2.1 NC nilpotent algebras

Let $\unicode[STIX]{x1D6EC}$ be an algebra. We regard $\unicode[STIX]{x1D6EC}$ as a Lie algebra by setting $[a,b]=ab-ba$ . The subspace $\unicode[STIX]{x1D6EC}_{k}^{\text{Lie}}\subset \unicode[STIX]{x1D6EC}$ is defined to be spanned by the elements of the form

$$\begin{eqnarray}\displaystyle [x_{1},[x_{2},\ldots ,[x_{k-1},x_{k}]\cdots \,]] & & \displaystyle \nonumber\end{eqnarray}$$

for $x_{i}\in \unicode[STIX]{x1D6EC}$ , $1\leqslant i\leqslant k$ . The NC filtration of $\unicode[STIX]{x1D6EC}$ is the decreasing filtration

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}=F^{0}\unicode[STIX]{x1D6EC}\supset F^{1}\unicode[STIX]{x1D6EC}\supset \cdots \supset F^{d}\unicode[STIX]{x1D6EC}\supset \cdots \,, & & \displaystyle \nonumber\end{eqnarray}$$

where $F^{d}\unicode[STIX]{x1D6EC}$ is the two-sided ideal of $\unicode[STIX]{x1D6EC}$ defined by

$$\begin{eqnarray}\displaystyle F^{d}\unicode[STIX]{x1D6EC}:=\mathop{\sum }_{m\geqslant 0}\mathop{\sum }_{i_{1}+\cdots +i_{m}=m+d}\unicode[STIX]{x1D6EC}\cdot \unicode[STIX]{x1D6EC}_{i_{1}}^{\text{Lie}}\cdot \unicode[STIX]{x1D6EC}\cdot \cdots \cdot \unicode[STIX]{x1D6EC}_{i_{m}}^{\text{Lie}}\cdot \unicode[STIX]{x1D6EC}. & & \displaystyle \nonumber\end{eqnarray}$$

Note that $\unicode[STIX]{x1D6EC}/F^{1}\unicode[STIX]{x1D6EC}$ is the abelianization $\unicode[STIX]{x1D6EC}^{\text{ab}}$ of $\unicode[STIX]{x1D6EC}$ . We set $\unicode[STIX]{x1D6EC}^{{\leqslant}d}:=\unicode[STIX]{x1D6EC}/F^{d+1}\unicode[STIX]{x1D6EC}$ , and denote $N^{{\leqslant}d}:=\unicode[STIX]{x1D6EC}^{{\leqslant}d}\otimes _{\unicode[STIX]{x1D6EC}}N$ for $N\in \unicode[STIX]{x1D6EC}~\text{mod}$ .

Definition 2.1. (i) An algebra $\unicode[STIX]{x1D6EC}$ is called NC nilpotent of degree $d$ (respectively NC nilpotent) if $F^{d+1}\unicode[STIX]{x1D6EC}=0$ (respectively  $F^{n}\unicode[STIX]{x1D6EC}=0$ for $n\gg 0$ ).

(ii) The NC completion of an algebra $\unicode[STIX]{x1D6EC}$ is

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}_{[[\text{ab}]]}:=\lim _{\longleftarrow }\unicode[STIX]{x1D6EC}^{{\leqslant}d}. & & \displaystyle \nonumber\end{eqnarray}$$

(iii) An algebra $\unicode[STIX]{x1D6EC}$ is called NC complete if the natural map $\unicode[STIX]{x1D6EC}\rightarrow \unicode[STIX]{x1D6EC}_{[[\text{ab}]]}$ is an isomorphism.

For an algebra $\unicode[STIX]{x1D6EC}$ , its subquotient

$$\begin{eqnarray}\displaystyle \operatorname{gr}_{F}^{\bullet }(\unicode[STIX]{x1D6EC}):=\bigoplus _{d\geqslant 0}F^{d}\unicode[STIX]{x1D6EC}/F^{d+1}\unicode[STIX]{x1D6EC} & & \displaystyle \nonumber\end{eqnarray}$$

is a commutative algebra.

2.2 NC schemes

Let $\unicode[STIX]{x1D6EC}$ be an NC complete algebra. The affine NC scheme

(2.2.1) $$\begin{eqnarray}\displaystyle \operatorname{Spf}\unicode[STIX]{x1D6EC}=(\operatorname{Spec}\unicode[STIX]{x1D6EC}^{\text{ab}},{\mathcal{O}}_{Y}) & & \displaystyle\end{eqnarray}$$

is a ringed space defined in the following way. For any multiplicative set $S\subset \unicode[STIX]{x1D6EC}^{\text{ab}}$ , its pull-back by the natural surjection $\unicode[STIX]{x1D6EC}^{{\leqslant}d}{\twoheadrightarrow}\unicode[STIX]{x1D6EC}^{\text{ab}}$ determines the multiplicative set in $\unicode[STIX]{x1D6EC}^{{\leqslant}d}$ , which satisfies the Ore localization condition (cf. [Reference KapranovKap98, Proposition 2.1.5]). Therefore, similarly to the case of usual affine schemes, the NC nilpotent algebra $\unicode[STIX]{x1D6EC}^{{\leqslant}d}$ determines the sheaf of algebras ${\mathcal{O}}_{Y}^{{\leqslant}d}$ on $\operatorname{Spec}\unicode[STIX]{x1D6EC}^{\text{ab}}$ . The sheaf of algebras ${\mathcal{O}}_{Y}$ is defined by

$$\begin{eqnarray}\displaystyle {\mathcal{O}}_{Y}:=\lim _{\longleftarrow }{\mathcal{O}}_{Y}^{{\leqslant}d}. & & \displaystyle \nonumber\end{eqnarray}$$

The category of NC schemes is the full subcategory of ringed spaces consisting of NC schemes. For an NC scheme $Y$ , the category $\operatorname{Coh}(Y)$ is defined to be the category of coherent left ${\mathcal{O}}_{Y}$ -modules on $Y$ . Note that the structure sheaf ${\mathcal{O}}_{Y}$ has a filtration by sheaves of two-sided ideals $F^{n}{\mathcal{O}}_{Y}\subset {\mathcal{O}}_{Y}$ , which defines a topology on the set of sections of ${\mathcal{O}}_{Y}$ . For a sheaf of two-sided ideals ${\mathcal{J}}\subset {\mathcal{O}}_{Y}$ , we denote by $\overline{{\mathcal{J}}}\subset {\mathcal{O}}_{Y}$ its topological closure. Then the quotient ${\mathcal{O}}_{Y}/\overline{{\mathcal{J}}}$ is the sheaf of NC complete algebras and hence determines the NC closed subscheme of $Y$ . In particular, the quotient ${\mathcal{O}}_{Y}^{{\leqslant}d}:={\mathcal{O}}_{Y}/F^{d+1}{\mathcal{O}}_{Y}$ defines the NC subscheme $Y^{{\leqslant}d}\subset Y$ , and $Y^{\text{ab}}:=Y^{{\leqslant}0}$ is a usual scheme.

Definition 2.4. An NC structure on a commutative scheme $M$ is an NC scheme

(2.2.2) $$\begin{eqnarray}\displaystyle M^{\operatorname{nc}}=(M,{\mathcal{O}}_{M}^{\operatorname{nc}}) & & \displaystyle\end{eqnarray}$$

such that $({\mathcal{O}}_{M}^{\operatorname{nc}})^{\text{ab}}={\mathcal{O}}_{M}$ .

Here, in the right-hand side of (2.2.2), $M$ is just regarded as a topological space. In general, it is not easy to construct non-trivial NC structures on a given algebraic variety. Instead, we introduce the following weaker notion of NC structures.

Definition 2.5. Let $M$ be a commutative scheme. A quasi-NC structure on $M$ consists of an affine open cover $\{U_{i}\}_{i\in \mathbb{I}}$ of $M$ with affine $U_{ij}:=U_{i}\cap U_{j}$ , affine NC structures $(U_{i},{\mathcal{O}}_{U_{i}}^{\text{nc}})$ on $U_{i}$ for each $i\in \mathbb{I}$ and isomorphisms of affine NC schemes

(2.2.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{ij}:(U_{ij},{\mathcal{O}}_{U_{j}}^{\text{nc}}|_{U_{ij}})\stackrel{\cong }{\rightarrow }(U_{ij},{\mathcal{O}}_{U_{i}}^{\text{nc}}|_{U_{ij}}) & & \displaystyle\end{eqnarray}$$

satisfying $\unicode[STIX]{x1D719}_{ij}^{\text{ab}}=\operatorname{id}$ .

Remark 2.6. A quasi-NC structure gives rise to the NC structure if and only if the isomorphisms $\unicode[STIX]{x1D719}_{ij}$ in (2.2.3) satisfy the cocycle condition, i.e.

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{ij}\circ \unicode[STIX]{x1D719}_{jk}\circ \unicode[STIX]{x1D719}_{ki}=\text{id}. & & \displaystyle \nonumber\end{eqnarray}$$

Remark 2.7. Although (2.2.3) may not satisfy the cocycle condition, Lemma 2.2 implies that $\operatorname{gr}_{F}^{\bullet }(\unicode[STIX]{x1D719}_{ij})$ always satisfies the cocycle condition. Hence, $\operatorname{gr}_{F}^{\bullet }({\mathcal{O}}_{U_{i}}^{\operatorname{nc}})$ glue together to give a sheaf of algebras on $M$ . In particular, we have the commutative scheme over $M$

$$\begin{eqnarray}\displaystyle \mathop{\bigcup }_{i\in \mathbb{I}}\operatorname{Spec}\,(\operatorname{gr}_{F}^{\bullet }({\mathcal{O}}_{U_{i}}^{\operatorname{nc}}))\rightarrow M. & & \displaystyle \nonumber\end{eqnarray}$$

2.3 Smooth NC schemes

Let ${\mathcal{N}}_{d}$ be the category of NC nilpotent algebras of degree $d$ and ${\mathcal{N}}$ the category of NC nilpotent algebras. We have the following inclusions:

(2.3.1) $$\begin{eqnarray}\displaystyle {\mathcal{C}}om:={\mathcal{N}}_{0}\subset {\mathcal{N}}_{1}\subset \cdots \subset {\mathcal{N}}_{d}\subset \cdots \subset {\mathcal{N}}. & & \displaystyle\end{eqnarray}$$

Let $\unicode[STIX]{x1D6EC}^{\prime }{\twoheadrightarrow}\unicode[STIX]{x1D6EC}$ be a surjection in ${\mathcal{N}}$ and take the exact sequence

(2.3.2) $$\begin{eqnarray}\displaystyle 0\rightarrow J\rightarrow \unicode[STIX]{x1D6EC}^{\prime }\rightarrow \unicode[STIX]{x1D6EC}\rightarrow 0. & & \displaystyle\end{eqnarray}$$

The sequence (2.3.2) is called a central extension if $J^{2}=0$ and $J$ lies in the center of $\unicode[STIX]{x1D6EC}^{\prime }$ . In particular, $J$ is a $\unicode[STIX]{x1D6EC}^{\text{ab}}$ -module.

Definition 2.8. For functors $h_{1},h_{2}:{\mathcal{N}}\rightarrow {\mathcal{S}}\!\mathit{et}$ , a natural transform $\unicode[STIX]{x1D719}:h_{1}\rightarrow h_{2}$ is called formally smooth if for any central extension (2.3.2) in ${\mathcal{N}}$ , the map

$$\begin{eqnarray}\displaystyle h_{1}(\unicode[STIX]{x1D6EC}^{\prime })\rightarrow h_{2}(\unicode[STIX]{x1D6EC}^{\prime })\times _{h_{2}(\unicode[STIX]{x1D6EC})}h_{1}(\unicode[STIX]{x1D6EC}) & & \displaystyle \nonumber\end{eqnarray}$$

is surjective.

An NC scheme $Y$ defines a covariant functor

$$\begin{eqnarray}\displaystyle h_{Y}:{\mathcal{N}}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle \nonumber\end{eqnarray}$$

sending $\unicode[STIX]{x1D6EC}$ to $\operatorname{Hom}(\operatorname{Spf}\unicode[STIX]{x1D6EC},Y)$ . In the situation of Definition 2.8, a functor $h:{\mathcal{N}}\rightarrow {\mathcal{S}}\!\mathit{et}$ is called formally smooth if the natural transform $h\rightarrow h_{\operatorname{Spec}\mathbb{C}}$ is formally smooth, i.e. for any central extension (2.3.2), the map $h(\unicode[STIX]{x1D6EC}^{\prime })\rightarrow h(\unicode[STIX]{x1D6EC})$ is surjective. If the same condition holds only for central extensions (2.3.2) in ${\mathcal{N}}_{d}$ , we call the functor $h|_{{\mathcal{N}}_{d}}$ formally $d$ -smooth.

It is easy to see that if an NC scheme $Y$ is smooth, then $Y^{{\leqslant}d}$ is $d$ -smooth. In particular, $Y_{\text{ab}}$ is a smooth scheme in the usual sense. If $Y$ is NC smooth, we say that $Y$ is an NC smooth thickening of $Y_{\text{ab}}$ . By [Reference KapranovKap98, Theorem 1.6.1], any smooth affine scheme has an NC smooth thickening, which is unique up to non-canonical isomorphisms. In particular, any smooth quasi-projective variety has a smooth quasi-NC structure in the sense of Definition 2.5.

2.4 (Quasi) NC structures via functors

We introduce the notion of an NC hull to construct quasi-NC structures.

We have the following lemma.

