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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях) Smooth DG algebras and twisted tensor product D. O. Orlov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Англоязычная версия:
Russian Mathematical Surveys, 2023, Volume 78, Issue 5, Pages DOI: https://doi.org/10.4213/rm10139e 
Реферативные базы данных:
     Dedicated to the blessed memory of Igor Rostislavovich Shafarevich on the occasion of his 100th birthday IntroductionThe main objects of study in algebraic geometry are smooth projective varieties. The property of being projective, or more precisely of being proper, can easily be seen on the level of the category of perfect complexes on a variety. Namely, morphisms between perfect complexes in this case are finite-dimensional vector spaces. This property is generalized to noncommutative varieties and to differential graded (DG) algebras directly. Another important property is the property of smoothness. Smoothness, like the property of being regular, is also fundamental and can be extended to both noncommutative varieties and DG algebras. The main goal of this paper is to study smooth finite-dimensional algebras and DG algebras. For finite-dimensional DG algebras the property of properness is already fulfilled. Regularity (or smoothness) for finite-dimensional algebras is in fact equivalent to the finiteness of the global dimension with the additional separability property for the semisimple part if we are talking about smoothness. We show that these properties can also be extended to DG algebras. A natural question arises: what kind of algebras or DG algebras of finite global dimension do we know and how to construct them? Algebras of finite global dimension can naturally be obtained from a directed quiver $Q$ with arbitrary relations $I$. Recall that a quiver $Q$ is said to be directed if a total order is fixed on its (finite) set of vertices, and for any arrow the target vertex is strictly greater than the source vertex. Any algebra of the form $A=\Bbbk Q/I$, where $\Bbbk Q$ is the path algebra of the quiver $Q$ over a field $\Bbbk$, and $I$ is an ideal of relations, has a finite global dimension. In this case the category of perfect complexes $ \operatorname{perf}\!{-}A$, which is equivalent to the derived category of finite-dimensional modules ${\mathcal D}^b({\rm mod}{-}A)$, is a triangulated category with a full exceptional collection. From the point of view of DG categories the DG category of perfect complexes $ \mathscr{P}\!\mathit{erf}{-} A$ can be obtained as a gluing of several categories of the form $ \mathscr{P}\!\mathit{erf}{-} \Bbbk$ via perfect bimodules. Such categories always turn out to be smooth. Thus, the procedure of gluing smooth categories via perfect bimodules is the main operation used for obtaining new smooth categories (see [20]–[22]). However, it is important to study more general algebras and DG algebras of finite global dimension: in particular, ones that cannot be obtained by the gluing procedure on the level of the category of perfect complexes. Algebras of such type with two simple modules appeared in [8], and it was shown in [9] and [18] that the categories of perfect complexes for these algebras do not have full exceptional collections. Another series of algebras of finite global dimension with two simple modules was constructed in [13]. One of the interesting features of these algebras is that their nilpotency index is equal to $4$, while the global dimension can be arbitrarily large. A question that also arises is to find other operations that allow one to construct smooth algebras and DG algebras, starting from known and elementary smooth algebras. A well-known example of such an operation is the usual tensor product. The tensor product of two smooth DG $\Bbbk$-algebras $\mathscr A\otimes_{\Bbbk}\mathscr B$ is also smooth (see [16]). However, this operation has strong limitations and increases naturally the rank of the semisimple part. Moreover, it can be defined only over a central subring. In this paper we consider the operation of twisted tensor product of noncommutative algebras and DG algebras over various subalgebras that do not belong to the centre, and apply this operation to the construction of new smooth finite-dimensional algebras and DG algebras. Note that the twisted tensor product appeared in the literature before. For example, [3] defined twisted tensor product over a central subring, and in [4], for a pair of finite-dimensional algebras with a common semisimple part, a certain operation was introduced, which should be understood as the twisted tensor product of these algebras over a semisimple subalgebra. In this paper we give sufficient conditions for a (DG) twisted tensor product of DG algebras to be smooth (see Theorems 2.16 and 3.13). In the case where these DG algebras admit augmentations, one can present an explicit twisted map for them, which defines a certain twisted tensor product of these DG algebras and allows one to construct new examples of smooth DG algebras (see (5) in Construction 2.12). In section 4 we consider examples of algebras of finite global dimension with two simple modules and show how they can be obtained as twisted tensor products. Furthermore, we give new examples of families of algebras with two simple modules that have a finite global dimension (see Theorem 4.11) and show that the categories of perfect modules over these algebras do not have exceptional objects (see Corollary 5.4). In contrast to already known algebras of such type, these families of algebras have non-trivial moduli spaces. We also show that all these algebras can be obtained by taking a sequence of twisted tensor products of elementary smooth algebras (see Theorem 4.16). In the last section we consider the Grothendieck groups of smooth DG algebras and show that any matrix in $\operatorname{SL}(n,{\mathbb Z})$ can be realized as the matrix of the Euler bilinear form (13) for some smooth DG algebra (see Corollary 5.2). The author is very grateful to Anton Fonarev and Alexander Kuznetsov for useful discussions and valuable comments. 1. Preliminaries1.1. Differential graded algebrasLet $\Bbbk$ be a field. Recall that a differential graded $\Bbbk$-algebra (${}={}$ DG algebra) $\mathscr R=(R,d_{\mathscr R})$ is a ${\mathbb Z}$-graded associative $\Bbbk$-algebra $R =\bigoplus\limits_{q\in {\mathbb Z}} R^q$ endowed with a $\Bbbk$-linear differential $d_{\mathscr R}\colon R \to R$ (that is, a homogeneous map $d_{\mathscr R}$ of degree $1$ with $d_{\mathscr R}^2=0$) that satisfies the graded Leibniz rule
$$
\begin{equation*}
d_{\mathscr R}(xy)=d_{\mathscr R}(x)y+(-1)^q x d_{\mathscr R}(y) \quad \text{for all}\ \ x\in R^q,\ \ y\in R.
\end{equation*}
\notag
$$
We consider DG algebras with an identity element $1\in R^0$. In this case, $d_{\mathscr R}(1)=0$. Notice that any ordinary associative $\Bbbk$-algebra $\Lambda$ can be considered as a DG algebra $\mathscr R$ such that $R^0=\Lambda$ and $R^q=0$ for $q \ne 0$. A differential graded module $\mathsf M$ over $\mathscr R$ (${}={}$ DG $\mathscr R$-module) is a ${\mathbb Z}$-graded right $R$-module $M=\bigoplus\limits_{q\in{\mathbb Z}} M^q$ endowed with a $\Bbbk$-linear differential $d_{\mathsf M}\colon M \to M$ of degree $1$ satisfying the equality $d_{\mathsf M}^2=0$ and the graded Leibniz rule, that is,
$$
\begin{equation*}
d_{\mathsf M}(mr)=d_{\mathsf M}(m)r+(-1)^q md_{\mathscr R}(r) \quad\text{for all}\ \ m\in M^q,\ \ r\in R.
\end{equation*}
\notag
$$
Let $\mathsf M$ and $\mathsf N$ be two DG modules. We can define the complex of $\Bbbk$-vector spaces $\mathsf{Hom}_{\mathscr R}(\mathsf M,\mathsf N)$ as the graded vector space
$$
\begin{equation*}
\operatorname{Hom}_{R}^{\rm gr}(M,N):= \bigoplus_{q\in{\mathbb Z}}\operatorname{Hom}_{R}(M,N)^q,
\end{equation*}
\notag
$$
where $\operatorname{Hom}_{R}(M,N)^q$ is the space of homogeneous homomorphisms of $R$-modules of degree $q$. The differential $D$ of the complex $\mathsf{Hom}_{\mathscr R}(\mathsf M,\mathsf N)$ is defined by the following rule:
$$
\begin{equation*}
D(f)=d_{\mathsf N}\circ f-(-1)^qf\circ d_{\mathsf M}\quad\text{for each}\ \ f\in \operatorname{Hom}_{R}(M,N)^q.
\end{equation*}
\notag
$$
Thus all (right) DG $\mathscr R$-modules form a DG category $ {\mathscr M}\!\mathit{od}{-} \mathscr R$. Let $ \mathscr A\!\mathit{c}{-} \mathscr R$ be the full DG subcategory consisting of all acyclic DG modules, that is, DG modules with trivial cohomology. The homotopy category ${{\mathcal H}^0}( {\mathscr M}\!\mathit{od}{-} \mathscr R)$ has the natural structure of a triangulated category, and the homotopy subcategory of acyclic complexes ${{\mathcal H}^0}( \mathscr A\!\mathit{c}{-} \mathscr R)$ forms a full triangulated subcategory in it. The derived category ${\mathcal D}(\mathscr R)$ is defined as the Verdier quotient
$$
\begin{equation*}
{\mathcal D}(\mathscr R):={{\mathcal H}^0} ( {\mathscr M}\!\mathit{od}{-} \mathscr R)/{{\mathcal H}^0} ( \mathscr A\!\mathit{c}{-} \mathscr R).
\end{equation*}
\notag
$$
It is well known that the derived category ${\mathcal D}(\mathscr R)$ is equivalent to the homotopy category ${{\mathcal H}^0}( \mathscr S\!\mathscr F{-} \mathscr R)$, where $ \mathscr S\!\mathscr F{-} \mathscr R\subset {\mathscr M}\!\mathit{od}{-} \mathscr R$ is the DG subcategory of semi-free modules. Recall that a DG module $\mathsf P$ is called semi-free if it has a filtration $0=\mathsf \Phi_0\subset \mathsf \Phi_1\subset \cdots=\mathsf P= \bigcup \mathsf \Phi_n$ with free quotients $\mathsf \Phi_{i+1}/\mathsf \Phi_i$ (see [12]). We will also need notions of semi-projective and semi-flat modules. Definition 1.1. A DG $\mathscr R$-module $\mathsf M$ is called semi-projective (DG projective) if the following equivalent conditions hold: Definition 1.2. A DG $\mathscr R$-module $\mathsf M$ is called semi-flat (DG flat) if the following equivalent conditions hold: It is easy to see that any semi-projective module is semi-flat, and the homotopy category of semi-projective module is equivalent to the category ${{\mathcal H}^0}( \mathscr S\!\mathscr F{-} \mathscr R)\cong {\mathcal D}(\mathscr R)$. 1.2. Categories of perfect modules and functorsDenote by $\mathscr{S\!F}_{{\rm fg}}{-}\mathscr R\subset \mathscr S\!\mathscr F{-} \mathscr R$ the full DG subcategory of finitely generated semi-free DG modules, that is, semi-free DG modules such that $\mathsf \Phi_n=\mathsf P$ for some $n$, and $\mathsf \Phi_{i+1}/\mathsf \Phi_i$ is a finite direct sum of $\mathscr R[m]$. The DG category of perfect modules $ \mathscr{P}\!\mathit{erf}{-} \mathscr R$ is the full DG subcategory of $ \mathscr S\!\mathscr F{-} \mathscr R$ consisting of all DG modules that are isomorphic to direct summands of objects of $\mathscr{S\!F}_{{\rm fg}}{-}\mathscr R$ in the homotopy category ${{\mathcal H}^0}( \mathscr S\!\mathscr F{-} \mathscr R)$. The homotopy category ${{\mathcal H}^0}( \mathscr{P}\!\mathit{erf}{-} \mathscr R)$, which we denote by $ \operatorname{perf}\!{-} \mathscr R$, is called the triangulated category of perfect modules. It is equivalent to the triangulated subcategory of compact objects ${\mathcal D}(\mathscr R)^c\subset {\mathcal D}(\mathscr R)$ (see [12]). In other words, the category $ \operatorname{perf}\!{-} \mathscr R$ is the subcategory in ${\mathcal D}(\mathscr R)$ that is (classically) generated by the DG algebra $\mathscr R$ itself in the following sense. Definition 1.3. A set $S$ of objects of a triangulated category ${\mathcal T}$ generates ${\mathcal T}$ (classically) if the smallest full triangulated subcategory of ${\mathcal T}$ containing $S$ and closed under taking direct summands coincides with the whole of the category ${\mathcal T}$. In the case where $S$ consists of a single object $E\in {\mathcal T}$, the object $E$ is called a classical generator of ${\mathcal T}$. A classical generator, which generates a triangulated category in a finite number of steps, is called a strong generator. More precisely, let ${\mathcal I}_1$ and ${\mathcal I}_2$ be two full subcategories of a triangulated category ${\mathcal T}$. Denote by ${\mathcal I}_1*{\mathcal I}_2$ the full subcategory of ${\mathcal T}$ consisting of all objects $M$ for which there is an exact triangle $M_1\to M\to M_2$ with $M_i\in {\mathcal I}_i$. For any subcategory ${\mathcal I}\subset{\mathcal T}$ denote by $\langle {\mathcal I}\rangle$ the smallest full subcategory of ${\mathcal T}$ containing ${\mathcal I}$ and closed under taking finite direct sums or direct summands and shifts. We put ${\mathcal I}_1 \diamond{\mathcal I}_2= \langle {\mathcal I}_1*{\mathcal I}_2\rangle$ and define $\langle {\mathcal I}\rangle_k=\langle{\mathcal I}\rangle_{k-1} \diamond\langle {\mathcal I}\rangle$ recursively. If ${\mathcal I}$ consists of a single object $E$, we denote $\langle {\mathcal I}\rangle$ by $\langle E\rangle_1$ and put recursively $\langle E\rangle _k=\langle E\rangle_{k-1}\diamond\langle E\rangle_1$. Definition 1.4. An object $E\in{\mathcal T}$ is called a strong generator if $\langle E\rangle_n={\mathcal T}$ for some $n\in{\mathbb N}$. It is easy to see that if a triangulated category has a strong generator, then all of its classical generators are also strong. Let $\mathscr R$ and $\mathscr S$ be two DG algebras, and let $\mathsf f\colon\mathscr R \to \mathscr S$ be a morphism of DG algebras. It induces the restriction DG functor $\mathsf f_*\colon {\mathscr M}\!\mathit{od}{-} \mathscr S\to {\mathscr M}\!\mathit{od}{-} \mathscr R$ between the DG categories of DG modules. The restriction functor $\mathsf f_*$ has left and right adjoint functors $\mathsf f^*$, $\mathsf f^{!}$ that are defined as follows:
$$
\begin{equation*}
\mathsf f^*\mathsf M=\mathsf M\otimes_{\mathscr R} \mathscr S,\quad \mathsf f^{!}=\mathsf{Hom}_{ {\mathscr M}\!\mathit{od}{-} \mathscr R} (\mathscr S,\mathsf M).
