,d^l)$ and by using Corollary <ref> and <ref> we get the relation

\[
\langle (s^n,d^n),(s^l,d^l)\rangle_{A[T]}~~=~~ \dim\Hom_{A[T]}(\tilde{N},\tilde{M}) ~-~ \dim \Ext_{A[T]}(\tilde{N},\tilde{M}).
\]

From $\mu(s^n,d^n) > \mu(s^l,d^l)$ we deduce $\Hom_{A[T]}(\tilde{N},\tilde{M})=0$, in particular

\[
\langle (s^n,d^n),(s^l,d^l)\rangle_{A[T]}~~=~~ ~-~ \dim \Ext_{A[T]}(\tilde{N},\tilde{M}).
\]

So the equation

\[
\sum_{n<l}\langle (s^n,d^n),(s^l,d^l)\rangle_{A[T]} ~~=~~0
\]

is fulfilled if and only if

\[
\langle (s^n,d^n),(s^l,d^l)\rangle_{A[T]} ~=~0
\]

for all $1\leq n<l\leq r$.

We carry on with Example <ref> here. Obviously the $\mathcal{G}_{(3,1)}$-orbit of $T = (k^3 \xto{ [1~0~0] }k)$ is dense in $\Rep_{(3,1)}(Q)$. Therefore we have $\Ext_{kQ}(T,T)=0$.

Applying the recursive criterion we derive that $(2,4,1)$ is semi-stable. In fact, by using the first criterion in Theorem <ref>, we see that in this case the semi-stability notion equals to the stability notion. From the geometry of the moduli (Theorem <ref>) we can conclude $\dim \M_{(2,4,1)}^\st(A[T]) = 4$.

2) Analogous statements as in 1) hold for the dimension vector $(3,6,2)$. And we can deduce $\dim\,\M_{(3,6,2)}^\st\big(A[T]\big) = 6$.

§ GENERATING SEMI-INVARIANTS

In this section we assume that $Q$ is an acyclic quiver. We determine a set of functions generating the ring of semi-invariants for given dimension vector $(s,d)\in\cN \hat{Q}_0$ on $\Rep_{(s,d)}(A[T])$ under the $\G_{(s,d)}$ base change action.

We start with a general observation:

Let $\G$ be a linear reductive group and $\mathcal{X}$ an affine $\G$-variety. Let $\mathcal{A}\subseteq \mathcal{X}$ be a closed and $\G$-stable subset. Then, for every semi-invariant function $f:\mathcal{A}\to k$ there is a semi-invariant $\tilde{f}:\mathcal{X}\to k$ such that $\tilde{f}\vert_{\mathcal{A}} = f$ holds.

Let $\chi$ be a character of $\G$.
We consider the action of $\G$ on $\mathcal{X}\times k$ given by

\[
g.(x,\lambda)~:=~(gx, \chi(g)\lambda),
\]

where $(x,\lambda)\in \mathcal{X}\times k,\,g\in \G$.

Since $\mathcal{A}\times k ~\subseteq~ \mathcal{X}\times k$ is closed and $\G$-stable, the categorical quotient $(\mathcal{A}\times k)// \G ~\subseteq~ (\mathcal{X}\times k)//\G$ is closed, i.e.

\[
k[\mathcal{X}\times k]^{\G} \to k[\mathcal{A}\times k]^{\G} ,\, f \mapsto f\vert_{\mathcal{A}\times k}
\]

is surjective.

In the following, to simplify the notation we denote by $\hat{A}$ the path algebra of the one-point extended quiver $\hat{Q}$.

Let $\tilde{N}$ be a representation of $\hat{Q}$ with projective resolution

\[
0\to \bop_{v\in \hat{Q}_0} \hat{A}e_v^{b(v)} \xto\theta \bop_{v\in \hat{Q}_0}\hat{A}e_v^{a(v)} \to \tilde{N}\to 0.
\]

For $\tilde{M}\in\Rep_{(s,d)}(\hat{Q})$ we apply the functor $\Hom_{\hat{A}}(?,\tilde{M})$ to this resolution and get

\[
0\to \Hom_{\hat{A}}(\tilde{N},\tilde{M})\to \Hom_{\hat{A}}(\bop_{v\in \hat{Q}_0}\hat{A}e_v^{a(v)},\tilde{M}) \xto{ \Hom_{\hat{A}}(\theta,\tilde{M})}\]

\[
\Hom_{\hat{A}}(\bop_{v\in \hat{Q}_0}\hat{A}e_v^{b(v)},\tilde{M})
\to\Ext_{\hat{A}}(\tilde{N},\tilde{M})\to 0.
\]