Proof. For $i=1,2$ , let $\unicode[STIX]{x1D719}_{i}:h_{Y_{i}}\rightarrow h$ be affine NC hulls, and write $Y_{i}=\operatorname{Spf}\unicode[STIX]{x1D6EC}_{i}$ . Note that, by the formal smoothness of NC hulls, the natural map

(2.4.1) $$\begin{eqnarray}\displaystyle \lim _{\longleftarrow }h_{Y_{i}}(\unicode[STIX]{x1D6EC}^{{\leqslant}d})\rightarrow \lim _{\longleftarrow }h(\unicode[STIX]{x1D6EC}^{{\leqslant}d}) & & \displaystyle\end{eqnarray}$$

is surjective for any NC complete algebra $\unicode[STIX]{x1D6EC}$ . Since the left-hand side of (2.4.1) coincides with $\operatorname{Hom}(\unicode[STIX]{x1D6EC}_{i},\unicode[STIX]{x1D6EC})$ , there exist algebra homomorphisms $u:\unicode[STIX]{x1D6EC}_{1}\rightarrow \unicode[STIX]{x1D6EC}_{2}$ , $v:\unicode[STIX]{x1D6EC}_{2}\rightarrow \unicode[STIX]{x1D6EC}_{1}$ which commute with $\unicode[STIX]{x1D719}_{1}$ , $\unicode[STIX]{x1D719}_{2}$ . Then the compositions

(2.4.2) $$\begin{eqnarray}\displaystyle v\circ u:\unicode[STIX]{x1D6EC}_{1}\rightarrow \unicode[STIX]{x1D6EC}_{1},\quad u\circ v:\unicode[STIX]{x1D6EC}_{2}\rightarrow \unicode[STIX]{x1D6EC}_{2} & & \displaystyle\end{eqnarray}$$

satisfy $(v\circ u)^{\text{ab}}=\operatorname{id}$ , $(u\circ v)^{\text{ab}}=\operatorname{id}$ . Therefore, Lemma 2.2 implies that (2.4.2) are algebra isomorphisms and hence $u$ , $v$ are isomorphisms.◻

Let $M$ be a commutative scheme and $h:{\mathcal{N}}\rightarrow {\mathcal{S}}\!\mathit{et}$ a functor. If $h|_{{\mathcal{C}}om}=h_{M}$ , we have the natural transform $h\rightarrow h_{M}$ applying $h$ to $\unicode[STIX]{x1D6EC}\rightarrow \unicode[STIX]{x1D6EC}^{\text{ab}}$ for $\unicode[STIX]{x1D6EC}\in {\mathcal{N}}$ . For a subscheme $U\subset M$ , we define

$$\begin{eqnarray}\displaystyle h|_{U}:=h\times _{h_{M}}h_{U}. & & \displaystyle \nonumber\end{eqnarray}$$

The following corollary obviously follows from Lemma 2.11.

We next describe some condition for the functor $h_{Y}\rightarrow h$ to give an NC hull. For a functor $h:{\mathcal{N}}\rightarrow {\mathcal{S}}\!\mathit{et}$ and a Cartesian diagram of algebras in ${\mathcal{N}}$

we consider the natural map

(2.4.3) $$\begin{eqnarray}\displaystyle h(\unicode[STIX]{x1D6EC}_{12})\rightarrow h(\unicode[STIX]{x1D6EC}_{1})\times _{h(\unicode[STIX]{x1D6EC})}h(\unicode[STIX]{x1D6EC}_{2}). & & \displaystyle\end{eqnarray}$$

Proof. The proof is similar to the classical argument on the existence of pro-representable hulls (cf. [Reference SchlessingerSch68, Theorem 2.11]). We consider a central extension in ${\mathcal{N}}$

$$\begin{eqnarray}\displaystyle 0\rightarrow J\rightarrow \unicode[STIX]{x1D6EC}^{\prime }\stackrel{p}{\rightarrow }\unicode[STIX]{x1D6EC}\rightarrow 0 & & \displaystyle \nonumber\end{eqnarray}$$

and show that the map

(2.4.4) $$\begin{eqnarray}\displaystyle h_{Y}(\unicode[STIX]{x1D6EC}^{\prime })\rightarrow h(\unicode[STIX]{x1D6EC}^{\prime })\times _{h(\unicode[STIX]{x1D6EC})}h_{Y}(\unicode[STIX]{x1D6EC}) & & \displaystyle\end{eqnarray}$$

is surjective. An element of the right-hand side of (2.4.4) is given by a commutative diagram

(2.4.5)

We need to find a dotted arrow $\unicode[STIX]{x1D6FE}$ so that both of the lower and upper triangles in (2.4.5) are commutative. Since $h_{Y}$ is smooth, there is a dotted arrow $\unicode[STIX]{x1D6FE}$ in (2.4.5) so that the lower triangle is commutative. The set of possible choices of such a $\unicode[STIX]{x1D6FE}$ is a principal homogeneous space over $\text{Der}({\mathcal{O}}_{Y^{\text{ab}}},J)$ . Here we regard $J$ as an ${\mathcal{O}}_{Y^{\text{ab}}}$ -module by the algebra homomorphism $\unicode[STIX]{x1D6FC}^{\text{ab}}:{\mathcal{O}}_{Y}^{\text{ab}}\rightarrow \unicode[STIX]{x1D6EC}^{\text{ab}}$ induced by $\unicode[STIX]{x1D6FC}$ .

On the other hand, we have the isomorphism

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}^{\prime }\times _{\unicode[STIX]{x1D6EC}}\unicode[STIX]{x1D6EC}^{\prime }\stackrel{\cong }{\rightarrow }\unicode[STIX]{x1D6EC}^{\prime }\times _{\unicode[STIX]{x1D6EC}^{\text{ab}}}(\unicode[STIX]{x1D6EC}^{\text{ab}}\oplus J) & & \displaystyle \nonumber\end{eqnarray}$$

given by $(x,y)\mapsto (x,x^{\text{ab}}+y-x)$ . Therefore, the assumption on the map (2.4.3) implies the surjection

(2.4.6) $$\begin{eqnarray}\displaystyle h(\unicode[STIX]{x1D6EC}^{\prime })\times _{h(\unicode[STIX]{x1D6EC}^{\text{ab}})}h(\unicode[STIX]{x1D6EC}^{\text{ab}}\oplus J){\twoheadrightarrow}h(\unicode[STIX]{x1D6EC}^{\prime })\times _{h(\unicode[STIX]{x1D6EC})}h(\unicode[STIX]{x1D6EC}^{\prime }). & & \displaystyle\end{eqnarray}$$

Let $\unicode[STIX]{x1D702}\in h(\unicode[STIX]{x1D6EC})$ be the element corresponding to the composition $\unicode[STIX]{x1D719}\circ \unicode[STIX]{x1D6FC}$ in the diagram (2.4.5). The surjection (2.4.6) restricts to the surjection

(2.4.7) $$\begin{eqnarray}\displaystyle h(p)^{-1}(\unicode[STIX]{x1D702})\times h(q)^{-1}(\unicode[STIX]{x1D702}^{\text{ab}}){\twoheadrightarrow}h(p)^{-1}(\unicode[STIX]{x1D702})\times h(p)^{-1}(\unicode[STIX]{x1D702}). & & \displaystyle\end{eqnarray}$$

Here $q:\unicode[STIX]{x1D6EC}^{\text{ab}}\oplus J\rightarrow \unicode[STIX]{x1D6EC}^{\text{ab}}$ is the projection. Since $\unicode[STIX]{x1D702}^{\text{ab}}\in h(\unicode[STIX]{x1D6EC}^{\text{ab}})$ is identified with the morphism $\unicode[STIX]{x1D6FC}^{\text{ab}}:{\mathcal{O}}_{Y}^{\text{ab}}\rightarrow \unicode[STIX]{x1D6EC}^{\text{ab}}$ , we have

$$\begin{eqnarray}\displaystyle h(q)^{-1}(\unicode[STIX]{x1D702}^{\text{ab}})=\text{Der}({\mathcal{O}}_{Y^{\text{ab}}},J). & & \displaystyle \nonumber\end{eqnarray}$$

Therefore, the surjection (2.4.7) shows that $\text{Der}({\mathcal{O}}_{Y^{\text{ab}}},J)$ acts on $h(p)^{-1}(\unicode[STIX]{x1D702})$ transitively.

Note that if $\unicode[STIX]{x1D6FE}$ is a dotted arrow in (2.4.5) so that the lower triangle is commutative, then we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}\circ \unicode[STIX]{x1D6FE}\in h(p)^{-1}(\unicode[STIX]{x1D702}),\quad \unicode[STIX]{x1D6FD}\in h(p)^{-1}(\unicode[STIX]{x1D702}). & & \displaystyle \nonumber\end{eqnarray}$$

Therefore, by acting $\text{Der}({\mathcal{O}}_{Y^{\text{ab}}},J)$ on $\unicode[STIX]{x1D6FE}$ , we can also make the upper triangle of (2.4.5) commutative.◻

Finally, we state the construction of NC structures via functors. In the following proposition, we put ${\mathcal{N}}_{\infty }={\mathcal{N}}$ .

2.5 NC hull and pro-representable hull

Let

$$\begin{eqnarray}\displaystyle {\mathcal{N}}^{\text{loc}}\subset {\mathcal{N}} & & \displaystyle \nonumber\end{eqnarray}$$

be the subcategory of local finite-dimensional $\mathbb{C}$ -algebras, i.e. an object of ${\mathcal{N}}^{\text{loc}}$ is a finite-dimensional $\mathbb{C}$ -algebra $\unicode[STIX]{x1D6EC}$ having the unique two-sided maximal ideal $\mathbf{n}\subset \unicode[STIX]{x1D6EC}$ . Note that a complete local $\mathbb{C}$ -algebra $\widehat{R}$ defines the functor

$$\begin{eqnarray}\displaystyle h_{\widehat{R}}:{\mathcal{N}}^{\text{loc}}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle \nonumber\end{eqnarray}$$

by sending $\unicode[STIX]{x1D6EC}$ to the set of local $\mathbb{C}$ -algebra homomorphisms $\widehat{R}\rightarrow \unicode[STIX]{x1D6EC}$ . We recall the notion of a pro-representable hull in [Reference SchlessingerSch68].

The proof similar to Lemma 2.11 shows that a pro-representable hull is unique up to non-canonical isomorphisms (cf. [Reference SchlessingerSch68, Proposition 2.9]).

Let $h:{\mathcal{N}}\rightarrow {\mathcal{S}}\!\mathit{et}$ be a functor, and suppose that it has an NC hull $h_{Y}\rightarrow h$ for an NC scheme  $Y$ . Note that $h_{Y}(\mathbb{C})\rightarrow h(\mathbb{C})$ is bijective, and they are identified with the set of closed points in  $Y^{\text{ab}}$ . Given a closed point $y\in Y^{\text{ab}}$ , we define the functor

(2.5.1) $$\begin{eqnarray}\displaystyle h_{y}^{\text{loc}}:{\mathcal{N}}^{\text{loc}}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle\end{eqnarray}$$

by sending $(\unicode[STIX]{x1D6EC},\mathbf{n})$ to the preimage of $y\in h(\mathbb{C})$ under the map

$$\begin{eqnarray}\displaystyle h(\unicode[STIX]{x1D6EC})\rightarrow h(\unicode[STIX]{x1D6EC}/\mathbf{n})=h(\mathbb{C}). & & \displaystyle \nonumber\end{eqnarray}$$

Let $m_{y}^{\text{ab}}\subset {\mathcal{O}}_{Y^{\text{ab}}}$ be the ideal sheaf which defines $y$ , and $m_{y}\subset {\mathcal{O}}_{Y}$ the two-sided ideal sheaf given by the pull-back of $m_{y}^{\text{ab}}$ by the surjection ${\mathcal{O}}_{Y}{\twoheadrightarrow}{\mathcal{O}}_{Y^{\text{ab}}}$ . We denote by $\widehat{{\mathcal{O}}}_{Y,y}$ the completion of ${\mathcal{O}}_{Y}$ by $m_{y}$ . We also set

$$\begin{eqnarray}\displaystyle {\mathcal{A}}\mathit{rt}^{\text{loc}}:={\mathcal{N}}^{\text{loc}}\cap {\mathcal{C}}om, & & \displaystyle \nonumber\end{eqnarray}$$

i.e. ${\mathcal{A}}\mathit{rt}^{\text{loc}}$ is the category of local Artinian $\mathbb{C}$ -algebras.

3.1 Representations of quivers

Recall that a quiver consists of data

$$\begin{eqnarray}\displaystyle Q=(Q_{0},Q_{1},h,t), & & \displaystyle \nonumber\end{eqnarray}$$

where $Q_{0}$ and $Q_{1}$ are finite sets (called the sets of vertices and arrows, respectively) and $h,t:Q_{1}\rightarrow Q_{0}$ are maps. The maps $h,t$ indicate the vertices at the head and tail of each arrow, respectively.

Definition 3.1. A representation of a quiver $Q$ over an NC scheme $(Y,{\mathcal{O}}_{Y})$ consists of data

(3.1.1) $$\begin{eqnarray}\displaystyle {\mathcal{W}}=(\{{\mathcal{W}}_{v}\}_{v\in Q_{0}},\{\unicode[STIX]{x1D719}_{a}\}_{a\in Q_{1}}) & & \displaystyle\end{eqnarray}$$

where each ${\mathcal{W}}_{v}$ is an object of $\operatorname{Coh}(Y)$ , and $\unicode[STIX]{x1D719}_{a}:{\mathcal{W}}_{t(a)}\rightarrow {\mathcal{W}}_{h(a)}$ is a morphism of coherent left ${\mathcal{O}}_{Y}$ -modules.