\end{equation*}
\notag
$$
The DG functor $\mathsf f_*$ preserves acyclic DG modules and induces a derived functor ${\mathbf R} f_*\colon {\mathcal D}(\mathscr S)\to {\mathcal D}(\mathscr R)$, and the DG functor $\mathsf f^*$ preserves semi-free DG modules. The existence of semi-free resolutions allows us to define a derived functor ${\mathbf L} f^*$ from ${\mathcal D}(\mathscr R)$ to ${\mathcal D}(\mathscr S)$ (see [12]). For example, the derived functor ${\mathbf L} f^*\colon {\mathcal D}(\mathscr R)\to {\mathcal D}(\mathscr R)$ is isomorphic to the induced homotopy functor ${\mathcal H}^0(\mathsf f^*)$ for the extension DG functor $\mathsf f^*\colon \mathscr S\!\mathscr F{-} \mathscr R\to \mathscr S\!\mathscr F{-} \mathscr S$. More generally, let $\mathsf T$ be an $\mathscr R$-$\mathscr S$-bimodule, that is (by definition), a DG module over $\mathscr R^{\circ}\otimes_{\Bbbk}\mathscr S$. For each DG $\mathscr R$-module $\mathsf M$ we obtain a DG $\mathscr S$-module $\mathsf M\otimes_{\mathscr R}\mathsf T$. The DG functor
$$
\begin{equation*}
(-)\otimes_{\mathscr R}\mathsf T\colon {\mathscr M}\!\mathit{od}{-} \mathscr R \to {\mathscr M}\!\mathit{od}{-} \mathscr S
\end{equation*}
\notag
$$
admits a right adjoint $\mathsf{Hom}_{\mathscr S} (\mathsf T, -)$. These functors induce an adjoint pair of derived functors $(-)\stackrel{{\mathbf L}}{\otimes}_{\mathscr R}\mathsf T$ and ${\mathbf R} \operatorname{Hom}_{\mathscr S} (\mathsf T,-)$ between the derived categories ${\mathcal D}(\mathscr R)$ and ${\mathcal D}(\mathscr S)$ (see [12]). 1.3. The properties of DG algebrasLet us now discuss some basic properties of DG algebras and categories of perfect modules over DG algebras. Definition 1.5. Let $\mathscr R$ be a DG $\Bbbk$-algebra. Then All these properties are properties of the DG category $ \mathscr{P}\!\mathit{erf}{-} \mathscr R$. It is easy to see that $\mathscr R$ is proper if and only if $\bigoplus\limits_{m\in{\mathbb Z}}\operatorname{Hom}(X, Y[m])$ is finite-dimensional for any two objects $X,Y\in \operatorname{perf}\!{-} \mathscr R$. It was proved in [17] that smoothness is invariant under Morita equivalence, that is, it is a property of the DG category $ \mathscr{P}\!\mathit{erf}{-} \mathscr R$. It is also known that any smooth DG algebra is regular (see [16]). Definition 1.6. A DG $\mathscr R$-module $\mathsf M$ is called cohomologically finite-dimensional if it is perfect as a complex of $\Bbbk$-vector spaces, that is, $\bigoplus\limits_{p\in{\mathbb Z}}H^p(\mathsf M)$ is a finite-dimensional vector space. Denote by ${\mathcal D}_{\mathrm{cf}}(\mathscr R)\subset {\mathcal D}(\mathscr R)$ the full triangulated subcategory of cohomologically finite-dimensional DG modules. It is obvious that $ \operatorname{perf}\!{-} \mathscr R\subseteq {\mathcal D}_{\mathrm{cf}}(\mathscr R)$ for any proper DG algebra $\mathscr R$. On the other hand, if $\mathscr R$ is smooth, then there is the reverse inclusion ${\mathcal D}_{\mathrm{cf}}(\mathscr R) \subseteq \operatorname{perf}\!{-} \mathscr R$ (see, for example, [14]). Thus, for any smooth and proper DG algebra $\mathscr R$ we have an equivalence ${\mathcal D}_{\mathrm{cf}}(\mathscr R)\cong \operatorname{perf}\!{-} \mathscr R$. Moreover, the following proposition is a particular case of Theorem 1.3 from [1]. Proposition 1.7 [1]. Let $\mathscr R$ be a proper and regular DG algebra. Then ${\mathcal D}_{\mathrm{cf}}(\mathscr R)\cong \operatorname{perf}\!{-} \mathscr R$. However, it should be noted that regularity itself, in contrast to smoothness, does not imply that all cohomologically finite-dimensional modules are perfect. The simplest example is a gluing of the field $\Bbbk$ with itself via an infinite-dimensional vector space. Proposition 1.8. Let $\mathscr R$ be a DG $\Bbbk$-algebra. Then the following conditions are equivalent: Proof. If $\mathscr R$ is smooth, then $\mathscr R^{\circ}$ and $\mathscr R^{\circ}\otimes_{\Bbbk}\mathscr R$ are smooth too (see, for example, [16; Lemma 3.3]). Hence $(1)\Rightarrow (2)$. Further, $(2)\Rightarrow (3)$ by Proposition 1.7, because smoothness of $\mathscr R$ implies regularity by [16; Lemma 3.6]. If $(3)$ holds, then $\mathscr R$ is proper and it is perfect as a bimodule. Therefore, $\mathscr R$ is smooth. Thus, $(3)\Rightarrow (1)$. This finishes the proof of the proposition. Definition 1.9. Let $\mathscr R$ be a proper DG algebra. We say that $\mathscr R$ has a regular (or smooth) realization if there is a proper and regular (or smooth) DG algebra $\mathscr S$ and an $\mathscr R$-$\mathscr S$-bimodule $\mathsf T$ such that the functor ${\mathbf L} F^*\cong (-)\stackrel{{\mathbf L}}{\otimes}_\mathscr R \mathsf T \colon \operatorname{perf}\!{-} \mathscr R\to \operatorname{perf}\!{-} \mathscr S$ is fully faithful. Remark 1.10. Note that there is an example of a proper DG algebra that does not have a smooth realization. Such an example can be found in Efimov’s paper [5] (Theorem 5.4 and Proposition 5.1). Proposition 1.11. Let $\mathscr R$ be a proper DG algebra. Assume that $\mathscr R$ has a regular (or smooth) realization. Then $\mathscr R$ is regular (or smooth) if and only if $ \operatorname{perf}\!{-} \mathscr R\cong {\mathcal D}_{\mathrm{cf}}(\mathscr R)$. Proof. In one direction, this follows from Proposition 1.7.
Assume now that $ \operatorname{perf}\!{-} \mathscr R\cong {\mathcal D}_{\mathrm{cf}}(\mathscr R)$, and consider a regular (or smooth) realization
$$
\begin{equation*}
{\mathbf L} F^*\cong (-)\stackrel{{\mathbf L}}{\otimes}_\mathscr R \mathsf T\colon \operatorname{perf}\!{-} \mathscr R\hookrightarrow \operatorname{perf}\!{-} \mathscr S.
\end{equation*}
\notag
$$
Since the DG algebra $\mathscr S$ is regular (or smooth), we also have an equivalence $ \operatorname{perf}\!{-} \mathscr S\cong {\mathcal D}_{\mathrm{cf}}(\mathscr S)$. We know that $\mathsf T={\mathbf L} F^*(\mathscr R)\in \operatorname{perf}\!{-} \mathscr S$. Hence the right adjoint functor ${\mathbf R} \operatorname{Hom}_{\mathscr S} (\mathsf T, -)\colon {\mathcal D}(\mathscr S)\to {\mathcal D}(\mathscr R)$ sends ${\mathcal D}_{\mathrm{cf}}(\mathscr S)$ to ${\mathcal D}_{\mathrm{cf}}(\mathscr R)$. Therefore, we obtain a projection
$$
\begin{equation*}
{\mathbf R} \operatorname{Hom}_{\mathscr S} (\mathsf T, -)\colon \operatorname{perf}\!{-} \mathscr S\to \operatorname{perf}\!{-} \mathscr R.
\end{equation*}
\notag
$$
If now $\mathscr S$ is regular, then $\mathscr S$ is a strong generator for $ \operatorname{perf}\!{-} \mathscr S$. Hence the object ${\mathbf R}\operatorname{Hom}_{\mathscr S}(\mathsf T,\mathscr S)$ is a strong generator for $ \operatorname{perf}\!{-} \mathscr R$, and $\mathscr R$ is regular too. If, in addition, $\mathscr S$ is smooth, then $\mathscr R$ is also smooth by [17; 3.24]. The proposition is proved. 2. Morphisms of DG algebras and twisted tensor product2.1. Morphisms of DG algebra and smoothnessNow let $\mathsf f\colon\mathscr R \to \mathscr S$ be a morphism of DG algebras. As above, it produces the derived functors
$$
\begin{equation*}
{\mathbf L} \mathsf f^*\colon {\mathcal D}(\mathscr R)\to {\mathcal D}(\mathscr S)\quad\text{and}\quad {\mathbf R}\mathsf f_*\colon {\mathcal D}(\mathscr S)\to {\mathcal D}(\mathscr R),
\end{equation*}
\notag
$$
which are called the inverse image and the direct image functors, respectively. These derived functors induce functors
$$
\begin{equation*}
{\mathbf L} \mathsf f^*\colon \operatorname{perf}\!{-} \mathscr R\to \operatorname{perf}\!{-} \mathscr S\quad\text{and}\quad {\mathbf R}\mathsf f_*\colon {\mathcal D}_{\mathrm{cf}}(\mathscr S)\to {\mathcal D}_{\mathrm{cf}}(\mathscr R).