The condition $\langle \dimv\,\tilde{N}, (s,d)\rangle_{\hat{Q}}=0$ is equivalent to

\[
\sum_{v\in \hat{Q}_0}a(v)\hat{d}_v~=~\sum_{v\in \hat{Q}_0}b(v)\hat{d}_v,
\]

that is, in this case we end up with a linear map between vector spaces of equal dimension. Schofield and Van den Bergh proved ([13]) that all semi-invariant
functions arise as linear combination of functions

\[
\omega_{\tilde{N}}:\Rep_{(s,d)}(\hat{Q})\to k,\, \tilde{M}\mapsto\det(\Hom_{{\hat{A}}}(\theta,\tilde{M}))
\]

induced by representations $\tilde{N}$ of $\hat{Q}$ with $\langle \dimv\,\tilde{N}, (s,d)\rangle_{\hat{Q}}=0$.

Using the relation $\hat{A}\epi A[T]$ we can conclude with Lemma <ref>:
$ $
The functions $\omega_{\tilde{N}}$ for representations $\tilde{N}$ of $A[T]$ such that $\langle \dimv\,\tilde{N},(s,d)\rangle_{\hat{Q}}=0$
generate the ring of invariants on $\Rep_{(s,d)}(A[T])$.

We can improve this description further by using the canonical exact sequence for one-point extensions and the explicit description of the standard projective resolution in Theorem <ref>: Let $\tilde{N}$ be a representation of $A[T]$. Then we have the canonical exact sequence

\[
\begin{tikzcd}
{0} \arrow[r]  & \tilde{N}_{\mathrm{full}} \arrow[r] & \tilde{N}  \arrow[r] &  \tilde{N}_{0}  \arrow[r] &  {0}   \end{tikzcd}\]

which in more detail reads

\[\begin{tikzcd}
0\arrow[r]  & T\otimes_k W \arrow[r] \arrow[d] & T\otimes_k W  \arrow[r]\arrow[d]&  0 \arrow[d] \arrow[r] & 0   \\
0 \arrow[r]  & \im{f} \arrow[r] & N  \arrow[r]&  \coker{f}   \arrow[r]&  0.
\end{tikzcd}
\]

Obviously $\tilde{N}_{0}$ can be interpreted as a representation of $Q$, with standard resolution of length $1$. Thus, we arrive at the following commutative diagram with exact columns and rows:

\[
\begin{tikzcd}
& \bop_{v\in \hat{Q}_0} Be_v^{b(v)}\ar[d,"\varphi_1"]\ar[r,"\theta"] &  \bop_{v\in \hat{Q}_0}Be_v^{a(v)}  \ar[r] \ar[d,"\varphi_0"] & \tilde{N} \ar[d] \ar[r] & 0\\
0\ar[r]& \bop_{v\in \hat{Q}_0} Be_v^{c(v)}\ar[r,"\rho"] \ar[d] & \bop_{v\in \hat{Q}_0}Be_v^{e(v)} \ar[r]\ar[d] & \tilde{N}_{0} \ar[r] \ar[d] & 0. \\
& 0 & 0 & 0
\end{tikzcd}
\]

Thus, if $\langle \dimv\,\tilde{N},(s,d)\rangle_{\hat{Q}}=0$ and  $\omega_{\tilde{N}} \neq 0$ hold, we can conclude from the diagram that the determinants $\omega_{\tilde{N}_{0}}$ and $\omega_{\tilde{N}_{\mathrm{full}}}$ can be formed. This discussion shows:
The ring of semi-invariant functions on $\Rep_{(s,d)}(A[T])$ is generated by the functions $\omega = \omega_{L} \cdot \omega_{\tilde{N}}$ induced by representations $L$ of $Q$ such that $\langle \dimv \,L,d\rangle_{Q}=0$ and full representations $\tilde{N}$ of $A[T]$ such that $\langle \dimv\,\tilde{N},(s,d)\rangle_{\hat{Q}}=0$.

From the homological properties we can further conclude:
For a character $\chi$ of $\G_{(s,d)}$, a representation $\tilde{M}=(M,V,f:T\otimes_kV\to M)\in\Rep_{(s,d)}(A[T])$ is $\chi$-semi-stable iff there is a non-trivial finite-dimensional representation $\tilde{N}=(N,W,g:T\otimes_kW\to N)$ of $A[T]$ such that

a) $\Hom_{A[T]}(\tilde{N},\tilde{M}) = 0$, and

b) $\langle \dimv \tilde{N},(s,d)\rangle_{A[T]} = \dim W \cdot \sum_{\substack{\alpha\in Q_1\\ \alpha:i\to j}} \dim T_i \cdot d_j$.