Given a representation (3.1.1) of $Q$ over $Y$ , we set

$$\begin{eqnarray}\displaystyle {\mathcal{W}}_{\bullet }:=\bigoplus _{v\in Q_{0}}{\mathcal{W}}_{v}. & & \displaystyle \nonumber\end{eqnarray}$$

Let $\mathbb{C}[Q]$ be the path algebra of $Q$ . By the definition, we have the natural algebra homomorphism

(3.1.2) $$\begin{eqnarray}\displaystyle \mathbb{C}[Q]\rightarrow \operatorname{Hom}_{{\mathcal{O}}_{Y}}({\mathcal{W}}_{\bullet },{\mathcal{W}}_{\bullet }) & & \displaystyle\end{eqnarray}$$

sending $a\in Q_{1}$ to $\unicode[STIX]{x1D719}_{a}$ . Recall that a quiver with relation is a pair $(Q,I)$ , where $Q$ is a quiver and $I\subset \mathbb{C}[Q]$ is a two-sided ideal.

For another representation ${\mathcal{W}}^{\prime }$ of $(Q,I)$ over $Y$ , the set of morphisms $\operatorname{Hom}({\mathcal{W}},{\mathcal{W}}^{\prime })$ consists of coherent left ${\mathcal{O}}_{Y}$ -module homomorphisms ${\mathcal{W}}_{v}\rightarrow {\mathcal{W}}_{v}^{\prime }$ for each $v\in Q_{0}$ which commute with $\unicode[STIX]{x1D719}_{a}$ and $\unicode[STIX]{x1D719}_{a}^{\prime }$ . The category

(3.1.3) $$\begin{eqnarray}\displaystyle \text{Rep}((Q,I)/Y) & & \displaystyle\end{eqnarray}$$

is defined to be the category of representations of $(Q,I)$ over $Y$ . For objects ${\mathcal{W}},{\mathcal{W}}^{\prime }$ in (3.1.3), we also have the sheaf homomorphisms ${\mathcal{H}}om({\mathcal{W}},{\mathcal{W}}^{\prime })$ on $Y$ by associating an open subset $U\subset Y$ with $\operatorname{Hom}({\mathcal{W}}|_{U},{\mathcal{W}}^{\prime }|_{U})$ , which is an object of $\operatorname{Coh}(Y)$ if $Y$ is a commutative scheme. If $I=\{0\}$ , or $Y=\operatorname{Spf}\unicode[STIX]{x1D6EC}$ for $\unicode[STIX]{x1D6EC}\in {\mathcal{N}}$ , we simply write (3.1.3) as $\text{Rep}(Q/Y)$ or $\text{Rep}((Q,I)/\unicode[STIX]{x1D6EC})$ , respectively. If further $Y=\operatorname{Spec}\mathbb{C}$ , we write (3.1.3) as $\text{Rep}(Q,I)$ , and $\text{Rep}(Q)$ if $I=\{0\}$ .

Let $Q$ be a quiver and ${\mathcal{W}}$ a representation of it over an NC scheme $Y$ . We call it flat if each ${\mathcal{W}}_{v}$ is a flat left ${\mathcal{O}}_{Y}$ -module. This is equivalent to that each ${\mathcal{W}}_{v}$ is a locally free left ${\mathcal{O}}_{Y}$ -module. We prepare the following lemma.

Lemma 3.4. (i) Let ${\mathcal{W}}$ be a flat representation of $(Q,I)$ over a commutative scheme $T$ and ${\mathcal{J}}$ a coherent sheaf on $T$ . Suppose that for any $t\in T$ , we have $\operatorname{Hom}({\mathcal{W}}|_{t},{\mathcal{W}}|_{t})=\mathbb{C}$ . Then the morphism

$$\begin{eqnarray}\displaystyle {\mathcal{J}}\rightarrow {\mathcal{H}}om({\mathcal{W}},{\mathcal{J}}\otimes _{{\mathcal{O}}_{T}}{\mathcal{W}}) & & \displaystyle \nonumber\end{eqnarray}$$

given by $u\mapsto u\otimes \operatorname{id}$ is an isomorphism.

(ii) Let $0\rightarrow J\rightarrow \unicode[STIX]{x1D6EC}^{\prime }\rightarrow \unicode[STIX]{x1D6EC}\rightarrow 0$ be a central extension in ${\mathcal{N}}$ and ${\mathcal{W}}$ a flat representation of $(Q,I)$ over $\unicode[STIX]{x1D6EC}^{\prime }$ . Then any automorphism of ${\mathcal{W}}$ which is the identity over $\unicode[STIX]{x1D6EC}$ is given by the left multiplication of a central element $1+u$ for $u\in J$ .

Proof. (i) We may assume that $I=\{0\}$ . For $t\in T$ , we have the isomorphism in $D(\text{Rep}(Q))$ :

$$\begin{eqnarray}\displaystyle {\mathcal{O}}_{t}\,\stackrel{\mathbf{L}}{\otimes }_{{\mathcal{O}}_{T}}\,\mathbf{R}{\mathcal{H}}om({\mathcal{W}},{\mathcal{W}})\cong \mathbf{R}\!\operatorname{Hom}({\mathcal{W}}|_{t},{\mathcal{W}}|_{t}). & & \displaystyle \nonumber\end{eqnarray}$$

Hence, we have the spectral sequence

$$\begin{eqnarray}\displaystyle E_{2}^{p,q}={\mathcal{T}}\!or_{-p}^{{\mathcal{O}}_{T}}({\mathcal{O}}_{t},{\mathcal{E}}xt^{q}({\mathcal{W}},{\mathcal{W}}))\Rightarrow \operatorname{Ext}^{p+q}({\mathcal{W}}|_{t},{\mathcal{W}}|_{t}). & & \displaystyle \nonumber\end{eqnarray}$$

Since $\mathbb{C}[Q]$ is hereditary, we have $\operatorname{Ext}^{{\geqslant}2}({\mathcal{W}}|_{t},{\mathcal{W}}|_{t})=0$ . Using the above spectral sequence and the assumption $\operatorname{Hom}({\mathcal{W}}|_{t},{\mathcal{W}}|_{t})=\mathbb{C}$ , we see that

$$\begin{eqnarray}\displaystyle {\mathcal{E}}xt^{{\geqslant}2}({\mathcal{W}},{\mathcal{W}})=0,\quad E_{2}^{-1,1}=0,\quad E_{2}^{0,0}=\mathbb{C}. & & \displaystyle \nonumber\end{eqnarray}$$

The vanishing of $E_{2}^{-1,1}$ shows that ${\mathcal{E}}xt^{1}({\mathcal{W}},{\mathcal{W}})$ is flat over ${\mathcal{O}}_{T}$ , and $E_{2}^{0,0}=\mathbb{C}$ shows that the natural map ${\mathcal{O}}_{T}\rightarrow {\mathcal{H}}om({\mathcal{W}},{\mathcal{W}})$ is an isomorphism.Footnote ¹ In particular, for ${\mathcal{J}}\in \operatorname{Coh}(T)$ , we have the isomorphism

$$\begin{eqnarray}\displaystyle {\mathcal{H}}^{0}({\mathcal{J}}\,\stackrel{\mathbf{L}}{\otimes }_{{\mathcal{O}}_{T}}\,\mathbf{R}{\mathcal{H}}om({\mathcal{W}},{\mathcal{W}}))\cong {\mathcal{J}}. & & \displaystyle \nonumber\end{eqnarray}$$

Then the result follows by taking the zeroth cohomology of the following isomorphism in $D(\operatorname{Coh}(T))$ :

$$\begin{eqnarray}\displaystyle {\mathcal{J}}\,\stackrel{\mathbf{L}}{\otimes }_{{\mathcal{O}}_{T}}\,\mathbf{R}{\mathcal{H}}om({\mathcal{W}},{\mathcal{W}})\cong \mathbf{R}{\mathcal{H}}om({\mathcal{W}},{\mathcal{J}}\,\stackrel{\mathbf{L}}{\otimes }_{{\mathcal{O}}_{T}}\,{\mathcal{W}}). & & \displaystyle \nonumber\end{eqnarray}$$

(ii) Let $g$ be an automorphism of ${\mathcal{W}}$ which is the identity over $\unicode[STIX]{x1D6EC}$ . We have the commutative diagram

By the above commutative diagram, the isomorphism $g$ is written as $\operatorname{id}+\,\unicode[STIX]{x1D704}\circ \unicode[STIX]{x1D6FC}\circ p$ for some morphism $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6EC}\otimes _{\unicode[STIX]{x1D6EC}^{\prime }}{\mathcal{W}}\rightarrow J\otimes _{\unicode[STIX]{x1D6EC}^{\text{ab}}}{\mathcal{W}}^{\text{ab}}$ . The morphism $\unicode[STIX]{x1D6FC}$ descends to the morphism ${\mathcal{W}}^{\text{ab}}\rightarrow J\otimes _{\unicode[STIX]{x1D6EC}^{\text{ab}}}{\mathcal{W}}^{\text{ab}}$ , which is given by $u\otimes \operatorname{id}$ for some $u\in J$ by (i). Therefore, $g$ is the left multiplication of $1+u$ .◻

Let ${\mathcal{W}}$ be a representation of $Q$ over an NC scheme $Y$ . We consider the natural morphism of sheaves of ${\mathcal{O}}_{Y}$ bi-modules

$$\begin{eqnarray}\displaystyle {\mathcal{W}}_{\bullet }\otimes _{\mathbb{C}}\operatorname{Hom}_{{\mathcal{O}}_{Y}}({\mathcal{W}}_{\bullet },{\mathcal{W}}_{\bullet })\otimes _{\mathbb{C}}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}({\mathcal{W}}_{\bullet },{\mathcal{O}}_{Y})\rightarrow {\mathcal{O}}_{Y} & & \displaystyle \nonumber\end{eqnarray}$$

given by $e\otimes f\otimes g\mapsto g\circ f(e)$ . By composing it with (3.1.2) and the inclusion $I\subset \mathbb{C}[Q]$ , we obtain the morphism of sheaves of ${\mathcal{O}}_{Y}$ bi-modules

(3.1.4) $$\begin{eqnarray}\displaystyle {\mathcal{W}}_{\bullet }\otimes _{\mathbb{C}}I\otimes _{\mathbb{C}}{\mathcal{H}}om_{{\mathcal{O}}_{Y}}({\mathcal{W}}_{\bullet },{\mathcal{O}}_{Y})\rightarrow {\mathcal{O}}_{Y}. & & \displaystyle\end{eqnarray}$$

If ${\mathcal{W}}$ is flat, then ${\mathcal{J}}_{I}$ is locally the two-sided ideal generated by the matrix components of the morphism (3.1.2) restricted to $I$ . In particular, the map (3.1.2) is zero on $I$ if and only if ${\mathcal{J}}_{I}=0$ . If $Z$ is the closed NC subscheme of $Y$ defined by ${\mathcal{O}}_{Z}:={\mathcal{O}}_{Y}/{\mathcal{J}}_{I}$ , the representation of $Q$

$$\begin{eqnarray}\displaystyle {\mathcal{W}}|_{Z}:=(\{{\mathcal{O}}_{Z}\otimes _{{\mathcal{O}}_{Y}}{\mathcal{W}}_{v}\}_{v\in Q_{0}},\{\operatorname{id}\otimes \,\unicode[STIX]{x1D719}_{a}\}_{a\in Q_{1}}) & & \displaystyle \nonumber\end{eqnarray}$$

over $Z$ is a representation of $(Q,I)$ .

3.2 Moduli spaces of representations of quivers

Let $(Q,I)$ be a quiver with relation. We set $\unicode[STIX]{x1D6E4}_{Q}:=\mathbb{Z}^{Q_{0}}$ , and define its inner product by

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}\cdot \unicode[STIX]{x1D6FE}^{\prime }=\mathop{\sum }_{v\in Q_{0}}\unicode[STIX]{x1D6FE}_{v}\cdot \unicode[STIX]{x1D6FE}_{v}^{\prime }. & & \displaystyle \nonumber\end{eqnarray}$$

For an object $W$ in $\text{Rep}(Q)$ , its dimension vector $\dim W$ is defined by

$$\begin{eqnarray}\displaystyle \dim W:=(\dim W_{v})_{v\in Q_{0}}\in \unicode[STIX]{x1D6E4}_{Q}. & & \displaystyle \nonumber\end{eqnarray}$$

We recall the notion of King’s $\unicode[STIX]{x1D703}$ -stability on $\text{Rep}(Q,I)$ .

Definition 3.6 [Reference KingKin94].