\end{equation*}
\notag
$$
Definition 2.1 [21; Definition 2.8]. A morphism of DG algebras $\mathsf f\colon\mathscr R \to \mathscr S$ is called a pp-morphism (perfect proper morphism) if the direct image functor ${\mathbf R}\mathsf f_*\colon {\mathcal D}(\mathscr S)\to {\mathcal D}(\mathscr R)$ sends perfect modules to perfect ones. The previous definition is equivalent to saying that $\mathscr S$ is perfect as a right DG $\mathscr R$-module. This also means that the inverse image functor ${\mathbf L} \mathsf f^*$ considered as a functor from $ \operatorname{perf}\!{-} \mathscr R$ to $ \operatorname{perf}\!{-} \mathscr S$ has a right adjoint ${\mathbf R} \mathsf f_*\colon \operatorname{perf}\!{-} \mathscr S\to \operatorname{perf}\!{-} \mathscr R$. Definition 2.2. Let $\mathscr R$ be a proper DG algebra. A morphism of DG algebras $\mathsf f\colon\mathscr R \to \mathscr S$ will be called respectable if objects of the form ${\mathbf R}\mathsf f_* \mathsf T$ for $\mathsf T\in{\mathcal D}_{\mathrm{cf}}(\mathscr S)$, generate the whole of the triangulated category ${\mathcal D}_{\mathrm{cf}}(\mathscr R)$. The morphism $\mathsf f$ will be called acceptable if the subcategory ${\mathcal D}\subseteq{\mathcal D}_{\mathrm{cf}}(\mathscr R)$ generated by the objects ${\mathbf R}\mathsf f_* \mathsf T$ for $\mathsf T\in{\mathcal D}_{\mathrm{cf}}(\mathscr S)$ contains the category $ \operatorname{perf}\!{-} \mathscr R$. The next lemma is almost obvious. Lemma 2.3. Let $\mathsf f\colon\mathscr R \to \mathscr S$ be a respectable pp-morphism of DG algebras. Assume that there is an inclusion ${\mathcal D}_{\mathrm{cf}}(\mathscr S)\subseteq \operatorname{perf}\!{-} \mathscr S$. Then there is an equivalence ${\mathcal D}_{\mathrm{cf}}(\mathscr R)\cong \operatorname{perf}\!{-} \mathscr R$. Proof. Since $\mathsf f$ is a pp-morphism, we have a functor ${\mathbf R} \mathsf f_*\colon \operatorname{perf}\!{-} \mathscr S\to \operatorname{perf}\!{-} \mathscr R$. If there is an inclusion ${\mathcal D}_{\mathrm{cf}}(\mathscr S)\subseteq \operatorname{perf}\!{-} \mathscr S$, then for any object $\mathsf T\subset {\mathcal D}_{\mathrm{cf}}(\mathscr S)$ the image ${\mathbf R}\mathsf f_*\mathsf T$ belongs to $\operatorname{perf}\!{-}\mathscr R$. By assumption, the morphism $\mathsf f$ is also respectable, that is, the category ${\mathcal D}_{\mathrm{cf}}(\mathscr R)$ is generated by objects of the form ${\mathbf R}\mathsf f_* \mathsf T$. Therefore, we obtain an inclusion ${\mathcal D}_{\mathrm{cf}}(\mathscr R)\subseteq \operatorname{perf}\!{-} \mathscr R$. On the other hand $\mathscr R$ is proper. Thus, we obtain an equivalence $ \operatorname{perf}\!{-} \mathscr R\cong{\mathcal D}_{\mathrm{cf}}(\mathscr R)$. The lemma is proved. Consider a commutative diagram of morphisms of DG algebras which satisfies the following two conditions:
$$
\begin{equation*}
\begin{array}{ll} \textrm{(Res)} & \begin{array}{ll} 1. & \textrm{the DG algebra $\mathscr C$ is semi-flat as a left DG $\mathscr A$-module} \\ &\textrm{(see Definition 1.2);} \\ 2. & \textrm{the canonical map $\mathscr R\otimes_{\mathscr A}\mathscr C\to\mathscr B$ is a quasi-isomorphism.} \end{array} \end{array}
\end{equation*}
\notag
$$
The following proposition allows us to deduce the smoothness of the DG algebra $\mathscr C$ in diagram (1). Proposition 2.4. Let (1) be a commutative diagram of morphisms of proper DG algebras that satisfies conditions (Res). Suppose the following conditions hold: Then there is an equivalence ${\mathcal D}_{\mathrm{cf}}(\mathscr C)\cong \operatorname{perf}\!{-} \mathscr C$. If, in addition, $\mathscr C$ has a regular (or smooth) realization, then $\mathscr C$ is also regular (or smooth). Proof. We know that $\mathscr R$ belongs to $ \operatorname{perf}\!{-} \mathscr A$ as a right DG $\mathscr A$-module because $\pi_A$ is a pp-morphism. Since $\mathscr C$ is semi-flat as a left DG $\mathscr A$-module, we obtain that $\mathscr B\cong \mathscr R\otimes_{\mathscr A}\mathscr C$ is also perfect as a right DG $\mathscr C$-module. Hence $p_B$ is a pp-morphism. Applying Lemma 2.3 to the morphism $p_B$, we obtain an equivalence ${\mathcal D}_{\mathrm{cf}}(\mathscr C)\cong \operatorname{perf}\!{-} \mathscr C$. If the DG algebra $\mathscr C$ has a regular (or smooth) realization, then by Proposition 1.11 it is regular (or smooth) itself. The proposition is proved. Corollary 2.5. Let (1) be a commutative diagram of morphisms of proper DG algebras that satisfies conditions (Res). Suppose the DG algebras $\mathscr A$ and $\mathscr B$ are regular (or smooth) and the morphism $p_B$ is respectable. Then there is an equivalence ${\mathcal D}_{\mathrm{cf}}(\mathscr C)\cong \operatorname{perf}\!{-} \mathscr C$. If, in addition, $\mathscr C$ has a regular (or smooth) realization, then $\mathscr C$ is also regular (or smooth). Proof. Since $\mathscr A$ is regular (or smooth) and $\mathscr R$ is proper, the DG module $\mathscr R$ belongs to $ \operatorname{perf}\!{-} \mathscr A$ by Proposition 1.7. Thus, $\pi_A$ is a pp-morphism and Proposition 2.4 implies the corollary. 2.2. Twisted tensor products of algebrasIn this section we consider and study so-called twisted tensor products of algebras and DG algebras. Twisted tensor products of algebras over a central subring were defined in [3]. Some special examples of twisted tensor products of finite-dimensional algebras over a common semisimple part appeared in [4] (see also [23] for DG algebras). We consider twisted tensor products of noncommutative (DG) algebras over an arbitrary (not necessarily commutative) (DG) ring. Let $R$, $A$, $B$ be $\Bbbk$-algebras, and let $\epsilon_A\colon R\to A$ and $\epsilon_B\colon R\to B$ be morphisms of algebras. We say that $A$ and $B$ are $R$-rings (or rings over $R$). Definition 2.6. A twisted tensor product over $R$ of two $R$-rings $A$ and $B$ is an $R$-ring $C$ together with two $R$-rings morphisms $i_A\colon A\to C$ and $i_B \colon B \to C$ such that the canonical map $\phi\colon A\otimes_R B \to C$ defined by $\phi(a\otimes b):=i_A(a)\cdot i_B(b)$ is an isomorphism of $R$-bimodules. There is a direct way to describe twisted tensor products. Let $\phi\colon A\otimes_R B \to C$ be the canonical isomorphism used in the definition of the twisted tensor product. Then we can define $\tau \colon B\otimes_R A \to A\otimes_R B$ by the rule $\tau(b \otimes a):=\phi^{-1}(i_B(b) \cdot i_A(a))$. Conversely, let $\tau\colon B\otimes_R A\to A\otimes_R B$ be an $R$-bilinear map for which
$$
\begin{equation}
\tau(1\otimes a)=a\otimes 1\quad\text{and}\quad \tau(b\otimes 1)=1\otimes b.
\end{equation}
\tag{2}
$$
In this case we can define a multiplication $\mu_{\tau}:=(\mu_A\otimes \mu_B)\circ (1\otimes\tau\otimes 1)$ on the $R$-bimodule $A\otimes_R B$. The multiplication $\mu_{\tau}$ is associative if and only if there is an equality
$$
\begin{equation}
\tau\circ (\mu_B\otimes \mu_A)=(\mu_A\otimes \mu_B) \circ (1\otimes\tau\otimes 1) \circ (\tau \otimes \tau) \circ (1\otimes\tau\otimes 1)
\end{equation}
\tag{3}
$$
holds for maps from $B\otimes_R B \otimes_R A \otimes_R A$ to $A \otimes_R B$ (see, for example, [3]). This means that the diagram is commutative. Definition 2.7. An $\mathscr R$-bilinear map $\tau$ that satisfies conditions (2) and (3) is called a twisting map for $A$ and $B$ over $R$, and we denote the $R$-ring $(A\otimes_R B,\mu_{\tau})$ by $A\otimes_R^{\tau}B$. Example 2.8. Any cyclic division algebra where $a,b,\zeta\in \Bbbk^*$ and $\zeta^n=1$, can be represented as a twisted tensor product $\Bbbk(\sqrt[n]{a})\otimes_{\Bbbk}^{\tau}\Bbbk(\sqrt[n]{b})$ with $\tau(y^k\otimes x^l)=\zeta^{kl}\cdot x^l\otimes y^k$. Example 2.9. The matrix algebra $M(n,\Bbbk)$ can be represented as a twisted tensor product $(\Bbbk[x]/x^n)\otimes_{\Bbbk}^{\tau}(\Bbbk[y]/y^n)$ for an appropriate twisting map $\tau$. Suppose the $R$-ring $A$ has an $R$-augmentation, that is, a morphism $\pi_A\colon A\to R$ such that $\pi_A\circ\epsilon_A$ is the identity map. Denote by $I_A=\operatorname{Ker}\pi_{A} \subset A$ the augmentation ideal. Definition 2.10. Suppose $A$ has an augmentation. A twisted tensor product $A\otimes_R^{\tau}B$ will be called right fixed (with respect to $\pi_A$) if $B$ is flat as a left $R$-module and the map $p_B\colon A\otimes_R^{\tau}B\to B$ induced by the augmentation $\pi_A\colon A\to R$ is a morphism of rings. Remark 2.11. In this case $p_B$ is surjective and the tensor product $I_A\otimes_R B$ is a two-sided ideal as the kernel of the morphism $p_B$. Therefore, the twisting map $\tau$ should send $B\otimes_R I_A$ to $I_A\otimes_R B$. Construction 2.12. Now we give a main example of a twisting map, which we use in what follows. Suppose that both $R$-rings $A$ and $B$ have $R$-augmentations $\pi_A\colon A\to R$ and $\pi_B\colon B\to R$. In this case there is a special twisting map given by the following rule: For this twisted tensor product we have $(1\otimes b)(a\otimes 1)=\mathbf{v}(b\otimes a)=0$, whenever $a\in I_A$ and $b\in I_B$. 2.3. The twisted tensor product of DG algebrasThe notion of a twisted tensor product can be easily extended to the case of DG algebras. Let $\mathscr R$, $\mathscr A$, $\mathscr B$ be DG $\Bbbk$-algebras, and let $\epsilon_A\colon \mathscr R\to \mathscr A$ and $\epsilon_B\colon \mathscr R\to \mathscr B$ be morphisms of DG algebras. In this case we say that the DG algebras $\mathscr A$ and $\mathscr B$ are DG $\mathscr R$-rings (or DG rings over $\mathscr R$). Definition 2.13. A twisted tensor product over $\mathscr R$ of two DG $\mathscr R$-rings $\mathscr A$ and $\mathscr B$ is a DG $\mathscr R$-ring $\mathscr C$ together with two $\mathscr R$-rings morphisms $i_A \colon \mathscr A\to \mathscr C$ and $i_B \colon \mathscr B \to \mathscr C$ such that the canonical map $\phi\colon \mathscr A\otimes_{\mathscr R} \mathscr B \to \mathscr C$ defined by $\phi(a\otimes b):=i_A(a)\cdot i_B(b)$ is an isomorphism. It follows from this definition that the differential of $\mathscr C$ is uniquely determined by the Leibniz rule, because we have $d_{\mathscr C}(a\otimes b)=d_{\mathscr A}(a)\otimes b+(-1)^{\deg(a)} a\otimes d_{\mathscr B}(b)$. The twisting map $\tau$ associated with a twisted tensor product of DG rings satisfies conditions (2) and (3) and, additionally,