Take the standard resolution as described as in Theorem <ref>

\[
P_\bullet(\tilde{N})\to \tilde{N } \to 0
\]

and consider the cochain complex $\mathsf{C}:=\Hom_{A[T]}\big(P_\bullet(\tilde{N}),\tilde{M}\big)$:

\[
\mathsf{C}:0 \to \Hom_{\tilde{R}}(\tilde{N},\tilde{M}) \xto{\hat{\phi}} \Hom_A\big(\Omega_A\otimes_A N,M\big)\oplus \Hom_R\big(T\otimes_k W,M\big)
\]

\[
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\to\Hom_A\big(\Omega_A\otimes_A (T\otimes_k W),M\big)\to 0.
\]

Then you have $H^i(\mathsf{C})=\Ext_{A[T]}^i(\tilde{N},\tilde{M})~~~(i=0,1,2)$.

In this way, we achieve the relation:

$\dim \Hom_{A[T]}(\tilde{N},\tilde{M})-\dim\Ext_{A[T]}(\tilde{N},\tilde{M}) + \dim
\Ext^2_{A[T]}(\tilde{N},\tilde{M})$

$\dim\Hom_{\tilde{R}}(\tilde{N},\tilde{M}) - \dim\Hom_A\big(\Omega_A\otimes_A N,M\big)-\dim\Hom_R\big(T\otimes_k W,M\big)$

$~~~~~~~+ \dim\Hom_A\big(\Omega_A\otimes_A (T\otimes_k W),M\big)$.

Both conditions in the theorem are equivalent to $\hat{\phi}$ being an isomorphism.

We carry on with Example <ref> here. Long calculations yields to following generating and algebraic independent semi-invariant functions:

1) The regular maps $\Rep_{(2,4,1)}\voll(A[T]) \to k$ defined by

\hbar_0: (A,B,C,M)  \mapsto   -\det \left(
\begin{array}{c|c|c}
A & B & C\\
\hline
0 & \begin{pmatrix}MA \\ 0 \end{pmatrix} &
\begin{pmatrix}0 \\ MA \end{pmatrix}
\end{array}
\right),

$\hbar_1: (A,B,C,M) \mapsto \det(A\,\vert \, B)$, $\hbar_2: (A,B,C,M) \mapsto \det(A\,\vert \, C)$,

$\hbar_3: (A,B,C,M) \mapsto \det(A+C\,\vert\,B)$, $\hbar_4: (A,B,C,M) \mapsto \det(A\,\vert\,B+C)$, and $\hbar_{5}: (A,B,C,M) \mapsto \det(A+C\,\vert \, B+C)$
give rise to the geometric quotient $\hat{\pi}:\Rep_{(2,4,1)}^\st(A[T]) \to \cP^4$ by $\G_d$ in the following way

\[
\hat{\pi}:=\big(\hbar_0\hbar_1,\hbar_0\hbar_2,\hbar_0\hbar_3,\hbar_0\hbar_4,\hbar_0\hbar_5\big).
\]

Thus, we have $\M_{(2,4,1)}^\st\big(A[T]\big) ~=~\cP^4$.

2)In the case of the dimension vector $(3,6,2)$ the regular maps
$\Rep_{(3,6,2)}\voll(A[T]) \to k$ defined by

\hbar_0: (A,B,C,M) \mapsto
\det{\left(
\begin{array}{c|c|c||c|c|c}
MA &   &  &   & -MA &   \\
\hline
& MA &  &  &  & -MA \\
\hline
&  &  MA & & & -MA \\
\hline
\hline
A &   & C &  & B & \\
\hline
& B &   &  A & & C
\end{array}\right)}, \\

$\hbar_1: (A,B,C,M) \mapsto  \det(A\,\vert \, B)$, $\hbar_2: (A,B,C,M) \mapsto \det(A\,\vert \, C)$,

$\hbar_3: (A,B,C,M) \mapsto \det(A+C\,\vert\,B)$, $\hbar_4: (A,B,C,M) \mapsto \det(A+B\,\vert\,C)$,

$\hbar_5: (A,B,C,M) \mapsto \det(A\,\vert\,B+C)$, $\hbar_6: (A,B,C,M)  \mapsto \det(A+B\,\vert\,B+C)$, and $\hbar_{7}: (A,B,C,M)\mapsto \det(A+C\,\vert \, B+C)$ give rise to the geometric quotient $\hat{\pi}:\Rep_{(3,6,2)}^\st(A[T]) \to \cP^6$ by $\G_d$ in the following way

\[
\hat{\pi}:=\big(\hbar_0\hbar_1,\hbar_0\hbar_2,\hbar_0\hbar_3,\hbar_0\hbar_4,\hbar_0\hbar_5, \hbar_0\hbar_6, \hbar_0\hbar_7\big),
\]

This shows that $\M_{(3,6,2)}^\st\big(A[T]\big) ~=~\cP^6$.