For $\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6E4}_{Q}$ , a representation $W$ of $(Q,I)$ is called $\unicode[STIX]{x1D703}$ -(semi)stable if $\unicode[STIX]{x1D703}\cdot \dim W=0$ and, for any subobject $0\neq W^{\prime }\subsetneq W$ in $\text{Rep}(Q,I)$ , we have the inequality

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}\cdot \dim W^{\prime }>({\geqslant})0. & & \displaystyle \nonumber\end{eqnarray}$$

For $\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6E4}_{Q}$ with $\unicode[STIX]{x1D703}\cdot \unicode[STIX]{x1D6FE}=0$ , the 2-functor

(3.2.1) $$\begin{eqnarray}\displaystyle \mathfrak{M}_{Q,I,\unicode[STIX]{x1D703}}^{}(\unicode[STIX]{x1D6FE}):{\mathcal{S}}\!\mathit{ch}/\mathbb{C}\rightarrow {\mathcal{G}}\mathit{roupoid} & & \displaystyle\end{eqnarray}$$

is defined by sending a $\mathbb{C}$ -scheme $T$ to the groupoid of flat representations ${\mathcal{W}}$ of $(Q,I)$ over $T$ such that ${\mathcal{W}}|_{t}$ for any $t\in T$ is a $\unicode[STIX]{x1D703}$ -semistable representation of $(Q,I)$ with dimension vector $\unicode[STIX]{x1D6FE}$ . The 2-functor (3.2.1) is known to be an algebraic stack of finite type over $\mathbb{C}$ . Indeed, let $W$ be the affine space given by

$$\begin{eqnarray}\displaystyle W:=\mathop{\prod }_{a\in Q_{1}}\operatorname{Hom}(W_{t(a)},W_{h(a)}). & & \displaystyle \nonumber\end{eqnarray}$$

By the construction of $W$ , there exists a tautological representation of $Q$ over $W$ . Let $M\subset W$ be the subscheme defined by the ideal of relations in $I$ , whose closed points correspond to $(Q,I)$ -representations. The subset of points $M^{ss}\subset M$ corresponding to $\unicode[STIX]{x1D703}$ -semistable representations forms an open subset of $M$ . Also, the group $G:=\prod _{v\in Q_{0}}\operatorname{GL}(W_{v})$ acts on $W$ by

$$\begin{eqnarray}\displaystyle (g_{v})\cdot (\unicode[STIX]{x1D719}_{a})=(g_{h(a)}^{-1}\circ \unicode[STIX]{x1D719}_{a}\circ g_{t(a)}). & & \displaystyle \nonumber\end{eqnarray}$$

The $G$ -action on $W$ preserves $M^{ss}$ , and the stack (3.2.1) is given by the quotient stack

$$\begin{eqnarray}\displaystyle \mathfrak{M}_{Q,I,\unicode[STIX]{x1D703}}^{}(\unicode[STIX]{x1D6FE})=[M^{ss}/G]. & & \displaystyle \nonumber\end{eqnarray}$$

On the other hand, let

(3.2.2) $$\begin{eqnarray}\displaystyle {\mathcal{M}}_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}):{\mathcal{S}}\!\mathit{ch}/\mathbb{C}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle\end{eqnarray}$$

be the functor defined by sending a $\mathbb{C}$ -scheme $T$ to the set of equivalence classes of $\unicode[STIX]{x1D703}$ -stable flat representations of $(Q,I)$ over $T$ . Here ${\mathcal{W}}$ and ${\mathcal{W}}^{\prime }$ are called equivalent if there is a line bundle ${\mathcal{L}}$ on $T$ such that ${\mathcal{W}}$ and ${\mathcal{W}}^{\prime }\otimes {\mathcal{L}}$ are isomorphic in $\text{Rep}((Q,I)/T)$ . The following result was proved by King.

Indeed, let $M^{s}\subset M^{ss}$ be the open subset consisting of $\unicode[STIX]{x1D703}$ -stable representations. For $w\in M^{s}$ , the subgroup of $G$ which fixes $w$ coincides with the diagonal subgroup $\mathbb{C}^{\ast }\subset G$ . Hence, the group $\overline{G}:=G/\mathbb{C}^{\ast }$ acts on $M^{s}$ without fixed points, and $M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ is given by

(3.2.3) $$\begin{eqnarray}\displaystyle M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})=M^{s}/\overline{G}. & & \displaystyle\end{eqnarray}$$

In particular, the open substack of $\mathfrak{M}_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ consisting of stable representations is a $\mathbb{C}^{\ast }$ -gerbe over (3.2.3). By the construction, we have the closed embedding

$$\begin{eqnarray}\displaystyle M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}){\hookrightarrow}M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}) & & \displaystyle \nonumber\end{eqnarray}$$

such that $M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ is defined by the ideal of relations in $I$ on the smooth moduli space $M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ . Since $M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ represents the functor (3.2.2), there exists a universal $(Q,I)$ -representation, i.e. a family of representations of $(Q,I)$

(3.2.4) $$\begin{eqnarray}\displaystyle {\mathcal{V}}=(\{{\mathcal{V}}_{v}\}_{v\in Q_{0}},\{\unicode[STIX]{x1D719}_{a}\}_{a\in Q_{1}}) & & \displaystyle\end{eqnarray}$$

over $M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ , such that the map $f\mapsto f^{\ast }{\mathcal{V}}$ gives the functorial isomorphism

$$\begin{eqnarray}\displaystyle \operatorname{Hom}(T,M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}))\stackrel{\cong }{\rightarrow }{\mathcal{M}}_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})(T) & & \displaystyle \nonumber\end{eqnarray}$$

for any $\mathbb{C}$ -scheme $T$ .

3.3 Construction of NC hull

Let $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{Q}$ be a primitive element. We consider the smooth moduli space $M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ of representations of $Q$ without relation (cf. Theorem 3.7), and a universal representation ${\mathcal{V}}$ on it given by (3.2.4) for $I=\{0\}$ . We take an affine open subset

$$\begin{eqnarray}\displaystyle U\subset M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}) & & \displaystyle \nonumber\end{eqnarray}$$

such that each ${\mathcal{V}}_{v}|_{U}$ is isomorphic to ${\mathcal{O}}_{U}\otimes _{\mathbb{C}}W_{v}$ , where $W_{v}$ is a $\mathbb{C}$ -vector space with dimension $\unicode[STIX]{x1D6FE}_{v}$ . Since $U$ is smooth, it admits an NC smooth thickening $U^{\operatorname{nc}}=(U,{\mathcal{O}}_{U}^{\text{nc}})$ , which is unique up to non-canonical isomorphisms (cf. [Reference KapranovKap98, Theorem 1.6.1]). We set

$$\begin{eqnarray}\displaystyle {\mathcal{V}}_{U,v}^{\text{nc}}:={\mathcal{O}}_{U}^{\text{nc}}\otimes _{\mathbb{C}}W_{v},\quad v\in Q_{0}, & & \displaystyle \nonumber\end{eqnarray}$$

which we regard as left ${\mathcal{O}}_{U}^{\operatorname{nc}}$ -modules. Since ${\mathcal{O}}_{U}^{\operatorname{nc}}{\twoheadrightarrow}{\mathcal{O}}_{U}$ is surjective, we can lift each universal morphism $\unicode[STIX]{x1D719}_{a}:{\mathcal{V}}_{t(a)}\rightarrow {\mathcal{V}}_{h(a)}$ restricted to $U$ to a left ${\mathcal{O}}_{U}^{\operatorname{nc}}$ -module homomorphism

(3.3.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{U,a}^{\operatorname{nc}}:{\mathcal{V}}_{U,t(a)}^{\operatorname{nc}}\rightarrow {\mathcal{V}}_{U,h(a)}^{\operatorname{nc}},\quad a\in Q_{1}. & & \displaystyle\end{eqnarray}$$

Then the data

(3.3.2) $$\begin{eqnarray}\displaystyle {\mathcal{V}}_{U}^{\operatorname{nc}}:=(\{{\mathcal{V}}_{U,v}^{\operatorname{nc}}\}_{v\in Q_{0}},\{\unicode[STIX]{x1D719}_{U,a}^{\operatorname{nc}}\}_{a\in Q_{1}}) & & \displaystyle\end{eqnarray}$$

is a representation of $Q$ over $U^{\operatorname{nc}}$ .

We define the functor

(3.3.3) $$\begin{eqnarray}\displaystyle h_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}):{\mathcal{N}}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle\end{eqnarray}$$

by sending $\unicode[STIX]{x1D6EC}\in {\mathcal{N}}$ to the isomorphism classes of triples $(f,{\mathcal{W}},\unicode[STIX]{x1D713})$ :

1. ∙ $f$ is a morphism $\operatorname{Spec}\unicode[STIX]{x1D6EC}^{\text{ab}}\rightarrow M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ of schemes;

2. ∙ ${\mathcal{W}}$ is a flat representation of $Q$ over $\unicode[STIX]{x1D6EC}$ ;

3. ∙ $\unicode[STIX]{x1D713}$ is an isomorphism $\unicode[STIX]{x1D713}:{\mathcal{W}}^{\text{ab}}\stackrel{\cong }{\rightarrow }f^{\ast }{\mathcal{V}}$ as representations of $Q$ over $\unicode[STIX]{x1D6EC}^{\text{ab}}$ .

An isomorphism $(f,{\mathcal{W}},\unicode[STIX]{x1D713})\rightarrow (f^{\prime },{\mathcal{W}}^{\prime },\unicode[STIX]{x1D713}^{\prime })$ exists if $f=f^{\prime }$ , and there is an isomorphism ${\mathcal{W}}\rightarrow {\mathcal{W}}^{\prime }$ as representations of $Q$ over $\unicode[STIX]{x1D6EC}$ commuting $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D713}^{\prime }$ . Note that we have

(3.3.4) $$\begin{eqnarray}\displaystyle h_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})|_{{\mathcal{C}}om}=h_{M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})}. & & \displaystyle\end{eqnarray}$$

Proposition 3.8. The natural transformation

(3.3.5) $$\begin{eqnarray}\displaystyle h_{U^{\operatorname{nc}}}\rightarrow h_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})|_{U} & & \displaystyle\end{eqnarray}$$

sending $g:\operatorname{Spf}\unicode[STIX]{x1D6EC}\rightarrow U^{\operatorname{nc}}$ to $(g^{\text{ab}},g^{\ast }{\mathcal{V}}_{U}^{\operatorname{nc}},\operatorname{id})$ is an NC hull of $h_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})|_{U}$ .

Proof. We write $h=h_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})|_{U}$ for simplicity. Since $h_{U^{\operatorname{nc}}}|_{{\mathcal{C}}om}=h_{U}$ , the natural transformation (3.3.5) is an isomorphism on ${\mathcal{C}}om$ by (3.3.4). Therefore, by Proposition 2.14, it is enough to show the following: for surjections $p_{1}:\unicode[STIX]{x1D6EC}_{1}{\twoheadrightarrow}\unicode[STIX]{x1D6EC}$ , $p_{2}:\unicode[STIX]{x1D6EC}_{2}{\twoheadrightarrow}\unicode[STIX]{x1D6EC}$ in ${\mathcal{N}}$ , $\unicode[STIX]{x1D6EC}_{12}:=\unicode[STIX]{x1D6EC}_{1}\times _{\unicode[STIX]{x1D6EC}}\unicode[STIX]{x1D6EC}_{2}$ , the natural map

(3.3.6) $$\begin{eqnarray}\displaystyle h(\unicode[STIX]{x1D6EC}_{12})\rightarrow h(\unicode[STIX]{x1D6EC}_{1})\times _{h(\unicode[STIX]{x1D6EC})}h(\unicode[STIX]{x1D6EC}_{2}) & & \displaystyle\end{eqnarray}$$

is surjective. The right-hand side of (3.3.6) consists of triples

(3.3.7) $$\begin{eqnarray}\displaystyle (f_{j},{\mathcal{W}}_{j},\unicode[STIX]{x1D713}_{j})\in h(\unicode[STIX]{x1D6EC}_{j}),\quad j=1,2 & & \displaystyle\end{eqnarray}$$

which are isomorphic over $\unicode[STIX]{x1D6EC}$ , i.e. $f:=f_{1}|_{\operatorname{ Spec}\unicode[STIX]{x1D6EC}^{\text{ab}}}=f_{2}|_{\operatorname{ Spec}\unicode[STIX]{x1D6EC}^{\text{ab}}}$ , and there is an isomorphism of representations of $Q$ over $\unicode[STIX]{x1D6EC}$

(3.3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}:\unicode[STIX]{x1D6EC}\otimes _{\unicode[STIX]{x1D6EC}_{1}}{\mathcal{W}}_{1}\stackrel{\cong }{\rightarrow }\unicode[STIX]{x1D6EC}\otimes _{\unicode[STIX]{x1D6EC}_{2}}{\mathcal{W}}_{2} & & \displaystyle\end{eqnarray}$$

which commutes with $\unicode[STIX]{x1D6EC}^{\text{ab}}\otimes _{\unicode[STIX]{x1D6EC}_{j}^{\text{ab}}}\unicode[STIX]{x1D713}_{j}$ for $j=1,2$ . Let us write ${\mathcal{W}}_{j}$ as

$$\begin{eqnarray}\displaystyle {\mathcal{W}}_{j}=(\{W_{j,v}\}_{v\in Q_{0}},\{\unicode[STIX]{x1D719}_{j,a}\}_{a\in Q_{1}}) & & \displaystyle \nonumber\end{eqnarray}$$

as representations of $Q$ over $\unicode[STIX]{x1D6EC}_{j}$ . Then the isomorphism $\unicode[STIX]{x1D6FE}$ consists of collections of isomorphisms of left $\unicode[STIX]{x1D6EC}$ -modules

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{v}:\unicode[STIX]{x1D6EC}\otimes _{\unicode[STIX]{x1D6EC}_{1}}W_{1,v}\stackrel{\cong }{\rightarrow }\unicode[STIX]{x1D6EC}\otimes _{\unicode[STIX]{x1D6EC}_{2}}W_{2,v} & & \displaystyle \nonumber\end{eqnarray}$$

for each $v\in Q_{1}$ , which commute with $\unicode[STIX]{x1D6EC}\otimes _{\unicode[STIX]{x1D6EC}_{j}}\unicode[STIX]{x1D719}_{j,a}$ . We set

$$\begin{eqnarray}\displaystyle W_{12,v}:=\{(x,y)\in W_{1,v}\times W_{2,v}:\unicode[STIX]{x1D6FE}_{v}\circ (1\otimes x)=1\otimes y\}. & & \displaystyle \nonumber\end{eqnarray}$$