$$
\begin{equation}
\tau(d_{\mathscr B}(b)\otimes a)+(-1)^{\deg(b)}\tau( b\otimes d_{\mathscr B}(a))=d_{\mathscr C}( \tau(b\otimes a)),
\end{equation}
\tag{6}
$$
which means that the map $\tau\colon \mathscr B\otimes_{\mathscr R} \mathscr A\to \mathscr A \otimes_{\mathscr R} \mathscr B$ should be a map of DG $\mathscr R$-bimodules. Suppose that the $\mathscr R$-ring $\mathscr A$ has an $\mathscr R$-augmentation, that is, a morphism $\pi_A\colon \mathscr A\to \mathscr R$ such that the composition $\pi_A\circ\epsilon_A$ is the identity map. Definition 2.14. A twisted tensor product $\mathscr A\otimes_{\mathscr R}^{\tau}\mathscr B$ will be called right fixed (with respect to $\pi_A$) if $\mathscr B$ is semi-flat as a left $\mathscr R$-module and the natural map $p_B\colon \mathscr A\otimes_{\mathscr R}^{\tau}\mathscr B\to \mathscr B$, which is induced by $\pi_A\colon \mathscr A\to \mathscr R$, is a morphism of DG algebras. When we have a right fixed twisted tensor product $\mathscr A\otimes_{\mathscr R}^{\tau}\mathscr B$, we can also consider a deformation of the differential that preserves the structure morphism $i_A\colon \mathscr A\to \mathscr A\otimes_{\mathscr R}^{\tau}\mathscr B$ and the projection $p_B\colon \mathscr A\otimes_{\mathscr R}^{\tau}\mathscr B\to \mathscr B$. Let $\mathscr A$ and $\mathscr B$ be $\mathscr R$-rings such that $\mathscr A$ has an augmentation $\pi_A\colon \mathscr A\to \mathscr R$ and $\mathscr B$ is semi-flat as the left DG $\mathscr R$-module. Consider the underlying algebras $A$, $B$, and $R$ and let $\tau\colon B\otimes_{R} A\to A\otimes_{R} B$ be a twisting map such that the twisted tensor product of algebras $A\otimes_{R}^{\tau}B$ is right fixed. Definition 2.15. We define a DG twisted tensor product $\mathscr C^{\nabla}=\mathscr A\otimes_{\mathscr R}^{\nabla,\tau}\mathscr B$ as a right fixed twisted tensor product of algebras $A\otimes_R^{\tau} B$ with a new differential $d_{\mathscr C^{\nabla}}$ such that $i_A\colon \mathscr A\to \mathscr C^{\nabla}$ and $p_B\colon \mathscr C^{\nabla}\to \mathscr B$ induced by $\pi_A\colon \mathscr A\to \mathscr R$ are morphisms of DG algebras. It follows from the definition that the differential $d_{\mathscr C^{\nabla}}$ has the properties
$$
\begin{equation*}
d_{\mathscr C^{\nabla}}(a\otimes 1)=d_{\mathscr A}(a)\otimes 1
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
d_{\mathscr C^{\nabla}}(a\otimes b)= d_{\mathscr A}(a)\otimes b+(-1)^{\deg (a)} (a\otimes d_{\mathscr B}(b)+a\cdot\nabla(1\otimes b)),
\end{equation*}
\notag
$$
where $\nabla(1\otimes b) \in I_A\otimes_R B$ and $I_A\subset A$ is the augmentation ideal. Theorem 2.16. Let $\mathscr R$ be a proper DG algebra. Let $\mathscr A$ and $\mathscr B$ be proper DG $\mathscr R$-rings such that $\mathscr A$ has an augmentation $\pi_A\colon \mathscr A\to \mathscr R$ and $\mathscr B$ is semi-flat as a left DG $\mathscr R$-module. Assume that the augmentation $\pi_A\colon \mathscr A\to \mathscr R$ is a pp-morphism and $\mathscr B$ is regular (or smooth). Let $\mathscr C^{\nabla}=\mathscr A\otimes_{\mathscr R}^{\nabla,\tau}\mathscr B$ be a DG twisted tensor product such that the morphism $p_B$ is respectable. Then there is an equivalence ${\mathcal D}_{\mathrm{cf}}(\mathscr C^{\nabla}) \cong \operatorname{perf}\!{-} \mathscr C^{\nabla}$. If, in addition, the DG algebra $\mathscr C^{\nabla}$ has a regular (or smooth) realization, then $\mathscr C^{\nabla}$ is also regular (or smooth). Proof. This follows directly from Proposition 2.4, because $\mathscr C^{\nabla}$ is semi-flat as the left DG $\mathscr A$-module and $\mathscr R\otimes_{\mathscr A}\mathscr C^{\nabla} \cong \mathscr R\otimes_{\mathscr A}\mathscr A\otimes_{\mathscr R}\mathscr B \cong \mathscr B$. Thus, the DG algebras $\mathscr R$, $\mathscr A$, $\mathscr B$, and $\mathscr C^{\nabla}$ form a commutative diagram (1) that satisfies properties (Res).
3. Finite-dimensional DG algebras and twisted tensor product3.1. Finite-dimensional DG algebrasLet $\mathscr R=(R,d_{\mathscr R})$ be a finite-dimensional DG algebra over a base field $\Bbbk$. Denote by $J\subset R$ the (Jacobson) radical of the $\Bbbk$-algebra $R$. The ideal $J$ is graded. On the other hand, the radical $J\subset R$ is in general not necessarily a DG ideal. In other words, $d_{\mathscr R}(J)$ is not necessarily a subspace of $J$. In fact, with any two-sided graded ideal $I\subset R $ we can associate two DG ideals $\mathsf I_{-}$ and $\mathsf I_{+}$ (see [22]). Definition 3.1. Let $\mathscr R=(R,d_{\mathscr R})$ be a finite-dimensional DG algebra and $I\subset R$ be a graded (two-sided) ideal. The internal DG ideal $\mathsf I_{-}=(I_{-},d_{\mathscr R})$ consists of all $r\in I$ such that $d_{\mathscr R}( r)\in I$, while the external DG ideal $\mathsf I_{+}=(I_{+},d_{\mathscr R})$ is the sum $I+d_{\mathscr R}(I)$. It is easy to see that $I_{-}$ and $I_{+}$ are indeed two-sided graded ideals of $R$. It is obvious that they are closed under the action of the differential $d_{\mathscr R}$. Thus, for any (two-sided) ideal $I\subset R$ we obtain two DG ideals $\mathsf I_{-}$ and $\mathsf I_{+}$ in the DG algebra $\mathscr R$. If the ideal $I\subset R$ is closed under the action of $d_{\mathscr R}$, then the DG ideals $\mathsf I_{-}$ and $\mathsf I_{+}$ coincide with $I$. Lemma 3.2. The natural morphism of DG ideals $\mathsf I_{-}\to\mathsf I_{+}$ is a quasi-isomorphism, and therefore the morphism of the DG algebras $\mathscr R/\mathsf I_{-}\to \mathscr R/\mathsf I_{+}$ is a quasi-isomorphism too. Proof. We have to check that the complex $\mathsf I_{+}/\mathsf I_{-}$ is acyclic. Let $x\in\mathsf I_{+}$ be an element such that $d_{\mathscr R}(x)=w$, where $w\in\mathsf I_{-}$. We know that $x=y+d_{\mathscr R}(z)$, where $y,z\in I$. Since $d_{\mathscr R}(y)=w$, the definition of $\mathsf I_{-}$ implies that $y\in\mathsf I_{-}$. Therefore, we obtain the equality $\bar{x}=d_{\mathscr R}(\bar{z})$ in $\mathsf I_{+}/\mathsf I_{-}$, where $\bar{x}$ and $\bar{z}$ are the images of $x$ and $z$ in the quotient $\mathsf I_{+}/\mathsf I_{-}$. The lemma is proved. Definition 3.3. Let $\mathscr R=(R,d_{\mathscr R})$ be a finite-dimensional DG algebra, and let $J\subset R$ be the radical. The DG ideals $\mathsf J_{-}$ and $\mathsf J_{+}$ are called the internal and external DG radicals of the DG algebra $\mathscr R$. Lemma 3.2 implies that a priori different DG ideals $\mathsf J_{-}$ and $\mathsf J_{+}$ give us quasi-isomorphic DG algebras $\mathscr R/\mathsf J_{-}$ and $\mathscr R/\mathsf J_{+}$. A useful property of the DG ideal $\mathsf J_{-}$ is its nilpotency (because it is a subideal of $J$), while a useful property of the DG ideal $\mathsf J_{+}$ is the semisimplicity of the underlying algebra (because it is a quotient of the semisimple algebra $S=R/J$). It was proved in [22] that the semisimplicity of the underlying algebra implies the semisimplicity of a DG algebra. Recall that a finite-dimensional DG algebra $\mathscr S$ is called simple if the DG category $ \mathscr{P}\!\mathit{erf}{-} \mathscr S$ of perfect DG modules is quasi-equivalent to $ \mathscr{P}\!\mathit{erf}{-} D$, where $D$ is a finite-dimensional division $\Bbbk$-algebra, and it is called semisimple if the DG category $ \mathscr{P}\!\mathit{erf}{-} \mathscr S$ is quasi-equivalent to a sum $ \mathscr{P}\!\mathit{erf}{-} D_1\oplus\cdots\oplus \mathscr{P}\!\mathit{erf}{-} D_m$, where all the $D_i$ are finite-dimensional division algebras over $\Bbbk$. In addition, $ \mathscr{P}\!\mathit{erf}{-} \mathscr S$ is called separable, if all division algebras $D_i$ are separable over $\Bbbk$ (see Definition 2.11 of [22]). Proposition 3.4 [22; Proposition 2.16]. Let $\mathscr S=(S,d_{\mathscr S})$ be a DG algebra over a field $\Bbbk$ such that $S$ is a semisimple algebra. Then $\mathscr S$ is a semisimple DG algebra. This proposition implies that both DG algebras $\mathscr R/\mathsf J_{-}$ and $\mathscr R/\mathsf J_{+}$ are semisimple, since they are quasi-isomorphic and the underlying algebra for $\mathscr R/\mathsf J_{+}$ is semisimple. 