§ HIGHER GEL'FAND MACPHERSON CORRESPONDENCE

We first recall the definition of quiver Grassmannians (see for example [1]). For a quiver $Q$, a representation $X$ of $Q$ of dimension vector $d$ and another dimension vector $e\leq d$, we define $\Gr^e_Q(X)$ as the set of subrepresentations $U$ of $X$ of dimension vector $d-e$. This carries a natural scheme structure as the geometric quotient by the base change group $\G_d$ of the set ${\rm Hom}_Q(X,e)_e$ of surjections (that is, rank $e$ maps) from $X$ to a representation of dimension vector $e$.

From now on, we assume $\End_A(T)=k$, $\Ext_A(T,T)=0$ and $\gamma_{T^s,d}=d$.

In this case, the regular map

\[
\phi:\Rep_{(s,d)}\voll(A[T]) \to \Hom_Q(T^s,d)_{d},
\big(M,f:T^s\to M\big)\mapsto f,
\]

which is always a locally trivial fiber bundle (<cit.>), has single element fibres, and thus is an isomorphism of varieties. Moreover, we have the $\G_d$-bundle

\[
\kappa: \Hom_Q(T^s,d)_{d} \to \Gr^d_Q(T^s),(\theta_i)_{i\in Q_0}\mapsto \ker{\oplus_{i\in Q_0}\theta_i}.
\]

All in all, we achieve a $\G_d$-bundle

\[
\psi: \Rep_{(s,d)}^\sst(A[T]) \to \Gr^{d,\sst}_Q(T^s),
\]

where $\Gr^{d,\sst}_Q(T^s):=\kappa\circ\phi\bigg(\Rep_{(s,d)}^\sst(A[T])\bigg)\subseteq \Gr^{d}_Q(T^s)$ is open.

We thus have an induced map

\[
\pi:\Gr^{d,\sst}_Q(T^s) \to \M_{(s,d)}^\sst\big(A[T]\big).
\]

The linear reductive group $\Aut_Q(T^s)$ acts naturally on $\Gr_Q^d(T^s)$ and for $\big[\tilde{M}\big]\in\M_{(s,d)}^\sst\big(A[T]\big)$ clearly we have

\[
\pi\inv\big([\tilde{M}]\big)=\Aut_Q(T^s).[(M,f)].
\]

We thus find:

$ $

There is an isomorphism of varieties:

\[
\M^\sst_{(s,d)}\big(A[T]\big)~\simeq~\Gr^{d,\sst}_Q(T^s) \big/\Aut_Q(T^s).
\]

By Theorem <ref> $\M^\sst_{(s,d)}\big(A[T]\big)$ is smooth, in particularly normal. Theorem <ref> shows that the quiver Grassmanian of a representation without self-extensions is irreducible. Since $\pi$ is surjective the claim follows from <cit.>.

§ MOTIVE OF THE MODULI SPACE

In this section, we assume $\Ext_A(T,T)=0$ and $\gamma_{T^s,d}=d$.

As an application of the arguments in Theorem <ref> and the explicit recursive formula given there, we derive a formula for the motive of the (smooth) moduli space $\M_{(s,d)}^\sst(\cC):=\M_{(s,d)}^\sst\big(A[T]\big)$ (over $\cC$).

To achieve this, we will follow closely the strategy of <cit.>, but replace counts of rational points over finite fields by motives as in the proof of <cit.>. We denote by $K_0({\rm Var}/\mathbb{C})$ the free abelian group generated by representatives $[X]$ of isomorphism classes of complex varieties $X$, modulo the relation $[X]=[C]+[U]$ if $C$ is isomorphic to a closed subvariety of $X$ with open complement isomorphic to $U$. Multiplication in $K_0({\rm Var}/\mathbb{C})$ is given by $[X]\cdot[Y]=[X\times Y]$. We denote by $\mathbb{L}$ the class of the affine line; the following calculations will be performed in the localization
$$\mathcal{K}=K_0({\rm Var}/\mathbb{C})[\mathbb{L}^{-1},(1-\mathbb{L}^n)^{-1}\, :\, n\geq 1].$$