Then $W_{12,v}$ is a projective left $\unicode[STIX]{x1D6EC}_{12}$ -module by [Reference MilnorMil71, Theorem 2.1]. Therefore, the data

(3.3.9) $$\begin{eqnarray}\displaystyle {\mathcal{W}}_{12}:=(\{W_{12,v}\}_{v\in Q_{0}},\{\unicode[STIX]{x1D719}_{1,a}\times \unicode[STIX]{x1D719}_{2,a}\}_{a\in Q_{1}}) & & \displaystyle\end{eqnarray}$$

determines a flat representation of $Q$ over $\unicode[STIX]{x1D6EC}_{12}$ . Also, since $\unicode[STIX]{x1D6EC}_{12}^{\text{ab}}=\unicode[STIX]{x1D6EC}_{1}^{\text{ab}}\times _{\unicode[STIX]{x1D6EC}^{\text{ab}}}\unicode[STIX]{x1D6EC}_{2}^{\text{ab}}$ , the morphisms $f_{1}$ , $f_{2}$ and $f$ induce the morphism of schemes

$$\begin{eqnarray}\displaystyle f_{12}:=f_{1}\times _{f}f_{2}:\operatorname{Spec}\unicode[STIX]{x1D6EC}_{12}^{\text{ab}}\rightarrow U. & & \displaystyle \nonumber\end{eqnarray}$$

Finally, since $\unicode[STIX]{x1D6FE}$ commutes with $\unicode[STIX]{x1D6EC}^{\text{ab}}\otimes _{\unicode[STIX]{x1D6EC}_{j}^{\text{ab}}}\unicode[STIX]{x1D713}_{j}$ , we have the isomorphism

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{12}:=\unicode[STIX]{x1D713}_{1}\times \unicode[STIX]{x1D713}_{2}:{\mathcal{W}}_{12}^{\text{ab}}\stackrel{\cong }{\rightarrow }f_{12}^{\ast }{\mathcal{V}} & & \displaystyle \nonumber\end{eqnarray}$$

of representations of $Q$ over $\unicode[STIX]{x1D6EC}_{12}^{\text{ab}}$ . Therefore, the triple $(f_{12},{\mathcal{W}}_{12},\unicode[STIX]{x1D713}_{12})$ determines an element of the left-hand side of (3.3.6), which is mapped to the triples (3.3.7) by the map (3.3.6). Therefore, the map (3.3.6) is surjective.◻

We next consider the moduli space $M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ in Theorem 3.7 for the non-zero relation $I\subset \mathbb{C}[Q]$ . We embed it into the smooth moduli space

$$\begin{eqnarray}\displaystyle M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})\subset M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}) & & \displaystyle \nonumber\end{eqnarray}$$

and take an affine open subset $U\subset M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ as before. Let $U^{\operatorname{nc}}$ be an NC smooth thickening of $U$ and ${\mathcal{V}}_{U}^{\operatorname{nc}}$ a lift of a universal representation (3.3.2) to $U^{\operatorname{nc}}$ . The representation ${\mathcal{V}}_{U}^{\operatorname{nc}}$ , together with the relation $I$ , determines the two-sided ideal ${\mathcal{J}}_{I,U}\subset {\mathcal{O}}_{U}^{\operatorname{nc}}$ of relations in $I$ (cf. Definition 3.5). We set

(3.3.10) $$\begin{eqnarray}\displaystyle V:=M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})\cap U,\quad {\mathcal{O}}_{V}^{\operatorname{nc}}:={\mathcal{O}}_{U}^{\operatorname{nc}}/\overline{{\mathcal{J}}}_{I,U},\quad V^{\operatorname{nc}}:=(V,{\mathcal{O}}_{V}^{\operatorname{nc}}). & & \displaystyle\end{eqnarray}$$

Since $({\mathcal{O}}_{V}^{\operatorname{nc}})^{\text{ab}}={\mathcal{O}}_{V}$ , the affine NC scheme $V^{\operatorname{nc}}$ is a closed NC subscheme of $U^{\operatorname{nc}}$ . The restriction ${\mathcal{V}}_{V}^{\operatorname{nc}}:={\mathcal{V}}_{U}^{\operatorname{nc}}|_{V^{\operatorname{nc}}}$ is a representation of $(Q,I)$ over $V^{\operatorname{nc}}$ .

We also define the functor

(3.3.11) $$\begin{eqnarray}\displaystyle h_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}):{\mathcal{N}}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle\end{eqnarray}$$

to be the subfunctor of $h_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ in (3.3.3), sending $\unicode[STIX]{x1D6EC}\in {\mathcal{N}}$ to the set of triples $(f,{\mathcal{W}},\unicode[STIX]{x1D713})\in h_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D6EC})$ such that $f$ factors through $M_{Q,\unicode[STIX]{x1D703},I}(\unicode[STIX]{x1D6FE})$ and ${\mathcal{W}}$ is a representation of $(Q,I)$ over $\unicode[STIX]{x1D6EC}$ . Note that we have

$$\begin{eqnarray}\displaystyle h_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})|_{{\mathcal{C}}om}=h_{M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})}. & & \displaystyle \nonumber\end{eqnarray}$$

Proposition 3.11. The natural transformation

(3.3.12) $$\begin{eqnarray}\displaystyle h_{V^{\operatorname{nc}}}\rightarrow h_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})|_{V} & & \displaystyle\end{eqnarray}$$

sending $g:\operatorname{Spf}\unicode[STIX]{x1D6EC}\rightarrow V^{\operatorname{nc}}$ to $(g^{\text{ab}},g^{\ast }{\mathcal{V}}_{V}^{\operatorname{nc}},\operatorname{id})$ is an NC hull of $h_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})|_{V}$ .

Proof. By Proposition 3.8, it is enough to note that the following diagram of functors:

is Cartesian. Indeed, for $g:\operatorname{Spf}\unicode[STIX]{x1D6EC}\rightarrow U^{\operatorname{nc}}$ , suppose that $g^{\ast }{\mathcal{V}}_{U}^{\operatorname{nc}}$ is a representation of $(Q,I)$ over  $\unicode[STIX]{x1D6EC}$ . This is equivalent to that the natural morphism

$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D6EC}\otimes _{{\mathcal{O}}_{U}^{\operatorname{nc}}}{\mathcal{V}}_{U}^{\operatorname{nc}})\otimes _{\mathbb{C}}I\otimes _{\mathbb{C}}\operatorname{Hom}_{\unicode[STIX]{x1D6EC}}(\unicode[STIX]{x1D6EC}\otimes _{{\mathcal{O}}_{U}^{\operatorname{nc}}}{\mathcal{V}}_{U}^{\operatorname{nc}},\unicode[STIX]{x1D6EC})\rightarrow \unicode[STIX]{x1D6EC} & & \displaystyle \nonumber\end{eqnarray}$$

is a zero map, which is equivalent to that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}\otimes _{{\mathcal{O}}_{U}^{\operatorname{nc}}}{\mathcal{J}}_{I,U}\otimes _{{\mathcal{O}}_{U}^{\operatorname{nc}}}\unicode[STIX]{x1D6EC}\rightarrow \unicode[STIX]{x1D6EC} & & \displaystyle \nonumber\end{eqnarray}$$

induced by ${\mathcal{J}}_{I,U}\subset {\mathcal{O}}_{U}^{\operatorname{nc}}$ is a zero map. The last condition is also equivalent to that $g^{\ast }:{\mathcal{O}}_{U}^{\operatorname{nc}}\rightarrow \unicode[STIX]{x1D6EC}$ vanishes on ${\mathcal{J}}_{I,U}$ ; hence, $g$ factors through $V^{\operatorname{nc}}{\hookrightarrow}U^{\operatorname{nc}}$ .◻

Let $\{U_{i}\}_{i\in \mathbb{I}}$ be a sufficiently small affine open covering of $M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ with affine $U_{ij}$ , and set $V_{i}=M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})\cap U_{i}$ . By applying the above construction to $U=U_{i}$ , we obtain the NC schemes

(3.3.13) $$\begin{eqnarray}\displaystyle U_{i}^{\operatorname{nc}}:=(U_{i},{\mathcal{O}}_{U_{i}}^{\operatorname{nc}}),\quad V_{i}^{\operatorname{nc}}:=(V_{i},{\mathcal{O}}_{V_{i}}^{\operatorname{nc}}) & & \displaystyle\end{eqnarray}$$

and the representations ${\mathcal{V}}_{U_{i}}^{\operatorname{nc}}$ of $Q$ over $U_{i}^{\operatorname{nc}}$ , ${\mathcal{V}}_{V_{i}}^{\operatorname{nc}}:={\mathcal{V}}_{U_{i}}^{\operatorname{nc}}|_{V_{i}}$ of $(Q,I)$ over $V_{i}^{\operatorname{nc}}$ , respectively. We have the following corollary.

Corollary 3.12. There exist isomorphisms of NC schemes

(3.3.14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{ij}:V_{j}^{\operatorname{nc}}|_{V_{ij}}\stackrel{\cong }{\rightarrow }V_{i}^{\operatorname{nc}}|_{V_{ij}} & & \displaystyle\end{eqnarray}$$

giving a quasi-NC structure on $M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ , and isomorphisms of representations of $(Q,I)$ :

(3.3.15) $$\begin{eqnarray}\displaystyle g_{ij}:\unicode[STIX]{x1D719}_{ij}^{\ast }{\mathcal{V}}_{V_{i}}^{\operatorname{nc}}|_{V_{ij}}\stackrel{\cong }{\rightarrow }{\mathcal{V}}_{V_{j}}^{\operatorname{nc}}|_{V_{ij}}. & & \displaystyle\end{eqnarray}$$

3.4 Comparison with the formal deformations of quiver representations

Here we use the notation in § 2.5. For an object $W\in \text{Rep}(Q,I)$ , the formal non-commutative deformation functor

(3.4.1) $$\begin{eqnarray}\displaystyle \text{Def}_{W}^{\operatorname{nc}}:{\mathcal{N}}^{\text{loc}}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle\end{eqnarray}$$

is defined by sending $(\unicode[STIX]{x1D6EC},\mathbf{n})$ to the set of isomorphism classes $({\mathcal{W}},\unicode[STIX]{x1D713})$ , where ${\mathcal{W}}$ is a flat representation of $(Q,I)$ over $\unicode[STIX]{x1D6EC}$ , and $\unicode[STIX]{x1D713}:\unicode[STIX]{x1D6EC}/\mathbf{n}\otimes _{\unicode[STIX]{x1D6EC}}{\mathcal{W}}\stackrel{\cong }{\rightarrow }W$ is an isomorphism as representations of $(Q,I)$ . The non-commutative deformation functor (3.4.1) was studied by Laudal [Reference LaudalLau02], where the existence of a pro-representable hull was proved. We also refer to [Reference EriksenEri10, Reference SegalSeg08, Reference Efimov, Lunts and OrlovELO09, Reference Efimov, Lunts and OrlovELO10, Reference Efimov, Lunts and OrlovELO11] for details on formal non-commutative deformation theory.

Recall that the formal commutative deformation space of $W$ is given by the solution of the Mauer–Cartan equation of the dg-algebra $\mathbf{R}\!\operatorname{Hom}(W,W)$ , up to gauge equivalence. Let

(3.4.2) $$\begin{eqnarray}\displaystyle (\operatorname{Ext}^{\ast }(W,W),\{m_{n}\}_{n\geqslant 2}) & & \displaystyle\end{eqnarray}$$

be the minimal $A_{\infty }$ -algebra which is $A_{\infty }$ quasi-isomorphic to $\mathbf{R}\!\operatorname{Hom}(W,W)$ . Here we take the $\operatorname{Ext}$ -groups in the category $\text{Rep}(Q,I)$ . By [Reference SegalSeg08], the pro-representable hull of (3.4.1) is described in terms of the $A_{\infty }$ -algebra (3.4.2). Let

$$\begin{eqnarray}\displaystyle m_{n}:\operatorname{Ext}^{1}(W,W)^{\otimes n}\rightarrow \operatorname{Ext}^{2}(W,W) & & \displaystyle \nonumber\end{eqnarray}$$

be the $n$ th $A_{\infty }$ -product. Below, for a vector space $V$ , we denote by $\widehat{T}^{\bullet }(V)$ the completed tensor algebra given by

$$\begin{eqnarray}\displaystyle \widehat{T}^{\bullet }(V)=\mathop{\prod }_{n\geqslant 0}V^{\otimes n}. & & \displaystyle \nonumber\end{eqnarray}$$

Let

$$\begin{eqnarray}\displaystyle J_{W}\subset \widehat{T}^{\bullet }(\operatorname{Ext}^{1}(W,W)^{\vee }) & & \displaystyle \nonumber\end{eqnarray}$$

be the topological closure of the two-sided ideal of the completed tensor algebra of $\operatorname{Ext}^{1}(W,W)^{\vee }$ generated by the image of the map

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\geqslant 2}m_{n}^{\vee }:\operatorname{Ext}^{2}(W,W)^{\vee }\rightarrow \widehat{T}^{\bullet }(\operatorname{Ext}^{1}(W,W)^{\vee }). & & \displaystyle \nonumber\end{eqnarray}$$

The pro-representable hull of (3.4.1) is given by the quotient algebra

$$\begin{eqnarray}\displaystyle R_{W}^{\operatorname{nc}}:=\widehat{T}^{\bullet }(\operatorname{Ext}^{1}(W,W)^{\vee })/J_{W}. & & \displaystyle \nonumber\end{eqnarray}$$

Let us take an open neighborhood $[W]\in V\subset M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ , and a non-commutative thickening $V^{\operatorname{nc}}=(V,{\mathcal{O}}_{V}^{\operatorname{nc}})$ as in the previous subsection (3.3.10).