3.2. Finite-dimensional DG modulesNow consider the internal DG radical $\mathsf J_{-}$. It is nilpotent as a subideal of the radical $J$. Let $J\supset J^2\supset\cdots\supset J^n=0$ be the powers of the radical $J$, where $n$ is the index of nilpotency of $J$. Denote by $\mathsf J_p$ the internal DG ideals $(J^p)_{-}$ for $p=1,\dots,n$. This gives us a chain of DG ideals $\mathsf J_{-}=\mathsf J_{1}\supseteq\cdots\supseteq\mathsf J_n=0$, and it is easy to check that $\mathsf J_{p}\mathsf J_{q}\subseteq \mathsf J_{p+q}$. Lemma 3.5. Let $\mathscr R$ be a finite-dimensional DG algebra. Then the triangulated subcategory ${\mathcal D}\subseteq{\mathcal D}_{\mathrm{cf}}(\mathscr R)$ generated by the DG $\mathscr R$-module $\mathscr R/\mathsf J_{+}$ contains all finite-dimensional DG modules. Proof. For any finite-dimensional DG $\mathscr R$-module $\mathsf M$ there is a filtration where each quotient $\mathsf M\mathsf J_{p}/\mathsf M\mathsf J_{p+1}$ is a DG module over the semisimple DG algebra $\mathscr R/\mathsf J_{-}$. Thus, any quotient $\mathsf M\mathsf J_{p}/\mathsf M\mathsf J_{p+1}$ and therefore any finite-dimensional DG module $\mathsf M$ belongs to the triangulated subcategory ${\mathcal D}$ that is generated by the DG $\mathscr R$-module $\mathscr R/\mathsf J_{-}\cong \mathscr R/\mathsf J_{+}$. This completes the proof of the lemma. The following proposition is proved in [22]. Proposition 3.6 [22; Proposition 2.5]. Let $\mathscr R$ be a finite-dimensional DG algebra. Let $\mathsf M$ be a perfect DG $\mathscr R$-module. Then $\mathsf M$ is homotopy equivalent to a finite-dimensional semi-projective DG $\mathscr R$-module and the DG endomorphism algebra $\mathsf{End}_{\mathscr R}(\mathsf M)$ is quasi-isomorphic to a finite-dimensional DG algebra. In [22] we claimed that any perfect DG module is homotopy equivalent to a finite-dimensional homotopically projective DG module. But in fact we showed that it is a direct summand of a semi-free DG module, and so it is semi-projective by condition $3$) in Definition 1.1. Proposition 3.6 and Lemma 3.5 imply directly the following corollary. Corollary 3.7. Let $\mathscr R$ be a finite-dimensional DG algebra. Then the subcategory ${\mathcal D}\subseteq{\mathcal D}_{\mathrm{cf}}(\mathscr R)$ generated by the DG $\mathscr R$-module $\mathscr R/\mathsf J_{+}$ contains the category $ \operatorname{perf}\!{-} \mathscr R$, and, hence, for any DG ideal $\mathsf I\subseteq \mathsf J_{+}$ the morphism $\mathscr R\to \mathscr R/\mathsf I$ is acceptable. For an ordinary finite-dimensional algebra $R$ we can show more. The argument of Lemma 3.5 proves that the semisimple module $S=R/J$, where $J$ is the radical, generates the whole of the category ${\mathcal D}_{\mathrm{cf}}(R)$. Furthermore, the finite-dimensional modules generate the whole category ${\mathcal D}_{\mathrm{cf}}(R)$, because any object of ${\mathcal D}_{\mathrm{cf}}(R)$ can be obtained from its cohomology modules by a sequential application of mapping cones. Thus, we obtain the following proposition. Proposition 3.8. Let $R$ be a finite-dimensional algebra, and let $J\subset R$ be the radical. Then for any ideal $I\subseteq J$ the morphism $R\to R/I$ is respectable. This proposition can be generalized to the case of non-positive finite-dimensional DG algebras. Recall that a DG algebra $\mathscr R$ is called non-positive if $\mathscr R^i=0$ for all $i>0$. In this case the triangulated category ${\mathcal D}_{\mathrm{cf}}(\mathscr R)$ has a t-structure such that the forgetful functor to the derived category of $\Bbbk$-vector spaces ${\mathcal D}(\Bbbk)$ is exact. Therefore, by the same argument as above for ordinary algebras, the DG module $\mathscr R/\mathsf J_{+}$ generates the whole triangulated category ${\mathcal D}_{\mathrm{cf}}(\mathscr R)$, and we obtain the following proposition. Proposition 3.9. Let $\mathscr R$ be a non-positive finite-dimensional DG algebra. For any DG ideal $\mathsf I\subseteq \mathsf J_{+}$ the morphism $\mathscr R\to \mathscr R/\mathsf I$ is respectable. 3.3. Smooth finite-dimensional DG algebras and twisted tensor productsIn this section we consider the case of an arbitrary finite-dimensional DG algebra. The main problem here is that there are examples of finite-dimensional DG algebras $\mathscr R$ and a cohomologically finite-dimensional DG $\mathscr R$-module $\mathsf M$ that does not have a finite-dimensional model (see [5]). This means that the DG module $\mathsf M$ is not quasi-isomorphic to any finite-dimensional DG $\mathscr R$-module. In particular, we cannot conclude that the semisimple DG module $\mathscr R/\mathsf J_{+}$ generates the whole category ${\mathcal D}_{\mathrm{cf}}(\mathscr R)$. Let $\mathscr R$ be a finite-dimensional DG algebra. As above, consider the radical $J$ of the algebra $R$ and denote by $\mathsf J_p$ the internal DG ideals $(J^p)_{-}$ for $p=1,\dots,n$. We obtain a chain of DG ideals $\mathsf J_{-}=\mathsf J_{1}\supseteq\cdots\supseteq\mathsf J_n=0$. Consider DG $\mathscr R$-modules $\mathsf M_p=\mathscr R/\mathsf J_p$ for $p=1,\dots,n$ as objects of the DG category $ {\mathscr M}\!\mathit{od}{-} \mathscr R$ of all right DG $\mathscr R$-modules. Denote by $\mathscr E$ the DG algebra of endomorphisms $\mathsf{End}_{\mathscr R}(\mathsf M)$ of the DG $\mathscr R$-module $\mathsf M=\bigoplus\limits_{p=1}^{n}\mathsf M_p$ in $ {\mathscr M}\!\mathit{od}{-} \mathscr R$. Let us take the right DG $\mathscr E$-module $\mathsf P_n=\mathsf{Hom}_{\mathscr R}(\mathsf M,\mathsf M_n)$. It is semi-projective, and we have $\mathsf{End}_{\mathscr E}(\mathsf P_n)\cong\mathscr R$. Thus, the DG $\mathscr E$-module $\mathsf P_n$ is actually a DG $\mathscr R$-$\mathscr E$-bimodule, and it induces two functors
$$
\begin{equation*}
(-)\stackrel{{\mathbf L}}{\otimes}_{\mathscr R} \mathsf P_n\colon {\mathcal D}(\mathscr R)\to{\mathcal D}(\mathscr E) \quad\text{and}\quad {\mathbf R}\operatorname{Hom}_{\mathscr E}(\mathsf P_n,-)\colon {\mathcal D}(\mathscr E)\to{\mathcal D}(\mathscr R),
\end{equation*}
\notag
$$
which are adjoint to each other. The DG $\mathscr E$-module $\mathsf P_n$ is perfect, and therefore the derived functor $(-)\stackrel{{\mathbf L}}{\otimes}_{\mathscr R} \mathsf P_n$ sends perfect modules to perfect ones. Thus, the $\mathscr R$-$\mathscr E$-bimodule $\mathsf P_n$ produces a functor
$$
\begin{equation*}
(-)\stackrel{{\mathbf L}}{\otimes}_{\mathscr R} \mathsf P_n\colon \operatorname{perf}\!{-} \mathscr R\to \operatorname{perf}\!{-} \mathscr E.
\end{equation*}
\notag
$$
The following theorem was proved in [22]. Theorem 3.10 [22; Theorem 2.19]. Let $\mathscr R$ be a finite-dimensional DG algebra of nilpotency index $n$, and let $\mathscr E= \mathsf{End}_{\mathscr R}\Bigl(\,\bigoplus\limits_{p=1}^{n}\mathsf M_p\Bigr)$ be the DG endomorphism algebra defined above. Then the following properties hold: This theorem gives us a regular (or a smooth) realization for any finite-dimensional DG algebra $\mathscr R$ and allows us to prove the following proposition. Proposition 3.11. Let $\mathscr R$ be a finite-dimensional DG algebra, and let $\mathsf I\subseteq \mathsf J_{+}$ be a DG ideal. Assume that $\mathsf f\colon \mathscr R\to \mathscr R/\mathsf I$ is a pp-morphism and the DG algebra $\mathscr R/\mathsf I$ is regular. Then the morphism $\mathsf f$ is respectable and the DG algebra $\mathscr R$ is also regular. If, in addition, the semisimple DG algebra $\mathscr R/\mathsf J_{+}$ is separable, then $\mathscr R$ is smooth. Proof. Let ${\mathcal D}\subseteq{\mathcal D}_{\mathrm{cf}}(\mathscr R)$ be the subcategory generated by the DG $\mathscr R$-module $\mathscr R/\mathsf J_{+}$. By Corollary 3.7 the subcategory ${\mathcal D}$ contains the category $ \operatorname{perf}\!{-} \mathscr R$. On the other hand, since $\mathscr R/\mathsf I$ is regular, the DG $\mathscr R/\mathsf I$-module $\mathscr R/\mathsf J_{+}$ belongs to $ \operatorname{perf}\!{-} \mathscr R/\mathsf I$. Hence $\mathscr R/\mathsf J_{+}$ is also perfect as a DG $\mathscr R$-module, because $\mathsf f\colon \mathscr R\to \mathscr R/\mathsf I$ is a pp-morphism. Thus, we obtain an equivalence ${\mathcal D}\cong \operatorname{perf}\!{-} \mathscr R$.
Now let us consider the functor
$$
\begin{equation*}
{\mathbf R}\operatorname{Hom}_{\mathscr E}(\mathsf P_n, -)\colon {\mathcal D}_{\mathrm{cf}}(\mathscr E)\to {\mathcal D}_{\mathrm{cf}}(\mathscr R).
\end{equation*}
\notag
$$
The category ${\mathcal D}_{\mathrm{cf}}(\mathscr E)$ is equivalent to $ \operatorname{perf}\!{-} \mathscr E$. Consider the subcategory ${\mathcal D}'\subseteq{\mathcal D}_{\mathrm{cf}}(\mathscr R)$ that is generated by the image of the functor ${\mathbf R}\operatorname{Hom}_{\mathscr E}(\mathsf P_n, -)$. Actually, it is generated by the DG modules $\mathsf M_p=\mathscr R/\mathsf J_p$ for $p=1,\dots,n$, because
$$
\begin{equation*}
{\mathbf R}\operatorname{Hom}_{\mathscr E} (\mathsf P_n, \mathscr E)=\bigoplus_{p=1}^{n} \mathsf M_p .
\end{equation*}
\notag
$$
Thus, by Lemma 3.5, the subcategory ${\mathcal D}'$ is generated by $\mathscr R/\mathsf J_{+}\cong \mathscr R/\mathsf J_{-}=M_1$ and it coincides with the subcategory ${\mathcal D}\cong \operatorname{perf}\!{-} \mathscr R$. This implies that the functor ${\mathbf R}\operatorname{Hom}_{\mathscr E}(\mathsf P_n,-)$ sends $ \operatorname{perf}\!{-} \mathscr E$ to $ \operatorname{perf}\!{-} \mathscr R$. Hence, the full embedding $ \operatorname{perf}\!{-} \mathscr R\hookrightarrow \operatorname{perf}\!{-} \mathscr E$ has a right adjoint. The DG algebra $\mathscr E$ is regular, that is, $\mathscr E$ is a strong generator for $ \operatorname{perf}\!{-} \mathscr E$. Therefore, the object ${\mathbf R}\operatorname{Hom}_{\mathscr E}(\mathsf P_n,\mathscr E)$ is a strong generator for $ \operatorname{perf}\!{-} \mathscr R$, and $\mathscr R$ is regular too. Thus, $ \operatorname{perf}\!{-} \mathscr R$ is equivalent to ${\mathcal D}_{\mathrm{cf}}(\mathscr R)$ and $f$ is respectable.