At this point, we recall a notation from the subsection <ref>. Let

\[
H:=\left\{ (s,d)^* \,:\, \exists\, r\in\cN_{>1},\,\text{$(s,d)^* = \big( (s^1,d^1),\ldots, (s^r,d^r) \big)$ is of HN-type} \right.
\]

\[
\left. \text{with weight $(s,d)$ and $s^r\neq0$}  \right\}. ~~~~~~~
\]

In this section, we write $\Rep_{(s,d)}$ for $\Rep_{(s,d)}\voll(A[T])$ and $\M_{(s,d)}^\sst  $ for $\M_{(s,d)}^\sst\big(A[T]\big)$.

§.§ Motive

Obviously $H$ is finite and by Theorem <ref> we have

\[
\Rep_{(s,d)}~~=~~ \Rep_{(s,d)}^\sst ~\sqcup \bigsqcup_{(s,d)^*\in H} \Rep^{HN}_{(s,d)^*}
\]

and thus

\[
[ \Rep_{(s,d)}^\sst] ~~=~~ [ \Rep_{(s,d)}]
~-~ \sum_{(s,d)^*\in H} [ \Rep^{HN}_{(s,d)^*}]
\]

in $\mathcal{K}$. We then find

\[
\frac{[ \Rep_{(s,d)}^\sst]}{[\G_{(s,d)}]}  ~~=~~ \frac{[\Rep_{(s,d)}]}{[\G_{(s,d)}]}
~-~ \sum_{(s,d)^*\in H} \frac{[ \Rep^{HN}_{(s,d)^*}]}{[\G_{(s,d)}]} .
\]

Fix $(s,d)^*\in H$. As in the proof of Theorem <ref> we have

\begin{eqnarray*}
\Rep^{HN}_{(s,d)^*}  & \simeq & \G_{(s,d)}\times^{\mathcal{P}_{(s,d)^*}}\mathcal{Z}_0\,, \\
{[\mathcal{Z}_{0}]}& =& \mathbb{L}^{\sum_{n<l} \sum_{\alpha: i\to j} d_i^ld_j^n + s^l \langle t, d^n\rangle_Q} ~~\cdot~~\prod_{i=1}^r[\Rep_{(d^i,s^i)}^\sst],
\end{eqnarray*}

and we arrive at

\[
[\Rep^{HN}_{(s,d)^*}] ~~=~~
\frac{[ \G_{(s,d)}] }{[ \mathcal{P}_{(s,d)^*}]}
\mathbb{L}^{\sum_{n<l} \sum_{\alpha: i\to j} d_i^ld_j^n + s^l \langle t, d^n\rangle_Q} ~~\cdot~~\prod_{i=1}^r[\Rep_{(d^i,s^i)}^\sst].
\]

This provides us with the motivic HN-recursion:

\[
\frac{[ \Rep_{(s,d)}^\sst]}{[\G_{(s,d)}]}  ~~~=~~~ \frac{[ \Rep_{(s,d)}]}{[\G_{(s,d)}]}
~~~-~~ \sum_{(s,d)^*\in H}\frac{\mathbb{L}^{\sum_{n<l} \sum_{\alpha: i\to j} d_i^ld_j^n + s^l \langle t, d^n\rangle_Q}}{[ \Rep_{(s,d)^*}]}
~~\cdot~~ \prod_{i=1}^r [\Rep_{(d^i,s^i)}^\sst]\;\;\,.
\]

Using the arguments of <cit.>, we see that the following relation

\[
\sum_{i\in\cZ}\dim_{\cC} \mathbf{H}^i\big(\M_{(s,d)}^\sst,\mathbb{Q}\big)\mathbb{L}^{i/2}
~~=~~ [ \M_{(s,d)}^\sst  ]
\]

holds and we obtain:
Let $(s,d)$ be a dimension vector such that semi-stability and stability  coincide. Then, with $\mathcal{P}\G_{(s,d)}:=\G_{(s,d)}/k^\times$ we have:

\[
\sum_{i\in\cZ}\dim_{\cC}\mathbf{H}^i\big(\M_{(s,d)}^\sst,\mathbb{Q}\big)\mathbb{L}^{i/2}~=~
\frac{[ \Rep_{(s,d)}]}{[\mathcal{P}\G_{(s,d)}]}  ~~~~-
\]

\[
(\mathbb{L}-1)\sum_{(s,d)^*\in H} \frac{1}{[ \mathcal{P}_{(s,d)^*}]} \cdot \mathbb{L}^{\sum_{n<l} \sum_{\alpha: i\to j} d_i^ld_j^n + s^l \langle t, d^n\rangle_Q}
~~\cdot~~ \prod_{i=1}^r [\Rep_{(d^i,s^i)}^\sst ]\;\;\,.
\]

As in <cit.> we have

\[
[ \M_{(s,d)}^\sst] ~= ~\frac{[ \Rep_{(s,d)}^\sst]}{[\mathcal{P}\G_{(s,d)}]},
\]

and the claim follows from the previous theorem.

§.§ Applications and examples

Let $Q$ be of Dynkin type, let $\mathrm{Is}(Q,d)$ be the set theoretic quotient of $\Rep_{d}(Q)$ by the structure group $\G_{d}$ via the base change action, and for $M\in \Rep_{d}(Q)$ let

\[
\Hom_A^\mathrm{epi}(T,M):=\big\{ f\in\Hom_A(T,M) \,:\, \text{$f$ is surjective}\big\}.
\]

Then, we have

\[
[ \Rep_{(s,d)}]  ~~=~~
\sum_{ \substack{ [M]\in\mathrm{Is}(Q,d),\\ \exists\,T^s\epi M} } [ \mathcal{O}_M ]\, \cdot\, [\Hom_A^\mathrm{epi}(T^s,M) ].
\]

We consider the map

\[
\Rep_{(s,d)}\to \bigg\{ M\in \Rep_d(Q)\,:\, \exists\, T^s\epi M \bigg\},\,(M,f)\mapsto M.
\]

and note that $[\Hom_A^\mathrm{epi}(T,?)]$ is constant along orbits.

Overall, we find:
Let $Q$ be of Dynkin type, and let $(s,d)$ be a dimension vector such that semi-stability and stability coincide. Then we have:

\[
\sum_{i\in\cZ}\dim_{\cC}\mathbf{H}^i\big(\M_{(s,d)}^\sst,\mathbb{Q}\big)\mathbb{L}^{i/2}~=~
\frac{1}{[\mathcal{P}\G_{(s,d)}]} ~~\cdot~~
\sum_{ \substack{ [M]\in\mathrm{Is}(Q,d),\\ \exists\,T^s\epi M} } [ \mathcal{O}_M] \,\cdot\, [\Hom_A^\mathrm{epi}(T^s,M) ]~~-~~
\]

\[
(\mathbb{L}-1)~~\cdot~~ \sum_{(s,d)^*\in H} \underbrace{\frac{1}{[ \mathcal{P}_{(s,d)^*}]} \cdot \mathbb{L}^{\sum_{n<l} \sum_{\alpha: i\to j} d_i^ld_j^n + s^l \langle t, d^n\rangle_Q}
\cdot~~ \prod_{i=1}^r [\Rep_{(d^i,s^i)}^\sst]}_{=:\mathcal{S}_{(s,d)^*}}\;\;\,.
\]

$ $

\[
Q=\big( 1\to 2\big),~~T=\bigg( k^3\xto{[1,0,0]}k\bigg),
\]

and $(s,d)=(2,4,1)$. Then $H$ consists of

\[
(1,2,0)\, >_\mu(1,2,1),~~~~(1,1,1)\, >_\mu(1,3,0),~~~~(1,1,0)\, >_\mu(1,3,1).
\]

We calculate all components in the formula for each element in $H$:

\begin{eqnarray*}
\mathcal{S}_{(1,2,0)\, >_\mu(1,2,1)} & = & \frac{(\mathbb{L}^3-1)(\mathbb{L}^3-\mathbb{L})}{(\mathbb{L}-1)^4},\\
\mathcal{S}_{(1,1,1)\, >_\mu(1,3,0)} & = & \frac{1}{(\mathbb{L}-1)^2},\\
\mathcal{S}_{(1,1,0)\, >_\mu(1,3,1)} & = & \frac{(\mathbb{L}^3-1)}{\mathbb{L}(\mathbb{L}-1)^3}.\\
\end{eqnarray*}

We continue to calculate and get

\[
\frac{[ \Rep_{(s,d)}]}{[\mathcal{P}\G_{(s,d)}]}~=~
\frac{\mathbb{L}^2(\mathbb{L}^4-1)}{(\mathbb{L}-1)^2}~+~\frac{(\mathbb{L}^3-1)(\mathbb{L}^4-1)}{(\mathbb{L}-1)(\mathbb{L}^2-1)(\mathbb{L}^2-\mathbb{L})}.
\]

In total, we end up with:

\begin{eqnarray*}
\sum_{i\in\cZ}\dim_{\cC}\mathbf{H}^i\big(\M_{(s,d)}^\sst,\mathbb{Q}\big)\mathbb{L}^{i/2}
& = &
\frac{\mathbb{L}^2(\mathbb{L}^4-1)}{(\mathbb{L}-1)^2}~+~\frac{(\mathbb{L}^3-1)(\mathbb{L}^4-1)}{(\mathbb{L}-1)(\mathbb{L}^2-1)(\mathbb{L}^2-\mathbb{L})} \\
&  &~-~
\frac{(\mathbb{L}^3-1)(\mathbb{L}^3-\mathbb{L})}{(\mathbb{L}-1)^3}~-~\frac{1}{(\mathbb{L}-1)}~-~\frac{(\mathbb{L}^3-1)}{\mathbb{L}(\mathbb{L}-1)^2} \\
& = & 1+\mathbb{L}+\mathbb{L}^2+\mathbb{L}^3+\mathbb{L}^4.
\end{eqnarray*}

This result was to be expected if one remembers our explicit calculations of the moduli space.

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# BEAM-BEAM EFFECTS IN THE SPS PROTON-ANTI PROTON COLLIDER

K. Cornelis  CERN  Geneva  Switzerland