Proof. Let

$$\begin{eqnarray}\displaystyle h_{Q,I,\unicode[STIX]{x1D703}}^{\text{loc}}(\unicode[STIX]{x1D6FE})_{[W]}:{\mathcal{N}}^{\text{loc}}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle \nonumber\end{eqnarray}$$

be the functor constructed from $h_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ by (2.5.1), which has a pro-representable hull $h_{\widehat{{\mathcal{O}}}_{V,[W]}^{\operatorname{nc}}}\rightarrow h_{Q,I,\unicode[STIX]{x1D703}}^{\text{loc}}(\unicode[STIX]{x1D6FE})_{[W]}$ by Lemma 2.17. We construct the natural transform

(3.4.3) $$\begin{eqnarray}\displaystyle h_{Q,I,\unicode[STIX]{x1D703}}^{\text{loc}}(\unicode[STIX]{x1D6FE})_{[W]}\rightarrow \text{Def}_{W}^{\operatorname{nc}} & & \displaystyle\end{eqnarray}$$

by sending triples $(f,{\mathcal{W}},\unicode[STIX]{x1D713})$ of the $(\unicode[STIX]{x1D6EC},\mathbf{n})$ -point of the left-hand side of (3.4.3) to the pair $({\mathcal{W}},\unicode[STIX]{x1D6EC}/\mathbf{n}\otimes _{\unicode[STIX]{x1D6EC}}\unicode[STIX]{x1D713})$ of the right-hand side. By the uniqueness of a pro-representable hull, it is enough to show that (3.4.3) is formally smooth and bijective on $\mathbb{C}[t]/t^{2}$ . Note that since the functor (3.2.2) is represented by $M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ , the functor

$$\begin{eqnarray}\displaystyle \text{Def}_{W}:=\text{Def}_{W}^{\operatorname{nc}}|_{{\mathcal{A}}\mathit{rt}^{\text{loc}}} & & \displaystyle \nonumber\end{eqnarray}$$

is pro-represented by $\widehat{{\mathcal{O}}}_{V,[W]}$ . Therefore, by Lemma 2.17, the natural transform (3.4.3) is an isomorphism on ${\mathcal{A}}\mathit{rt}^{\text{loc}}$ and in particular bijective on $\mathbb{C}[t]/t^{2}$ . It remains to show that (3.4.3) is formally smooth.

Let $(f,{\mathcal{W}},\unicode[STIX]{x1D713})$ be a $(\unicode[STIX]{x1D6EC},\mathbf{n})$ -point of the left-hand side of (3.4.3), and suppose that ${\mathcal{W}}$ extends to a flat representation ${\mathcal{W}}^{\prime }$ of $(Q,I)$ over $\unicode[STIX]{x1D6EC}^{\prime }$ , where $\unicode[STIX]{x1D6EC}^{\prime }{\twoheadrightarrow}\unicode[STIX]{x1D6EC}$ is a surjection in ${\mathcal{N}}^{\text{loc}}$ . Then ${\mathcal{W}}^{^{\prime }\text{ab}}$ is a flat extension of ${\mathcal{W}}^{\text{ab}}$ to $\unicode[STIX]{x1D6EC}^{^{\prime }\text{ab}}$ . By the pro-representability of $\text{Def}_{W}$ , the morphism $f:\operatorname{Spec}\unicode[STIX]{x1D6EC}^{\text{ab}}\rightarrow V$ uniquely extends to a morphism of schemes $f^{\prime }:\operatorname{Spec}\unicode[STIX]{x1D6EC}^{^{\prime }\text{ab}}\rightarrow V$ such that there exists an isomorphism $\unicode[STIX]{x1D713}^{\prime }:{\mathcal{W}}^{^{\prime }\text{ab}}\stackrel{\cong }{\rightarrow }f^{^{\prime }\ast }{\mathcal{V}}$ . By Lemma 3.4(i), any automorphism of $f^{\ast }{\mathcal{V}}$ extends to an automorphism of $f^{^{\prime }\ast }{\mathcal{V}}$ and hence one can choose $\unicode[STIX]{x1D713}^{\prime }$ so that $\unicode[STIX]{x1D6EC}^{\text{ab}}\otimes _{\unicode[STIX]{x1D6EC}^{^{\prime }\text{ab}}}\unicode[STIX]{x1D713}^{\prime }=\unicode[STIX]{x1D713}$ holds. Then the triple $(f^{\prime },{\mathcal{W}}^{\prime },\unicode[STIX]{x1D713}^{\prime })$ is an extension of $(f,{\mathcal{W}},\unicode[STIX]{x1D713})$ to $\unicode[STIX]{x1D6EC}^{\prime }$ , showing that (3.4.3) is formally smooth.◻

By Corollary 3.12 together with the above lemma, we obtain the following result.

3.5 Partial NC thickening of moduli spaces of representations of quivers

It is not clear whether the quasi-NC structures in Corollary 3.12 glue together to give an NC structure. Here we discuss the possibility to extend a given $(d-1)$ th-order NC thickening to that of the $d$ th-order NC thickening. Let $U_{i}^{\operatorname{nc}}$ , $V_{i}^{\operatorname{nc}}$ be affine NC schemes given in (3.3.13), and take representations ${\mathcal{V}}_{U_{i}}^{\operatorname{nc}}$ , ${\mathcal{V}}_{V_{i}}^{\operatorname{nc}}$ of $(Q,I)$ as before. For $d\in \mathbb{Z}_{{\geqslant}0}$ , we set

$$\begin{eqnarray}\displaystyle U_{i}^{d}:=(U_{i}^{\operatorname{nc}})^{{\leqslant}d},\quad V_{i}^{d}:=(V_{i}^{\operatorname{nc}})^{{\leqslant}d},\quad {\mathcal{V}}_{U_{i}}^{d}:=({\mathcal{V}}_{U_{i}}^{\operatorname{nc}})^{{\leqslant}d},\quad {\mathcal{V}}_{V_{i}}^{d}:=({\mathcal{V}}_{V_{i}}^{\operatorname{nc}})^{{\leqslant}d}. & & \displaystyle \nonumber\end{eqnarray}$$

The isomorphisms (3.3.14) and (3.3.15) in Corollary 3.12 induce isomorphisms

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{ij}^{{\leqslant}d}:V_{j}^{d}|_{V_{ij}}\stackrel{\cong }{\rightarrow }V_{i}^{d}|_{V_{ij}},\quad g_{ij}^{{\leqslant}d}:\unicode[STIX]{x1D719}_{ij}^{{\leqslant}d\ast }{\mathcal{V}}_{V_{j}}^{d}|_{V_{ij}}\stackrel{\cong }{\rightarrow }{\mathcal{V}}_{V_{i}}^{d}|_{V_{ij}}, & & \displaystyle \nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{ij}^{{\leqslant}d}$ give a quasi-NC structure on $M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ , and $g_{ij}^{{\leqslant}d}$ are isomorphisms of representations of $(Q,I)$ . We put the following assumption.

We define the functor

$$\begin{eqnarray}\displaystyle h_{Q,I,\unicode[STIX]{x1D703}}^{d}(\unicode[STIX]{x1D6FE}):{\mathcal{N}}_{d}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle \nonumber\end{eqnarray}$$

by sending $\unicode[STIX]{x1D6EC}\in {\mathcal{N}}_{d}$ to the isomorphism classes of triples $(f,{\mathcal{W}},\unicode[STIX]{x1D713})$ :

1. ∙ $f$ is a morphism of NC schemes $f:\operatorname{Spf}\unicode[STIX]{x1D6EC}^{{\leqslant}d-1}\rightarrow M_{Q,I,\unicode[STIX]{x1D703}}^{d-1}(\unicode[STIX]{x1D6FE})$ ;

2. ∙ ${\mathcal{W}}$ is a flat representation of $(Q,I)$ over $\unicode[STIX]{x1D6EC}$ ;

3. ∙ $\unicode[STIX]{x1D713}:{\mathcal{W}}^{{\leqslant}d-1}\stackrel{\cong }{\rightarrow }f^{\ast }{\mathcal{V}}^{d-1}$ is an isomorphism of representations of $(Q,I)$ over $\unicode[STIX]{x1D6EC}^{{\leqslant}d-1}$ .

An isomorphism $(f,{\mathcal{W}},\unicode[STIX]{x1D713})\rightarrow (f^{\prime },{\mathcal{W}}^{\prime },\unicode[STIX]{x1D713}^{\prime })$ exists if $f=f^{\prime }$ , and there is an isomorphism ${\mathcal{W}}\rightarrow {\mathcal{W}}^{\prime }$ as representations of $(Q,I)$ over $\unicode[STIX]{x1D6EC}$ commuting with $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D713}^{\prime }$ .

Proposition 3.16. The natural transform

(3.5.1) $$\begin{eqnarray}\displaystyle h_{V_{i}^{d}}\rightarrow h_{Q,I,\unicode[STIX]{x1D703}}^{d}(\unicode[STIX]{x1D6FE})|_{V_{i}} & & \displaystyle\end{eqnarray}$$

sending $g:\operatorname{Spf}\unicode[STIX]{x1D6EC}\rightarrow V_{i}^{d}$ for $\unicode[STIX]{x1D6EC}\in {\mathcal{N}}_{d}$ to $(g^{{\leqslant}d-1},g^{\ast }{\mathcal{V}}_{V_{i}}^{d},\operatorname{id})$ is an isomorphism of functors.

Proof. Note that (3.5.1) is an isomorphism on ${\mathcal{N}}_{{<}d}$ . Similarly to the proof of Proposition 3.11, we have the Cartesian square

(3.5.2)

It is enough to show that the top arrow of (3.5.2) is an isomorphism. We write $h^{d}=h_{Q,\{0\},\unicode[STIX]{x1D703}}^{d}(\unicode[STIX]{x1D6FE})|_{U_{i}}$ for simplicity. By Proposition 2.15, it remains to show the following: for surjections $p_{1}:\unicode[STIX]{x1D6EC}_{1}{\twoheadrightarrow}\unicode[STIX]{x1D6EC}$ , $p_{2}:\unicode[STIX]{x1D6EC}_{2}{\twoheadrightarrow}\unicode[STIX]{x1D6EC}$ in ${\mathcal{N}}_{d}$ with $\unicode[STIX]{x1D6EC}\in {\mathcal{N}}_{d-1}$ , the natural map

(3.5.3) $$\begin{eqnarray}\displaystyle h^{d}(\unicode[STIX]{x1D6EC}_{12})\rightarrow h^{d}(\unicode[STIX]{x1D6EC}_{1})\times _{h^{d}(\unicode[STIX]{x1D6EC})}h^{d}(\unicode[STIX]{x1D6EC}_{2}) & & \displaystyle\end{eqnarray}$$

is bijective. We follow the same notation and argument as in the proof of Proposition 3.8, replacing $h$ with $h^{d}$ and $\ast ^{\text{ab}}$ with $\ast ^{{\leqslant}d-1}$ . The difference from the proof of Proposition 3.8 is that, since we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}\otimes _{\unicode[STIX]{x1D6EC}_{j}}{\mathcal{W}}_{j}=(\unicode[STIX]{x1D6EC}\otimes _{\unicode[STIX]{x1D6EC}_{j}}{\mathcal{W}}_{j})^{{\leqslant}d-1} & & \displaystyle \nonumber\end{eqnarray}$$

and the isomorphism $\unicode[STIX]{x1D6FE}$ in (3.3.8) should commute with $\unicode[STIX]{x1D6EC}\otimes _{\unicode[STIX]{x1D6EC}_{j}}\unicode[STIX]{x1D713}_{j}$ , $\unicode[STIX]{x1D6FE}$ is uniquely determined by the right-hand side of (3.5.3). Therefore, sending the triples (3.3.7) to $(f_{12},{\mathcal{W}}_{12},\unicode[STIX]{x1D713}_{12})$ is a well-defined map from the right-hand side to the left-hand side of (3.5.3), giving the inverse of (3.5.3).◻

By the above proposition together with Remark 2.13, we have the following corollary.

By the above corollary, if the local $(Q,I)$ representations ${\mathcal{V}}_{V_{i}}^{d}$ glue together to give a global $(Q,I)$ representation, then we can extend the $d$ th-order NC structure $M_{Q,I,\unicode[STIX]{x1D703}}^{d}(\unicode[STIX]{x1D6FE})$ to a $(d+1)$ th-order NC thickening. The obstruction of gluing ${\mathcal{V}}_{V_{i}}^{d}$ is given as follows. We set

$$\begin{eqnarray}\displaystyle {\mathcal{J}}^{d}:=\operatorname{Ker}({\mathcal{O}}_{M_{Q,I,\unicode[STIX]{x1D703}}^{d}(\unicode[STIX]{x1D6FE})}{\twoheadrightarrow}{\mathcal{O}}_{M_{Q,I,\unicode[STIX]{x1D703}}^{d-1}(\unicode[STIX]{x1D6FE})}). & & \displaystyle \nonumber\end{eqnarray}$$

Note that ${\mathcal{J}}^{d}$ is a coherent sheaf on $M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ .

Lemma 3.18. In the situation of Corollary 3.17, there is a class

(3.5.4) $$\begin{eqnarray}\displaystyle \text{ob}\in H^{2}(M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}),{\mathcal{J}}^{d}) & & \displaystyle\end{eqnarray}$$

such that $\text{ob}=0$ if and only if $\{{\mathcal{V}}_{V_{i}}^{d}\}_{i\in \mathbb{I}}$ glue to give a representation ${\mathcal{V}}^{d}$ of $(Q,I)$ over $M_{Q,I,\unicode[STIX]{x1D703}}^{d}(\unicode[STIX]{x1D6FE})$ .

Since Assumption 3.15 is always satisfied for $d=1$ , the affine NC structures $\{V_{i}^{1}\}_{i\in \mathbb{I}}$ glue together to give an NC structure $M_{Q,I,\unicode[STIX]{x1D703}}^{1}(\unicode[STIX]{x1D6FE})$ on $M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ . Indeed, one of such a thickening is explicitly constructed in the following way. We first construct a 1-smooth NC structure $M_{Q,\unicode[STIX]{x1D703}}^{1}(\unicode[STIX]{x1D6FE})$ on the smooth moduli space $M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ by

$$\begin{eqnarray}\displaystyle {\mathcal{O}}_{M_{Q,\unicode[STIX]{x1D703}}^{1}(\unicode[STIX]{x1D6FE})}={\mathcal{O}}_{M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})}\oplus \unicode[STIX]{x1D6FA}_{M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})}^{2}. & & \displaystyle \nonumber\end{eqnarray}$$

The multiplication is given

$$\begin{eqnarray}\displaystyle (x,f)\cdot (y,g)=(xy,xg+fy+dx\wedge dy). & & \displaystyle \nonumber\end{eqnarray}$$

The universal representations ${\mathcal{V}}_{U_{i}}^{1}$ may not glue together, but the ideals of relations ${\mathcal{J}}_{I,U_{i}}^{1}\subset {\mathcal{O}}_{U_{i}}^{1}$ coincide on $U_{ij}$ and hence determine the two-sided ideal ${\mathcal{J}}_{I}^{1}\subset {\mathcal{O}}_{M_{Q,\unicode[STIX]{x1D703}}^{1}(\unicode[STIX]{x1D6FE})}$ . Then the NC scheme

$$\begin{eqnarray}\displaystyle M_{Q,I,\unicode[STIX]{x1D703}}^{1}(\unicode[STIX]{x1D6FE})=(M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}),{\mathcal{O}}_{M_{Q,\unicode[STIX]{x1D703}}^{1}}(\unicode[STIX]{x1D6FE})/{\mathcal{J}}_{I}^{1}) & & \displaystyle \nonumber\end{eqnarray}$$

is isomorphic to $V_{i}^{1}$ on $V_{i}$ .

3.6 NC structures on framed quiver representations

As we observed in Remark 3.9, the issue for having the global NC structure is caused by some automorphisms of sheaves over non-commutative bases. One of the classical ways to kill automorphisms of sheaves is to add additional data called framing. Here we show that this classical idea also works for the construction problem of global NC structures. Let $(Q,I)$ be a quiver with relation. We fix a vertex $\star \in Q_{0}$ .

A framed representation $({\mathcal{W}},\unicode[STIX]{x1D70F})$ over $Y$ is called flat if ${\mathcal{W}}$ is flat. Let $({\mathcal{W}}^{\prime },\unicode[STIX]{x1D70F}^{\prime })$ be another framed representation of $(Q,I)$ over $Y$ . An isomorphism from $({\mathcal{W}},\unicode[STIX]{x1D70F})$ to $({\mathcal{W}}^{\prime },\unicode[STIX]{x1D70F}^{\prime })$ is an isomorphism $g:{\mathcal{W}}\stackrel{\cong }{\rightarrow }{\mathcal{W}}^{\prime }$ in $\text{Rep}((Q,I)/Y)$ such that $g_{\star }\circ \unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{\prime }$ .

For $\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{Q}$ with $\unicode[STIX]{x1D703}\cdot \unicode[STIX]{x1D6FE}=0$ , the functor

$$\begin{eqnarray}\displaystyle {\mathcal{M}}_{Q,I,\unicode[STIX]{x1D703}}^{\star }(\unicode[STIX]{x1D6FE}):{\mathcal{S}}\!\mathit{ch}/\mathbb{C}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle \nonumber\end{eqnarray}$$

is defined by sending a $\mathbb{C}$ -scheme $T$ to the set of isomorphism classes of flat framed representations $({\mathcal{W}},\unicode[STIX]{x1D70F})$ over $T$ such that for any $t\in T$ , the restriction $({\mathcal{W}}|_{t},\unicode[STIX]{x1D70F}|_{t})$ is a $\unicode[STIX]{x1D703}$ -stable framed representation of $(Q,I)$ with $\dim {\mathcal{W}}|_{t}=\unicode[STIX]{x1D6FE}$ .

Proof. Let $Q^{\diamond }$ be a quiver defined by adding one vertex $\diamond$ and one arrow $\diamond \rightarrow \star$ to the quiver $Q$ . Note that the relation $I$ naturally determines the relation in $Q^{\diamond }$ . By the construction, giving a framed representation of $(Q,I)$ is equivalent to giving a representation of $(Q^{\diamond },I)$ whose dimension vector at the vertex $\diamond$ is one.

For a rational number $0<\unicode[STIX]{x1D700}\ll 1$ , let $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D700}}^{\diamond }$ be an element of $\unicode[STIX]{x1D6E4}_{Q^{\diamond }}\otimes \mathbb{Q}$ given by

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D700},v}^{\diamond }=\unicode[STIX]{x1D703}_{v}+\unicode[STIX]{x1D700}\text{ for }v\in Q_{0},\quad \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D700},\diamond }^{\diamond }=-\unicode[STIX]{x1D700}\mathop{\sum }_{v\in Q_{0}}\unicode[STIX]{x1D6FE}_{v}. & & \displaystyle \nonumber\end{eqnarray}$$

Then $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D700}}^{\diamond }\cdot \unicode[STIX]{x1D6FE}^{\diamond }=0$ , where $\unicode[STIX]{x1D6FE}^{\diamond }=(1,\unicode[STIX]{x1D6FE})\in \unicode[STIX]{x1D6E4}_{Q^{\diamond }}$ and $1$ is the dimension vector at the vertex $\diamond$ . Let $({\mathcal{W}},\unicode[STIX]{x1D70F})$ be a flat framed representation of $(Q,I)$ over a $\mathbb{C}$ -scheme $T$ such that $\dim {\mathcal{W}}|_{t}=\unicode[STIX]{x1D6FE}$ for any $t\in T$ . Then we have the associated flat representation ${\mathcal{W}}^{\diamond }$ of $(Q^{\diamond },I)$ over $T$ by putting ${\mathcal{W}}_{\diamond }^{\diamond }={\mathcal{O}}_{T}$ and the morphism corresponding to $\diamond \rightarrow \star$ is $\unicode[STIX]{x1D70F}$ . It is easy to show that $({\mathcal{W}},\unicode[STIX]{x1D70F})$ is a family of framed $\unicode[STIX]{x1D703}$ -stable representations of $(Q,I)$ if and only if ${\mathcal{W}}^{\diamond }$ is a family of $\unicode[STIX]{x1D703}^{\diamond }$ -stable representations of $(Q^{\diamond },I)$ . Therefore, $({\mathcal{W}},\unicode[STIX]{x1D70F})\mapsto {\mathcal{W}}^{\diamond }$ gives the natural transform

(3.6.1) $$\begin{eqnarray}\displaystyle {\mathcal{M}}_{Q,I,\unicode[STIX]{x1D703}}^{\star }(\unicode[STIX]{x1D6FE})\rightarrow {\mathcal{M}}_{Q^{\diamond },I,\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D700}}^{\diamond }}(\unicode[STIX]{x1D6FE}^{\diamond }). & & \displaystyle\end{eqnarray}$$

For a $T$ -valued point ${\mathcal{W}}^{\diamond }$ of the right-hand side of (3.6.1), it is equivalent to a unique element ${\mathcal{W}}^{^{\prime }\diamond }$ such that ${\mathcal{W}}_{\diamond }^{^{\prime }\diamond }={\mathcal{O}}_{T}$ , up to isomorphisms. Therefore, the natural transform (3.6.1) is indeed an isomorphism of functors, and the result follows from Theorem 3.7.◻

Let $({\mathcal{V}},\unicode[STIX]{x1D704})$ be the universal framed representation of $(Q,I)$ over $M_{Q,I,\unicode[STIX]{x1D703}}^{\star }(\unicode[STIX]{x1D6FE})$ . We define the functor

(3.6.2) $$\begin{eqnarray}\displaystyle h_{Q,I,\unicode[STIX]{x1D703}}^{\star }(\unicode[STIX]{x1D6FE}):{\mathcal{N}}\rightarrow {\mathcal{S}}\!\mathit{et} & & \displaystyle\end{eqnarray}$$

by sending $\unicode[STIX]{x1D6EC}\in {\mathcal{N}}$ to the isomorphism classes of triples $(f,({\mathcal{W}},\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D713})$ :

1. ∙ $f$ is a morphism of schemes $\operatorname{Spec}\unicode[STIX]{x1D6EC}^{\text{ab}}\rightarrow M_{Q,I,\unicode[STIX]{x1D703}}^{\star }(\unicode[STIX]{x1D6FE})$ ;

2. ∙ $({\mathcal{W}},\unicode[STIX]{x1D70F})$ is a flat framed representation of $(Q,I)$ over $\unicode[STIX]{x1D6EC}$ ;

3. ∙ $\unicode[STIX]{x1D713}$ is an isomorphism $\unicode[STIX]{x1D713}:({\mathcal{W}}^{\text{ab}},\unicode[STIX]{x1D70F}^{\text{ab}})\stackrel{\cong }{\rightarrow }f^{\ast }({\mathcal{V}},\unicode[STIX]{x1D704})$ as framed representations of $(Q,I)$ over $\unicode[STIX]{x1D6EC}^{\text{ab}}$ .

We use the following lemma.

Using the above lemma, we show the following result on the existence of global NC structures on the moduli spaces of stable framed representations.

Proof. In order to simplify the notation, we write $M=M_{Q,I,\unicode[STIX]{x1D703}}^{\star }(\unicode[STIX]{x1D6FE})$ and $N=M_{Q,\{0\},\unicode[STIX]{x1D703}}^{\star }(\unicode[STIX]{x1D6FE})$ . Note that $N$ is non-singular. Let $U\subset N$ be a sufficiently small affine open subset, and write $h^{\star }=h_{Q,\{0\},\unicode[STIX]{x1D703}}^{\star }(\unicode[STIX]{x1D6FE})|_{U}$ . Let $U^{\operatorname{nc}}$ be an NC smooth thickening of $U$ . Similarly to Proposition 3.8, we can construct a natural transform $h_{U^{\operatorname{nc}}}\rightarrow h^{\star }$ , which is an isomorphism on ${\mathcal{C}}om$ . We show that $h_{U^{\operatorname{nc}}}\rightarrow h^{\star }$ is indeed an isomorphism on ${\mathcal{N}}$ . By Proposition 2.15, it is enough to show the following: for surjections $p_{j}:\unicode[STIX]{x1D6EC}_{j}{\twoheadrightarrow}\unicode[STIX]{x1D6EC}$ in ${\mathcal{N}}$ with $j=1,2$ , $\unicode[STIX]{x1D6EC}_{12}:=\unicode[STIX]{x1D6EC}_{1}\times _{\unicode[STIX]{x1D6EC}}\unicode[STIX]{x1D6EC}_{2}$ , the natural map

(3.6.3) $$\begin{eqnarray}\displaystyle h^{\star }(\unicode[STIX]{x1D6EC}_{12})\rightarrow h^{\star }(\unicode[STIX]{x1D6EC}_{1})\times _{h^{\star }(\unicode[STIX]{x1D6EC})}h^{\star }(\unicode[STIX]{x1D6EC}_{2}) & & \displaystyle\end{eqnarray}$$

is a bijection. Let us take an element of the right-hand side of (3.6.3), i.e. elements of $h^{\star }(\unicode[STIX]{x1D6EC}_{1})$ and $h^{\star }(\unicode[STIX]{x1D6EC}_{2})$ which are isomorphic over $\unicode[STIX]{x1D6EC}$ . Following the proof of Proposition 3.8, one can lift it to an element of the left-hand side of (3.6.3). By Lemma 3.23, the framed isomorphism over $\unicode[STIX]{x1D6EC}$ is uniquely determined and hence the similar argument of the proof of Proposition 3.16 shows that the above lift is uniquely determined. Thus, we obtain a map from the right-hand side to the left-hand side of (3.6.3), which obviously gives the inverse of (3.6.3). Therefore, $h_{U^{\operatorname{nc}}}\rightarrow h^{\ast }$ is an isomorphism.

By Remark 2.13, the affine NC structures $U^{\operatorname{nc}}$ on each affine open subset $U\subset N$ glue together to give the NC structure $(N,{\mathcal{O}}_{N}^{\operatorname{nc}})$ on $N$ . Again by Lemma 3.23, the local framed universal representations on $U^{\operatorname{nc}}$ also glue to give the global universal framed representation $({\mathcal{V}}^{\operatorname{nc}},\unicode[STIX]{x1D70F}^{\operatorname{nc}})$ on $N^{\operatorname{nc}}$ , which induces the functorial isomorphism $h_{N^{\operatorname{nc}}}\rightarrow h_{Q,\{0\},\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ by the above argument.