Suppose that the semisimple DG algebra $\mathscr R/\mathsf J_{+}$ is separable. Theorem 3.10 implies that the DG algebra $\mathscr E$ is smooth. Finally, by [17; 3.24] the DG algebra $\mathscr R$ is smooth too because $ \operatorname{perf}\!{-} \mathscr R$ is an admissible subcategory of a smooth category. The proposition is proved. Since $\mathscr R/\mathsf J_{+}$ is regular as a semisimple DG algebra, Propositions 3.11 and 1.7 imply the following. Corollary 3.12. A finite-dimensional DG algebra $\mathscr R$ is regular if and only if the DG $\mathscr R$-module $\mathscr R/\mathsf J_{+}$ is perfect. If, in addition, the semisimple DG algebra $\mathscr R/\mathsf J_{+}$ is separable, then $\mathscr R$ is smooth. Now we can apply these results to DG twisted tensor products of finite-dimensional DG algebras. Theorem 3.13. Let $\mathscr R$ be a finite-dimensional DG algebra. Let $\mathscr A$ and $\mathscr B$ be finite-dimensional $\mathscr R$-rings such that $\mathscr A$ has an augmentation $\pi_A\colon \mathscr A\to \mathscr R$ with an ideal $\mathsf I_{\mathscr A}$ and $\mathscr B$ is semi-flat as a left DG $\mathscr R$-module. Let $\mathscr C^{\nabla}=\mathscr A\otimes_{\mathscr R}^{\nabla,\tau}\mathscr B$ be a DG twisted tensor product. Assume that $\mathscr A$ and $\mathscr B$ are regular and the ideal $\mathsf I_{\mathscr A}\otimes_{\mathscr R}\mathscr B \subset \mathscr C^{\nabla}$ is contained in the external radical $(\mathsf J_{\mathscr C^{\nabla}})_{+}$. Then, the DG algebra $\mathscr C^{\nabla}=\mathscr A\otimes_{\mathscr R}^{\nabla,\tau}\mathscr B$ is also regular. If, in addition, the semisimple DG algebra $\mathscr B/(\mathsf J_{\mathscr B})_{+}$ is separable, then $\mathscr C^{\nabla}$ is smooth. Proof. Since $\mathscr A$ is regular and $\mathscr R$ is proper, the DG $\mathscr A$-module $\mathscr R$ belongs to $ \operatorname{perf}\!{-} \mathscr A$. Thus, $\pi_A$ is a pp-morphism. Since $\mathscr C^{\nabla}$ is semi-flat as the left DG $\mathscr A$-module, the DG $\mathscr C^{\nabla}$-module $\mathscr B\cong \mathscr R\otimes_{\mathscr A}\mathscr C^{\nabla}$ is perfect. This means that the morphism $p_B\colon \mathscr C^{\nabla}\to \mathscr B$ is also a pp-morphism. Applying Proposition 3.11 to $p_B$ we obtain that the morphism $p_B\colon \mathscr C^{\nabla}\to\mathscr B$ is respectable and the DG algebra $\mathscr C^{\nabla}$ is regular.
The preimage of the radical $J_{B}\subset\mathscr B$ under the surjective morphism $p_B$ is the sum of the radical $J_{C}$ and $\operatorname{Ker}p_B$. Since $\operatorname{Ker} p_B= \mathsf I_{\mathscr A}\otimes_{\mathscr R}\mathscr B$ is contained in the external radical $(\mathsf J_{\mathscr C^{\nabla}})_{+}$, the preimage of the external radical $(\mathsf J_{\mathscr B})_{+}$ coincides with $(\mathsf J_{\mathscr C^{\nabla}})_{+}$. When $\mathscr C^{\nabla}/(\mathsf J_{\mathscr C^{\nabla}})_{+}\cong \mathscr B/(\mathsf J_{\mathscr B})_{+}$ is separable, the DG algebra $\mathscr C^{\nabla}$ is also smooth. The theorem is proved. 4. Algebras and DG algebras with two simple modules4.1. Quivers with two vertices and Kronecker algebrasIn this section we consider some smooth basic algebras (and also DG algebras) with two simple modules, that is, $\Bbbk$-algebras $R$ such that the semisimple part $R/J$ is isomorphic to the algebra $\Bbbk\times \Bbbk$. The main goal is, on the one hand, to demonstrate how such algebras can be obtained from simpler ones using twisted tensor product and, on the other hand, to construct new smooth algebras. Let $Q$ be a quiver. It consists of data $(Q_0,Q_1,s,t)$, where $Q_0$ and $Q_1$ are finite sets of vertices and arrows, respectively, while $s,t \colon Q_1\to Q_0$ are maps associating to each arrow its source and target. The path algebra $\Bbbk Q$ is determined by the generators $e_q$ for $q\in Q_0$ and $a$ for $a\in Q_1$ with the following relations: $e^2_q=e_q$, $e_r e_q=0$ for $r\ne q$, and $e_{t(a)}a=a e_{s(a)}=a$. As a $\Bbbk$-vector space, the path algebra $\Bbbk Q$ has a basis consisting of the set of all paths in $Q$, where a path $\overline{p}$ is a (possibly empty) sequence $a_{l} a_{l-1}\cdots a_1$ of compatible arrows, that is, $s(a_{i+1})=t(a_i)$ for all $i=1,\dots,l-1$. For an empty path we have to choose a vertex of the quiver. The composition of two paths $\overline{p}_1$ and $\overline{p}_2$ is defined as $\overline{p}_2 \overline{p}_1$ if they are compatible and as $0$ if they are not compatible. To obtain a basic finite-dimensional algebra we have to consider a quiver with relations. A relation on a quiver $Q$ is a subspace of $\Bbbk Q$ spanned by linear combinations of paths having a common source and a common target and of length at least $2$. A quiver with relations is a pair $(Q,I)$, where $Q$ is a quiver and $I$ is a two-sided ideal of the path algebra $\Bbbk Q$ generated by relations. The quotient algebra $\Bbbk Q/I$ will be called the quiver algebra of the quiver with relations $(Q,I)$. It can be shown that any basic finite-dimensional algebra can be realized as an algebra of some quiver with relations $(Q,I)$ (see [7]). Let $Q_{n,m}$ be a quiver with two vertices $\boldsymbol{1}$ and $\boldsymbol{2}$, and with $n$ arrows $\{c_1,\dots,c_n\}$ from $\boldsymbol{1}$ to $\boldsymbol{2}$ and $m$ arrows $\{b_1,\dots,b_m\}$ from $\boldsymbol{2}$ to $\boldsymbol{1}$: Let $\Bbbk Q_{n,m}$ be the path algebra of the quiver $Q_{n,m}$. Denote by $B$ and $C$ the vector spaces generated by the arrows $\{b_1,\dots,b_m\}$ and $\{c_1,\dots,c_n\}$, respectively. Remark 4.1. We do not consider quivers with loops, because quiver algebras with relations for quivers with loops have an infinite global dimension (see [11; Corollary 5.6]). The first block of natural examples of smooth algebras is obtained by considering the path algebras of the quivers $Q_{n,0}$ (or $Q_{0,m}$). They are called Kronecker quivers, and their path algebras have global dimension $1$. Denote by $K_n$ the Kronecker algebras $\Bbbk Q_{n,0}$ with natural augmentations of the semisimple part. It is an easy exercise to check the following statement. Proposition 4.2. There is an isomorphism of algebras We can also consider a DG version of such algebras if we assume that each arrow $c_i$ has some degree $d_i\in{\mathbb Z}$. Denote by $K_n[d_1,\dots,d_n]$ the DG algebra of such a DG quiver. The same arguments show that any DG algebra $K_n[d_1,\dots,d_n]$ is equal to $K_1[d_n]\otimes^{\mathbf{v}}_S K_{n-1}[d_1,\dots,d_{n-1}]$. The triangulated category $ \operatorname{perf}\!{-} K_n[d_1,\dots,d_n]$ has a full exceptional collection consisting of two objects, and any proper category with an exceptional collection of two objects is equivalent to the category $ \operatorname{perf}\!{-} K_n[d_1,\dots,d_n]$ for some $n$ and $d_i$ (see [20; Proposition 3.8] for a general statement). Another generalization is to consider a DG twisted tensor product with non-trivial $\nabla$. The simplest example is the DG algebra $K_1\otimes^{\nabla,\mathbf{v}}_S K_{1}[-1]$ with $\nabla(c_2)=c_1$. The resulting DG algebra is quasi-isomorphic to $K_0=S$. 4.2. Green algebrasThe second block of examples of smooth algebras that we are going to discuss appeared in E. L. Green’s paper [8] and was discussed in detail by D. Happel in [9] (see also [10]). Let us take the quiver algebras of the quivers $Q_{n,n}$ or $Q_{n,n-1}$ with the following relations: 1) $c_jb_i=0$ for $j\leqslant i$; 2) $b_ic_j=0$ for $i<j$. Denote these algebras by $G_{2n}$ and $G_{2n-1}$, respectively. It was proved in [9] that the algebra $G_{k}$ has finite global dimension, which is equal to $k$. For small $k$ we have $G_0=K_0=S$ and $G_1=K_1$. Moreover, it is not very difficult to check that the category $ \operatorname{perf}\!{-} G_2$ is equivalent to $ \operatorname{perf}\!{-} K_2[0,1]$ and therefore has a full exceptional collection. On the other hand, it was proved in [9] and [18] that the algebras $G_{k}$ are not derived equivalent to quasi-hereditary algebras for $k\geqslant 3$. Therefore, the categories $ \operatorname{perf}\!{-} G_k$ do not have full exceptional collections when $k\geqslant 3$ (see [15]). Now let us show that each algebra $G_k$ can be realized as an iterated twisted tensor product of algebras of type $K_1$. Denote by $K_1^{\circ}$ the Kronecker algebra $\Bbbk Q_{0,1}$. This algebra is opposite to the algebra $K_1=\Bbbk Q_{1,0}$, and it is also isomorphic to $K_1$. Proof. The proof is a direct calculation. It goes by induction, where the first factors $K_1$ and $K_1^{\circ}$ are related to the arrows $c_{n+1}$ and $b_n$, respectively. There are some generalizations of these algebras. As above, we can consider a DG version by taking into account that each arrow $b_i$ and $c_j$ can have some degree. Another generalization is that, instead of algebras of type $K_1$ (and $K_1^{\circ}$), one can take algebras of type $K_q$. Thus, taking iterated twisted tensor products we can obtain algebras of the form
$$
\begin{equation}
G_{\langle p_n, q_n,\dots, p_1, q_1\rangle}=K_{p_n}^{\circ} \otimes^{\mathbf{v}}_S K_{q_n}\otimes^{\mathbf{v}}_S\cdots \otimes^{\mathbf{v}}_S K_{p_1}^{\circ}\otimes^{\mathbf{v}}_S K_{q_1}.