###### Abstract

During the proton-anti proton collider run several experiments were carried
out in order to understand the effect of the beam-beam interaction on
backgrounds and lifetimes. In this talk a selection of these experiments will
be presented. From these experiments, the importance of relative beam sizes
and tune ripple could be demonstrated.

##  1 GENERAL LAYOUT OF SPS COLLIDER OPERATIONS

In the first collider runs, the Super Proton Synchrotron (SPS) was operated
with three proton bunches against three anti-proton bunches, colliding in six
collision points. The proton bunch intensity at that time was close to
$2\times 10^{11}$ and the anti-proton intensities were about ten times less.
In this configuration, total tune shifts of 0.028 were sometimes obtained but
the anti-proton lifetimes in the beginning of the coast were poor. A
horizontal pretzel scheme was introduced to separate the beams in the unwanted
collision points (Fig. 1) and the SPS could then profit from the upgraded
anti-proton accumulation facility, operating with six against six bunches. In
this scheme the beams were separated in 9 of the 12 crossing points.

Figure 1: Schematic layout of the SPS pretzel scheme.

The beam separation was 6 $\sigma$ or better in all the crossing points,
except for one, where the separation was only 3.5 $\sigma$ (Fig. 2).

Figure 2: Beam half-separation in the 12 crossing points.

The same electrostatic separators were used during injection to separate the
beams at all crossing points, in order to keep emittances small during the
injection and acceleration process. The parameters of the SPS collider in the
final six-on-six operation are listed in Table 1.

Table 1: General parameters during the SPS collider run with six protons on six anti-protons. The beam emittance $\epsilon$ is defined as $\epsilon=4\gamma\sigma^{2}/\beta$.  Parameter | value
---|---
Beam injection energy | 26 GeV
Beam coast energy | 315 GeV
Proton intensity |  $1.7\times 10^{11}$ p/bunch, 6 bunches
Anti-proton intensity |  $0.8\times 10^{11}$ $\overline{\mathrm{p}}$/bunch, 6 bunches
$\epsilon_{x},\epsilon_{y}$ | 15–20 mm mrad
$\xi_{x,y}$ | 0.015–0.020(total)

The tune diagram with typical proton and anti-proton footprints during physics
runs is shown in Fig.3. The tune difference between protons and anti-protons
could be trimmed with sextupoles, placed where the beams were separated. The
core of the beam is at the higher tune values.

Figure 3: Working point during physics runs. The horizontal and vertical lines
represent respectively the 16th (.6875), the 13th (.6923) and the 10th (.700)
order resonance.

##  2 THE INFLUENCE OF BEAM SIZE

In some of the runs, where the machine went into coast with substantially
smaller pbar emittances, the protons had a substantially shorter lifetime and
gave a lot of proton background, and was in spite of the fact that the pbar
intensity was 10 times less than the proton intensity. This phenomenon could
be artificially reproduced by scraping one of the beams with a collimator at a
place where they are separated and observing the background and/or the
lifetime of the other beam. In Fig.4 a horizontal tune scan is shown looking
at the proton background before and after the pbar emittance was reduced by
30%. Although the intensity of the anti-protons was reduced by the scraping
process, the protons seem to suffer much more from the beam-beam interaction.
This experiment is showing that the high amplitude particles (amplitude
measured in units of $\sigma$ of the other beam) suffer from the high-order
resonances.

Figure 4: Proton background as a function of tune, before (Pbgd1) and after
(Pbgd2) the pbar emittance was reduced by 30%.

In fact it turned out to be very important for the lifetime and background
that the two beams had the same size. In case of unequal beam sizes, the
bigger beam would loose all the particles sitting outside the other beam
mainly, creating a low lifetime and high background in the beginning of the
coast. This is illustrated in Fig.5, where the evolution of the proton and
anti-proton emittances is plotted during a normal physics coast. By accident