Now we consider the subscheme $M\subset N$ . Let ${\mathcal{J}}_{I}\subset {\mathcal{O}}_{N^{\operatorname{nc}}}$ be the ideal of relation in $I$ , and set ${\mathcal{O}}_{M}^{\operatorname{nc}}={\mathcal{O}}_{N}^{\operatorname{nc}}/\overline{{\mathcal{J}}}_{I}$ . Then, as in the proof of Proposition 3.11, the NC scheme $(M,{\mathcal{O}}_{M}^{\operatorname{nc}})$ is the desired NC structure by the Cartesian square

3.7 An example

We describe an example of an NC thickening of the moduli space of representations of a quiver. We consider the quiver $Q$ described as

with relation $I$ given by

(3.7.1) $$\begin{eqnarray}\displaystyle xy=yx. & & \displaystyle\end{eqnarray}$$

We also set the vectors $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ in $\unicode[STIX]{x1D6E4}_{Q}$ to be

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}=(\unicode[STIX]{x1D6FE}_{\diamond },\unicode[STIX]{x1D6FE}_{\star })=(1,2),\quad \unicode[STIX]{x1D703}=(\unicode[STIX]{x1D703}_{\diamond },\unicode[STIX]{x1D703}_{\star })=(-2,1). & & \displaystyle \nonumber\end{eqnarray}$$

A representation $W$ of $(Q,I)$ with dimension vector $\unicode[STIX]{x1D6FE}$ is given by the diagram

(3.7.2)

where $f,A,B$ are linear maps satisfying $AB=BA$ . It is easy to see that a $(Q,I)$ -representation (3.7.2) is $\unicode[STIX]{x1D703}$ -stable if and only if $f(1)$ generates $\mathbb{C}^{2}$ as a $\mathbb{C}[x,y]$ -module. Hence, we have the natural identification

$$\begin{eqnarray}\displaystyle M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})=\operatorname{Hilb}^{2}(\mathbb{C}^{2}), & & \displaystyle \nonumber\end{eqnarray}$$

where the right-hand side is the Hilbert scheme of two points on $\mathbb{C}^{2}$ . On the other hand, the smooth moduli space $M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})$ is given by

$$\begin{eqnarray}\displaystyle M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})=(\mathbb{C}^{2}\times M_{2}(\mathbb{C})\times M_{2}(\mathbb{C}))^{s}/\text{GL}_{2}(\mathbb{C}). & & \displaystyle \nonumber\end{eqnarray}$$

Here $(-)^{s}$ means the $\unicode[STIX]{x1D703}$ -stable part, and the $\text{GL}_{2}(\mathbb{C})$ action is given by

$$\begin{eqnarray}\displaystyle g(v,A,B)=(g^{-1}v,g^{-1}Ag,g^{-1}Bg). & & \displaystyle \nonumber\end{eqnarray}$$

By the $\unicode[STIX]{x1D703}$ -stability, we have $v\neq 0$ and hence we have

(3.7.3) $$\begin{eqnarray}\displaystyle M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})=(\{(1,0)^{t}\}\times M_{2}(\mathbb{C})\times M_{2}(\mathbb{C}))^{s}/G, & & \displaystyle\end{eqnarray}$$

where $G$ is the stabilizer of $(1,0)^{t}$ :

$$\begin{eqnarray}\displaystyle G=\left(\begin{array}{@{}cc@{}}1 & u\\ 0 & v\end{array}\right),\quad u,v\in \mathbb{C}. & & \displaystyle \nonumber\end{eqnarray}$$

We omit $\{(1,0)^{t}\}$ in the notation of the right-hand side of (3.7.3). Then we have

$$\begin{eqnarray}\displaystyle (M_{2}(\mathbb{C})\times M_{2}(\mathbb{C}))^{s}=\left\{A=\left(\begin{array}{@{}cc@{}}a_{1} & a_{2}\\ a_{3} & a_{4}\end{array}\right),B=\left(\begin{array}{@{}cc@{}}b_{1} & b_{2}\\ b_{3} & b_{4}\end{array}\right):a_{3}\neq 0\text{ or }b_{3}\neq 0\right\}. & & \displaystyle \nonumber\end{eqnarray}$$

The open subsets $a_{3}\neq 0$ , $b_{3}\neq 0$ are $G$ -invariants and hence we obtain the open covering

$$\begin{eqnarray}\displaystyle M_{Q,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})=U_{A}\cup U_{B}, & & \displaystyle \nonumber\end{eqnarray}$$

where $U_{A}$ , $U_{B}$ are quotients of $a_{3}\neq 0$ , $b_{3}\neq 0$ , respectively. We also obtain the open cover

$$\begin{eqnarray}\displaystyle M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE})=V_{A}\cup V_{B},\quad V_{\ast }=U_{\ast }\cap M_{Q,I,\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6FE}). & & \displaystyle \nonumber\end{eqnarray}$$

For example, $U_{A}$ is given by

$$\begin{eqnarray}\displaystyle U_{A}=\left\{A=\left(\begin{array}{@{}cc@{}}0 & a_{2}\\ 1 & a_{4}\end{array}\right),B=\left(\begin{array}{@{}cc@{}}b_{1} & b_{2}\\ b_{3} & b_{4}\end{array}\right):a_{i},b_{i}\in \mathbb{C}\right\}, & & \displaystyle \nonumber\end{eqnarray}$$

so $U_{A}\cong \mathbb{C}^{6}$ . A smooth NC thickening of $U_{A}$ is given by

$$\begin{eqnarray}\displaystyle U_{A}^{\operatorname{nc}}=\text{Spf}~\mathbb{C}\langle a_{2},a_{4},b_{1},b_{2},b_{3},b_{4}\rangle _{[[\text{ab}]]}. & & \displaystyle \nonumber\end{eqnarray}$$

Then the ideal of relation (3.7.1) in ${\mathcal{O}}_{U_{A}}^{\operatorname{nc}}$ is determined by the relation

$$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}0 & a_{2}\\ 1 & a_{4}\end{array}\right)\left(\begin{array}{@{}cc@{}}b_{1} & b_{2}\\ b_{3} & b_{4}\end{array}\right)=\left(\begin{array}{@{}cc@{}}b_{1} & b_{2}\\ b_{3} & b_{4}\end{array}\right)\left(\begin{array}{@{}cc@{}}0 & a_{2}\\ 1 & a_{4}\end{array}\right), & & \displaystyle \nonumber\end{eqnarray}$$

where we regard $a_{i}$ and $b_{i}$ as non-commutative variables in ${\mathcal{O}}_{U_{A}}^{\operatorname{nc}}$ . By expanding the above matrix multiplications, we obtain

(3.7.4) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}a_{2}b_{3}-b_{2} & a_{2}b_{4}-b_{1}a_{2}-b_{2}a_{4}\\ b_{1}+a_{4}b_{3}-b_{4} & b_{2}+a_{4}b_{4}-b_{3}a_{2}-b_{4}a_{4}\end{array}\right)=0. & & \displaystyle\end{eqnarray}$$

By definition, the two-sided ideal ${\mathcal{J}}_{A}\subset {\mathcal{O}}_{U_{A}}^{\operatorname{nc}}$ of relations in $I$ is generated by the matrix components of the left-hand side of (3.7.4)

$$\begin{eqnarray}\displaystyle {\mathcal{J}}_{A}=(a_{2}b_{3}-b_{2},a_{2}b_{4}-b_{1}a_{2}-b_{2}a_{4},b_{1}+a_{4}b_{3}-b_{4},b_{2}+a_{4}b_{4}-b_{3}a_{2}-b_{4}a_{4}). & & \displaystyle \nonumber\end{eqnarray}$$

The quotient ${\mathcal{O}}_{V_{A}}^{\operatorname{nc}}={\mathcal{O}}_{U_{A}}^{\operatorname{nc}}/\overline{{\mathcal{J}}}_{A}$ gives an NC thickening of $V_{A}$ :

$$\begin{eqnarray}\displaystyle V_{A}^{\operatorname{nc}}=\text{Spf}\,\biggl(\frac{\mathbb{C}\langle a_{2},a_{4},b_{1},b_{3}\rangle }{([a_{2},b_{1}]+a_{2}[a_{4},b_{3}],[a_{2},b_{3}]+[a_{4},b_{1}]+a_{4}[a_{4},b_{3}])}\biggr)_{[[\text{ab}]]}. & & \displaystyle \nonumber\end{eqnarray}$$

Note that $V_{A}\cong \mathbb{C}^{4}$ , with coordinates $(a_{2},a_{4},b_{1},b_{3})$ . In the same way, we obtain an NC thickening of $V_{B}\cong \mathbb{C}^{4}$ :

$$\begin{eqnarray}\displaystyle V_{B}^{\operatorname{nc}}=\text{Spf}\,\biggl(\frac{\mathbb{C}\langle b_{2}^{\prime },b_{4}^{\prime },a_{1}^{\prime },a_{3}^{\prime }\rangle }{([b_{2}^{\prime },a_{1}^{\prime }]+b_{2}^{\prime }[b_{4}^{\prime },a_{3}^{\prime }],[b_{2}^{\prime },a_{3}^{\prime }]+[b_{4}^{\prime },a_{1}^{\prime }]+b_{4}^{\prime }[b_{4}^{\prime },a_{3}^{\prime }])}\biggr)_{[[\text{ab}]]}. & & \displaystyle \nonumber\end{eqnarray}$$

Note that

$$\begin{eqnarray}\displaystyle V_{A}\cap V_{B}=\{b_{3}\neq 0\}=\{a_{3}^{\prime }\neq 0\}. & & \displaystyle \nonumber\end{eqnarray}$$

The gluing isomorphism

$$\begin{eqnarray}\displaystyle V_{A}^{\operatorname{nc}}|_{V_{A}\cap V_{B}}\stackrel{\cong }{\rightarrow }V_{B}^{\operatorname{nc}}|_{V_{A}\cap V_{B}} & & \displaystyle \nonumber\end{eqnarray}$$

is calculated as

$$\begin{eqnarray}\displaystyle a_{3}^{\prime } & \mapsto & \displaystyle b_{3}^{-1},\nonumber\\ \displaystyle a_{1}^{\prime } & \mapsto & \displaystyle -b_{1}b_{3}^{-1},\nonumber\\ \displaystyle b_{4}^{\prime } & \mapsto & \displaystyle b_{1}+b_{3}^{-1}b_{1}b_{3}+b_{3}^{-1}a_{4}b_{3}^{2},\nonumber\\ \displaystyle b_{2}^{\prime } & \mapsto & \displaystyle a_{2}b_{3}^{2}-b_{1}b_{3}^{-1}b_{1}b_{3}-b_{1}b_{3}^{-1}a_{4}b_{3}^{2}.\nonumber\end{eqnarray}$$

4.1 Graded algebras

Let $(X,{\mathcal{O}}_{X}(1))$ be a polarized projective scheme over $\mathbb{C}$ . Then $X=\text{Proj}(A)$ for the graded $\mathbb{C}$ -algebra $A$ given by

$$\begin{eqnarray}\displaystyle A=\bigoplus _{i\geqslant 0}H^{0}(X,{\mathcal{O}}_{X}(i)). & & \displaystyle \nonumber\end{eqnarray}$$

Below, we set $\mathfrak{m}:=A_{{>}0}$ , the maximal ideal of $A$ .

For $\unicode[STIX]{x1D6EC}\in {\mathcal{N}}$ , we set $A_{\unicode[STIX]{x1D6EC}}:=A\otimes _{\mathbb{C}}\unicode[STIX]{x1D6EC}$ . Let $A_{\unicode[STIX]{x1D6EC}}~\text{mod}_{\operatorname{gr}}$ be the category of finitely generated graded left $A_{\unicode[STIX]{x1D6EC}}$ -modules. For $M\in A_{\unicode[STIX]{x1D6EC}}~\text{mod}_{\text{gr}}$ , we denote by $M_{i}$ the degree- $i$ part of $M$ , and $M(j)$ the graded left $A_{\unicode[STIX]{x1D6EC}}$ -module such that $M(j)_{i}=M_{j+i}$ . For an interval $\mathbb{I}\subset \mathbb{Z}$ , we define

$$\begin{eqnarray}\displaystyle A_{\unicode[STIX]{x1D6EC}}~\text{mod}_{\mathbb{I}}\subset A_{\unicode[STIX]{x1D6EC}}~\text{mod}_{\text{gr}} & & \displaystyle \nonumber\end{eqnarray}$$

to be the subcategory of graded left $A_{\unicode[STIX]{x1D6EC}}$ -modules $M$ such that $M_{i}=0$ for $i\notin I$ . For $q>p>0$ , we describe the category $A_{\unicode[STIX]{x1D6EC}}~\text{mod}_{[p,q]}$ in terms of a quiver with relation.

We define the quiver $Q_{[p,q]}$ whose set of vertices is

$$\begin{eqnarray}\displaystyle \{p,p+1,\ldots ,q\}. & & \displaystyle \nonumber\end{eqnarray}$$

The number of arrows of $Q_{[p,q]}$ from $i$ to $j$ is given by $\dim _{\mathbb{C}}\mathfrak{m}_{j-i}$ (cf. Figure 1). Below, we fix bases of $\mathfrak{m}_{k}$ for each $k\in \mathbb{Z}_{{\geqslant}1}$ , and identify the set of arrows from $i$ to $j$ with the set of basis elements of $\mathfrak{m}_{j-i}$ . The multiplication

(4.1.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D717}:\mathfrak{m}_{j-i}\otimes _{\mathbb{C}}\mathfrak{m}_{k-j}\rightarrow \mathfrak{m}_{k-i} & & \displaystyle\end{eqnarray}$$

in $A$ defines the relation in $Q_{[p,q]}$ , by defining the two-sided ideal $I\subset \mathbb{C}[Q_{[p,q]}]$ to be generated by