\end{equation}
\tag{8}
$$
In the case where $p_i=q_i=1$ for all $1\leqslant i\leqslant n$ we obtain a usual algebra $G_{2n}$. In the case where all the $p_i$ and $q_i$ are equal to $1$, except for the last one, $p_n$, which is equal to $0$, we obtain $G_{2n-1}$. Let us also consider another generalization of Green algebras to the case of $N$ simple modules. We fix a natural number $N\geqslant 2$ and consider the semisimple algebra $S=S_N=\underbrace{\Bbbk\times\dots\times\Bbbk}_N$. For $1\leqslant i\ne j\leqslant N$ denote by $K_{ij}[d]$ the finite-dimensional DG algebra with semisimple part equal to $S=S_N$ and with only one arrow from the vertex $i$ to the vertex $j$ of degree $d$ in the Jacobson radical. Taking iterated twisted tensor product over $S=S_N$ of such DG algebras with the twisting map given by formula (5) from Construction 2.12, we obtain new DG algebras, which will be called generalized Green DG algebras on $N$ vertices. By Definition 2.13 of the twisted tensor product of DG algebras, any such generalized Green DG algebra has a trivial differential. The next proposition follows directly from Theorem 3.13. Proposition 4.4. Any iterated twisted tensor product of DG algebras $K_{ij}[d]$ over the semisimple part $S=S_N$ via the twisting (5) is a smooth DG algebra. In section 5, we discuss bilinear forms on the Grothendieck group of these DG algebras. 4.3. New examples of smooth algebrasNow we introduce and describe new families of smooth algebras with two simple modules. We assume for simplicity that the base field $\Bbbk$ is infinite. We start with a quiver of the form $Q_{m,n}$, as in (7), and fix some new relations. These relations will also depend on some integer $k$, $0<k< n$. It is important to note that these families of algebras have non-trivial moduli spaces. Let $V_i\subset C$, $i=1,\dots,m$, be some subspaces of dimension $k$ and $W_i\subset C$, $i=1,\dots,m$, be some subspaces of codimension $k$. Denote by the resulting family of subspaces. We consider the following relations $I_{\mathfrak{F}}$ depending on the family $\mathfrak{F}$:
$$
\begin{equation*}
\begin{alignedat}{3} &(1)\quad &bcb'&=0 & \quad &\text{for any } c\in C,\ b, b'\in B; \\ &(2) \quad &w b_i&=0 & \quad &\text{for any } w\in W_i, \text{ where } i=1,\dots,m; \\ &(3) \quad &b_i v&=0 & \quad &\text{for any } v\in V_i, \text{ where } i=1,\dots,m. \end{alignedat}
\end{equation*}
\notag
$$
Denote by $R_{\mathfrak{F}}$ the quotient algebra $R_{\mathfrak{F}}=\Bbbk Q_{n,m}/I_{\mathfrak{F}}$. Let $S_1$ and $S_2$ be the right simple $R_{\mathfrak{F}}$-modules, and let $P_1=e_1R_{\mathfrak{F}}$ and $P_2=e_2 R_{\mathfrak{F}}$ be right projective $R_{\mathfrak{F}}$-modules. There are natural short exact sequences of right modules:
$$
\begin{equation}
0\to BR_{\mathfrak{F}}\to P_1\to S_1\to 0\quad \text{and}\quad 0\to CR_{\mathfrak{F}}\to P_2\to S_2\to 0,
\end{equation}
\tag{9}
$$
where
$$
\begin{equation*}
CR_{\mathfrak{F}}=\sum_{c\in C} cR_{\mathfrak{F}} \quad\text{and}\quad BR_{\mathfrak{F}}=\sum_{b\in B} bR_{\mathfrak{F}}
\end{equation*}
\notag
$$
are the right ideals generated by $C$ and $B$. Let us choose subspaces $V\subset C$ and $W\subset C$ of dimensions $k$ and $n-k$, respectively, such that $V\cap W_i=0$ and $W\cap V_i=0$ for all $i=1,\dots,m$. We also assume that $V\cap W=0$, so that $V\oplus W=C$. The relations $I_{\mathfrak{F}}$ imply the following decompositions. Lemma 4.5. The following decompositions of vector spaces hold: Lemma 4.5 implies that the right $R_{\mathfrak{F}}$-module $BR_{\mathfrak{F}}$ is isomorphic to the direct sum $\mathop\bigoplus\limits_{i=1}^{m} b_i R_{\mathfrak{F}}$. Moreover, the following statement holds. Lemma 4.6. For any subset $\{j_1,\dots,j_l\}\subseteq \{1,\dots,n\}$ the right ideal $b_{j_1}R_{\mathfrak{F}}+\cdots+b_{j_l}R_{\mathfrak{F}}\subset R_{\mathfrak{F}}$ is isomorphic to the direct sum $b_{j_1}R_{\mathfrak{F}}\oplus\cdots\oplus b_{j_l}R_{\mathfrak{F}}$ as a right $R_{\mathfrak{F}}$-module. For any element $r\in R_{\mathfrak{F}}$ denote by $\operatorname{rAnn}(r)\subset R_{\mathfrak{F}}$ the right annihilator of $r$. The following lemma is a direct consequence of Lemma 4.5. Lemma 4.7. For any $x\in C\setminus W_i$, there are isomorphisms of right modules $xb_i R_{\mathfrak{F}}\cong b_i R_{\mathfrak{F}}$ and $\operatorname{rAnn} (xb_i)\cong\operatorname{rAnn} (b_i)$. Let $U_{ij}\subseteq C$ denote the sum of the subspaces $V_i$ and $W_j$. We put
$$
\begin{equation*}
t_{ij}=\operatorname{codim} U_{ij}=\dim (V_i\cap W_j).
\end{equation*}
\notag
$$
Denote by $t$ the maximum of all $t_{ij}$. Now we can choose a sequence of $t$ elements $x_1,\dots,x_t\in C$ such that $x_1$ does not belong to any proper subspace $U_{ij}$ and $x_{q+1}$ does not belong to any subspace $U_{ij}+\langle x_1,\dots,x_q\rangle$ when $t_{ij}>q$. For any element $c\in C$ we denote by $T_c\subseteq \{1,\dots,m\}$ the subset consisting of all $l$ such that $c\in W_l$. Let us compute the right annihilators of certain elements. Lemma 4.8. The following isomorphisms of right modules hold: Proof. Equality 1) directly follows from relation (2) of $I_{\mathfrak{F}}$ and Lemma 4.6.
By Lemma 4.5, the vector space $CR_{\mathfrak{F}}$ can be decomposed as
$$
\begin{equation*}
C\oplus \bigoplus\limits_{j=1}^m V b_j\oplus \bigoplus\limits_{j=1}^m V b_j W,
\end{equation*}
\notag
$$
where $V\subset C$ and $W\subset C$ are of dimension $k$ and $n-k$, respectively, such that $V\cap W_i=0$ and $W\cap V_i=0$ for all $i=1,\dots,m$. Now, the annihilator $\operatorname{rAnn}(b_i)$ is equal to
$$
\begin{equation*}
e_1 R_{\mathfrak{F}} \oplus V_i \oplus \bigoplus\limits_{j=1}^m V b_j\oplus \bigoplus\limits_{j=1}^m V b_j W.
\end{equation*}
\notag
$$
Since $\dim (V_i\cap W_j)=t_{ij}$, the codimension of the subspace $V_ib_j$ in the space $V b_j$ is equal to $t_{ij}$. By the construction above, the vectors $x_sb_j$, where $s=1,\dots,t_{ij}$, give a basis for a complementary subspace in $V b_j$. Taking into account Lemmas 4.6 and 4.7, we obtain equalities 2). The lemma is proved. Using the previous lemma we obtain the following exact sequences:
$$
\begin{equation}
\begin{gathered} \, 0\to \bigoplus_{j\in T_{c}} b_j R_{\mathfrak{F}}\to P_1\to c R_{\mathfrak{F}}\to 0, \\ 0\to V_i R_{\mathfrak{F}} \oplus \mathop{\bigoplus}\limits_{j=1}^{m} (b_j R_{\mathfrak{F}})^{\oplus t_{ij}}\to P_2\to b_iR_{\mathfrak{F}}\to 0. \end{gathered}
\end{equation}
\tag{10}
$$
The first short sequence implies the following proposition. Proposition 4.9. Let $c$ be an element of $C$. If $T_{c}=\varnothing$, then $c R_{\mathfrak{F}}\cong P_1$ and $\operatorname{pd}(c R_{\mathfrak{F}})=0$. If $T_{c}\ne \varnothing$, then $\operatorname{pd}(cR_{\mathfrak{F}})= \max\{\operatorname{pd}(b_j R_{\mathfrak{F}})\mid j\in T_{c}\}+1$. Furthermore, for any subspace $U\subseteq C$ of dimension $d$ denote the dimension of the intersection $U\cap W_j$ by $t_{U,j}$. The right module $UR_{\mathfrak{F}}$ can be covered by the projective module $P_1^{d}$ and it is a minimal projective cover. It is easy to see that for any subspace $U\subseteq C$ the following short exact sequence holds:
$$
\begin{equation}
0\to \mathop{\bigoplus}\limits_{j=1}^{m} (b_j R_{\mathfrak{F}})^{\oplus t_{U,j}}\to P_1^{d} \to U R_{\mathfrak{F}}\to 0.
\end{equation}
\tag{11}
$$
In particular, for the right modules $V_i R_{\mathfrak{F}}$ and the module $CR_{\mathfrak{F}}$ we have the short exact sequences
$$
\begin{equation}
\begin{aligned} \, &0\to \mathop{\bigoplus}\limits_{j=1}^{m} (b_j R_{\mathfrak{F}})^{\oplus t_{ij}}\to P_1^{k}\to V_i R_{\mathfrak{F}}\to 0, \\ &0\to \bigoplus_{i=1}^{m} (b_i R_{\mathfrak{F}})^{\oplus (n-k)} \to P_1^{n }\to CR_{\mathfrak{F}}\to 0. \end{aligned}
\end{equation}
\tag{12}
$$
The following generalization of Proposition 4.9 holds. Proposition 4.10. Let $U\subseteq C$ be a subspace of dimension $d$. If $t_{U,j}\ne 0$ for some $j$, then otherwise, $UR_{\mathfrak{F}}\cong P_1^d$. For any $i$ the equality holds. Proof. The first statement follows from the short exact sequence (11). When $t_{U,j}=0$ for all $j$, we have an isomorphism $UR_{\mathfrak{F}}\cong P_1^d$. If some $t_{U,j}$ is distinct from zero, then $\operatorname{pd}(UR_{\mathfrak{F}}) =\operatorname{pd}\Bigl(\,\bigoplus\limits_{j=1}^{m} (b_j R_{\mathfrak{F}})^{\oplus t_{U,j}}\Bigr)+1$. Therefore, we obtain $\operatorname{pd}(UR_{\mathfrak{F}})= \max\{\operatorname{pd}(b_j R_{\mathfrak{F}})\mid t_{U,j}\ne 0 \}+1$.
The second short sequence of (10) implies that $\operatorname{pd}(b_iR_{\mathfrak{F}})= \operatorname{pd}(V_iR_{\mathfrak{F}})+1$ because $\operatorname{pd}(V_i R_{\mathfrak{F}})= \max\{\operatorname{pd}(b_j R_{\mathfrak{F}})|\mid t_{ij}\ne 0\}+1$ is larger than $\operatorname{pd}(b_j R_{\mathfrak{F}})$ for every $j$ such that $t_{ij}\ne 0$. The proposition is proved. To any algebra $R_{\mathfrak{F}}$ we attach a quiver $\Gamma_{\mathfrak{F}}$, which encodes intersections of the subspaces in the family $\mathfrak{F}$ and allows us to construct minimal resolutions for modules. Any such quiver $\Gamma_{\mathfrak{F}}$ has $2m$ vertices $b_1,\dots,b_m$ and $v_1,\dots,v_m$. The numbers of arrows between pairs of vertices in this quiver depend on the numbers $t_{ij}=\dim(V_i\mathrel{\cap} W_j)$. Namely, arrows in $\Gamma_{\mathfrak{F}}$ are defined by the following rule:
Using the quiver $\Gamma_{\mathfrak{F}}$, we can describe minimal projective resolutions for the modules $b_iR_{\mathfrak{F}}$. Denote by ${\mathcal{P}}_{\Gamma_{\mathfrak{F}}}$ the set of all paths in $\Gamma_{\mathfrak{F}}$. Let ${\mathcal{P}}_{b_i}\subset {\mathcal{P}}_{\Gamma_{\mathfrak{F}}}$ be the subset of all paths $p$ whose source $s(p)$ coincides with $b_i$. For any $i$ the set ${\mathcal{P}}_{b_i}$ decomposes into the disjoint union $\bigsqcup\limits_{s\geqslant0}{\mathcal{P}}^s_{b_i}$, where ${\mathcal{P}}_{b_i}^s$ is the set of all paths of length $s$. Iterating the short exact sequences from (10) and (12) and using the combinatorics of the quiver $\Gamma_{\mathfrak{F}}$, we can produce minimal resolutions of the right modules $b_i R_{\mathfrak{F}}$ for any $i$. They have the following form:
$$
\begin{equation*}
\cdots\to \bigoplus_{\substack{p\in {\mathcal{P}}_{b_i}^s \\ t(p)=v_j}}P_1^k \oplus\bigoplus_{\substack{p\in {\mathcal{P}}_{b_i}^s\\ t(p)=b_j }} P_2\to\cdots \to P_1^k \oplus \bigoplus\limits_{\substack{p\in {\mathcal{P}}_{b_i}^1\\ t(p)=b_j}} P_2 \to P_2\to b_iR_{\mathfrak{F}}\to 0.