the protons are smaller than the anti-protons at the beginning, but throughout
the coast the anti-protons mainly lose high amplitude particles and after
three to four hours the anti-proton emittance is reduced and matches the
proton emittance. The small growth of the proton emittance is due to intra-
beam scattering.

Figure 5: Evolution of the proton and anti-proton emittance during the first
200 min. of a coast.

##  3 THE INFLUENCE OF SEPARATION

Tune scans were also performed for different separations. Reducing the
separation in the parasitic crossing from 6 $\sigma$ to 3 $\sigma$, increases
the background on the 16th as well as on the 13th order resonance as can be
seen in Fig. 6. In another experiment only one bunch of protons was colliding
head-on with one bunch of anti-protons at two collision points. The beam was
then separated at one of the two points in steps of 0.1 $\sigma$. The result
is shown in Fig. 7. The background rises very quickly as a function of
separation, reaching a maximum at 0.3 $\sigma$. The background then decreases
very slowly as a function of further separation.

Figure 6: Tune scans with full (Pbgd1) and half separation (Pbgd2). Figure 7:
Proton background as a function of the separation.

##  4 THE EFFECT OF TUNE MODULATION

High-order resonances in non-linear fields manifest themselves as stable
islands in phase space. Most of the particles will have a varying amplitude
but the motion stays periodic and stable. Tune modulation will make islands
move inward and outward. For small frequencies, the particles will stay
trapped in the islands and they can be transported to very high amplitudes due
to tune variation. If the tune is varied fast enough there will be passages
created between the islands through which the particles can move very quickly
inward or outward (cf. empty bucket acceleration). At even higher frequencies
the separatrices of the island become transparent. Particles can be trapped in
an island at a low amplitude and leave the island again at a high amplitude.
Whatever the mechanism, tune modulation, together with non-linear resonances,
creates the possibilities for particles to move to higher amplitudes. This
could be very clearly observed in the SPS: in order to have good lifetimes
with colliding beams, the chromaticity had to be tuned as close as possible to
zero. Also, the noise on the main magnets had be reduced to a minimum in order
to preserve good lifetimes during collision. The effect of tune modulation can
be easily demonstrated by simulation. In Fig. 8 the results are shown of a
simple simulation using a head-on beam-beam kick separated by a linear
transfer matrix in which the phase advance is modulated with a frequency of
200 Hz. In the first case the tune modulation is $5\times 10^{-5}$ and in the
second case it is $3\times 10^{-4}$. The diffusion rate ($z$-axis) is shown as
a function of the unperturbed tune ($x$-axis) and the initial amplitude
($y$-axis) ranging from 0 to 10 $\sigma$. In both cases the 16th and 19th
order resonance can clearly be seen. In the first case, the particles on the
10th order resonance move out with a diffusion time constant of 1 minute or
faster only from 4 $\sigma$ onwards and on the 16th order only from 7 $\sigma$
onwards. In the second case, with the stronger tune modulation, the particles
on the 10th order move out with the same speed already at 2.5 $\sigma$ and on
the 16th order already at 5 $\sigma$.

Figure 8: Diffusion rates ($z$) as function of tune ($x$) and initial
amplitude ($y$: 0 to 10 $\sigma$). Total tune shift 0.012. Tune modulation 200
Hz, modulation amplitude 0.000 05 (above) and 0.0003 (below).

##  5 CONCLUSIONS

  * •

Experience with the SPS proton anti-proton collider showed that it is very
important to have the same beam sizes for both beams in order to obtain good
lifetimes and backgrounds;

  * •

The high order resonances have almost no effect on the particles with small
amplitudes;

  * •

The ’bad’ effect of miss-crossing reaches a maximum at a separation of 0.2 to
0.3 $\sigma$.

  * •

All tune modulation (chromaticity, power supply ripple, etc.) should be
reduced to a minimum.