\end{equation*}
\notag
$$
These resolutions and Proposition 4.10 imply the following theorem. Theorem 4.11. Let $Q_{n,m}$ and $\mathfrak{F}$ be as above. Then the algebra $R_{\mathfrak{F}}=\Bbbk Q_{n,m}/I_{\mathfrak{F}}$ has a finite global dimension if and only if the quiver $\Gamma_{\mathfrak{F}}$ does not have oriented cycles. Moreover, the global dimension of $R_{\mathfrak{F}}$ is $l+2$, where $l$ is the maximum length of a path in $\Gamma_{\mathfrak{F}}$. Proof. By Proposition 4.10 we have If there is a cycle in $\Gamma_{\mathfrak{F}}$, then $\operatorname{pd}(b_iR_{\mathfrak{F}})= \operatorname{pd}(b_iR_{\mathfrak{F}})+s$ for some $i$ and $s>0$. This implies that the projective dimension $\operatorname{pd}(b_iR_{\mathfrak{F}})$ is infinite.
On the other hand, if the quiver $\Gamma_{\mathfrak{F}}$ does not have oriented cycles, then the projective dimensions of the modules $b_iR_{\mathfrak{F}}$ are equal to the lengths of the longest paths $p$ in $\Gamma_{\mathfrak{F}}$ whose sources $s(p)$ are equal to $b_i$. Finally, short exact sequences (9) give us the following equalities for the simple modules: Therefore, the global dimension of $R_{\mathfrak{F}}$ is equal to $\operatorname{pd}(S_2)= \max\{\operatorname{pd}(b_j R_{\mathfrak{F}})\mid j\}+2=l+2$, where $l$ is the maximum length of a path in $\Gamma_{\mathfrak{F}}$. The theorem is proved.It is easy to see that the maximum length of a path in $\Gamma_{\mathfrak{F}}$ is odd and any odd positive integer up to $2m-1$ can be realized. Thus, we obtain the following corollary. Corollary 4.12. For any $1\leqslant s\leqslant m$, there is a family of subspaces of the space $C$ such that the algebra $R_{\mathfrak{F}}=\Bbbk Q_{n,m}/I_{\mathfrak{F}}$ has a global dimension equal to $2s+1$. For subspaces $V_i$ and $W_j$ in general position, we have
$$
\begin{equation*}
V_i\cap W_j=0\quad \text{for all}\ \ 1\leqslant i,j\leqslant m.
\end{equation*}
\notag
$$
In this case the maximum length of paths in $\Gamma_{\mathfrak{F}}$ is $1$, and we obtain the following corollary. Corollary 4.13. Suppose $V_i\cap W_j=0$ for any $i$ and $j$. Then $\operatorname{pd}(V_i R_{\mathfrak{F}})=0$ and $\operatorname{pd}(b_i R_{\mathfrak{F}})=1$ for each $i$, and the minimal projective resolutions of the simple modules have the following forms In this case the global dimension of the algebra $R_{\mathfrak{F}}$ is $3$. Example 4.14. Let $n=m\geqslant 2$ and $k=1$, so that $\dim V_i=1$. Let us choose some elements $a_i\in C$, for which $V_i=\langle a_i\rangle$. We define $W_i\subset C$ as the subspaces of codimension $1$ spaanned by the elements $\{a_n,\dots,a_{i+1},a_i-a_1,\dots,a_2-a_1\}$. The algebras $R_{\mathfrak{F}}$ obtained in this way are exactly those constructed in [13]. The global dimension of these algebras is equal to $2m+1$. Now we describe the algebras $R_{\mathfrak{F}}$ as iterated twisted tensor products of certain already known algebras. We assume that $R_{\mathfrak{F}}$ has a finite global dimension, that is, it is smooth. First consider the case $m=1$, where the family $\mathfrak{F}$ consists of two subspaces $V_1$ and $W_1$. The algebra $R_{\mathfrak{F}}$ has finite global dimension if and only if $V_1\cap W_1=0$. Denote by $K(V_1)$ and $K(W_1)$ the Kronecker subalgebras of $R_{\mathfrak{F}}$ generated by the subspaces $V_1$ and $W_1$, respectively. A simple direct calculation shows that for $m=1$ any smooth algebra $R_{\mathfrak{F}}$ can be represented as a twisted tensor product of the form $K(V_1)\otimes^{\mathbf{v}}_S K_{1}^{\circ}\otimes^{\mathbf{v}}_S K(W_1)$. Thus, in this case the algebra $R_{\mathfrak{F}}$ is a generalized Green algebra of the form $G_{\langle 0,k,1,(n-k)\rangle}$ as in (8). Now consider the general case. Since $\Gamma_{\mathfrak{F}}$ does not have oriented cycles, it is a directed quiver. Hence we may suppose, changing the numbering if necessary, that $V_m\cap W_i=0$ for all $i=1,\dots,m$. Consider the family $\mathfrak{G}$ of the subspaces $V_i$ and $W_i$ for $i< m$, and take the algebra $R_{\mathfrak{G}}$. There is a natural embedding of the Kronecker algebra $K(V_m)$ into $R_{\mathfrak{G}}$ that is given by the embedding $V_m\subset C$. Thus, $R_{\mathfrak{G}}$ can be regarded as a $K(V_m)$-ring. Proof. The indecomposable left projective $K(V_m)$-modules are $K(V_m)e_1=V_m\oplus\langle e_1 \rangle$ and $K(V_m)e_2=\langle e_2\rangle$. By Lemma 4.5 we can represent $R_{\mathfrak{G}}$ as the direct sum of the following spaces: $\Bigl(\,\bigoplus\limits_{j=1}^{m-1} V_m b_j W\oplus \bigoplus\limits_{j=1}^{m-1} b_j W\Bigr)$, $\Bigl(\,\bigoplus\limits_{j=1}^{m-1} V_m b_j\oplus \bigoplus\limits_{j=1}^{m-1} b_j\Bigr)$, $(V_m\oplus \langle e_1\rangle)$, and $(\langle e_2\rangle \oplus W)$ for some subspace $W\subset C$ of dimension $n-k$ such that $W\cap V_i=0$ for all $i$. Now, the first three spaces are projective left $K(V_m)$-modules of type $(K(V_m)e_1)^r$, while the last one is isomorphic to the projective module $(K(V_m)e_2)^{(n-k+1)}$. The proposition is proved. The subspace $W_m\subset C$ determines an augmentation $\pi\colon R_{\mathfrak{G}}\to K(V_m)$ by setting $\pi(W_m)=0$. We also consider the algebra $K(V_m)\otimes^{\mathbf{v}}_S K_{1}^{\circ}$. It is a $K(V_m)$-ring and has a natural augmentation $K(V_m)\otimes^{\mathbf{v}}_S K_{1}^{\circ}\to K(V_m)$. Theorem 4.16. Suppose the quiver $\Gamma_{\mathfrak{F}}$ does not have oriented cycles. Then the algebra $R_{\mathfrak{F}}$ is isomorphic to the twisted tensor product of the algebras $(K(V_m)\mathrel{\otimes^{\mathbf{v}}_S} K_{1}^{\circ})$ and $R_{\mathfrak{G}}$ over $K(V_m)$ via the twisting map $\mathbf{v}$ defined by (5), that is, Proof. For brevity denote the algebra $(K(V_m)\otimes^{\mathbf{v}}_SK_{1}^{\circ})$ by $K(V_m;1)$. We have an isomorphism $K(V_m; 1)\cong K(V_m)\oplus (V_m b_m\oplus \langle b_m\rangle)$ of right $K(V_m)$-modules, and $(V_m b_m\oplus \langle b_m\rangle)$ is isomorphic to $S_2^{k+1}$, where $S_2$ is the simple right $K(V_m)$-module associated with $e_2$.
Consider the tensor product $K(V_m; 1)\otimes_{K(V_m)}^{\mathbf{v}} R_{\mathfrak{G}}$ as a vector space. It is isomorphic to the direct sum of $R_{\mathfrak{G}}$ and the space $(V_m b_m\oplus \langle b_m\rangle)\otimes_{K(V_m)} R_{\mathfrak{G}}$. Using the decomposition of $R_{\mathfrak{G}}$ as a projective left $K(V_m)$-module that was described in the proof of Proposition 4.15, we obtain
$$
\begin{equation*}
\begin{aligned} \, (V_m b_m\oplus \langle b_m\rangle)\otimes_{K(V_m)} R_{\mathfrak{G}} &\cong (V_m b_m\oplus \langle b_m\rangle)\otimes_{\Bbbk} (\langle e_2 \rangle\oplus W) \\ &=(V_m b_m\oplus \langle b_m\rangle) \oplus (V_m b_m W\oplus b_m W), \end{aligned}
\end{equation*}
\notag
$$
because the tensor product $S_2\otimes_{K(V_m)} K(V_m)e_1$ is zero.
Applying the decomposition from Lemma 4.5 we can establish a natural isomorphism from $R_{\mathfrak{F}}$ to the tensor product $K(V_m;1)\otimes_{K(V_m)}^{\mathbf{v}} R_{\mathfrak{G}}$ that is an isomorphism of vector spaces. Thus, we have to show that the relations $I_{\mathfrak{F}}$ also hold for the twisted tensor product $K(V_m;1)\otimes_{K(V_m)}^{\mathbf{v}} R_{\mathfrak{G}}$. Since the isomorphism is compatible with the embeddings of the algebras $R_{\mathfrak{G}}$ and $K(V_m;1)$, it is sufficient to check only those relations $I_{\mathfrak{F}}$ in which $b_m$ is involved. We already know that $b_m V_m=0$ just like in the algebra $K(V_m;1)$. The relations $b_i C b_m=0$ for $i<m$ and $W_m b_m=0$ hold, because $W_m$ and $b_i C$ belong to the kernel of the augmentation $\pi\colon R_{\mathfrak{G}}\to K(V_m)$. Now we should check the relations $b_m C b_i=0$ for any $i\leqslant m$. We know that $b_m V_m=0$ and $W_i b_i=0$ for all $i$. By assumption, we have $C=V_m\oplus W_i$ for any $i\leqslant m$. Hence, $b_m C b_i=0$ for any $i\leqslant m$. The theorem is proved. Corollary 4.17. Any algebra $R_{\mathfrak{F}}$ can be obtained from algebras of type $K_1$ by applying successive twisted tensor product operations. Remark 4.18. Note that for $m=1$ the algebra $R_{\mathfrak{F}}=K(V_1) \otimes^{\mathbf{v}}_S K_{1}^{\circ}\otimes^{\mathbf{v}}_S K(W_1)$ can also be represented as a twisted tensor product $(K(V_1)\otimes^{\mathbf{v}}_S K_{1}^{\circ}) \otimes_{K(V_1)}^{\mathbf{v}} K(C)$, where the augmentation $K(C)\to K(V_1)$ is given by specifying $W_1\subset C$. Remark 4.19. Theorems 4.16 and 3.13 provide another proof that the algebra $R_{\mathfrak{F}}$ has a finite global dimension when the quiver $\Gamma_{\mathfrak{F}}$ does not have cycles. 5. Smooth algebras and Grothendieck group5.1. Grothendieck group and generalized Green DG algebrasLet $\mathscr R=(R,d_{\mathscr R})$ be a finite-dimensional DG algebra. Consider the Grothendieck group $K_0( \operatorname{perf}\!{-} \mathscr R)$. When $\mathscr R$ is smooth, the abelian group $K_0( \operatorname{perf}\!{-} \mathscr R)$ is free of finite rank; hence it is isomorphic to ${\mathbb Z}^N$ for some $N\geqslant 0$ [22; Corollary 2.21]. For any perfect DG modules $\mathsf E$ and $\mathsf F$ we denote by $\chi_{\mathscr R}(\mathsf E,\mathsf F)$ the alternating sum
$$
\begin{equation}
\chi_{\mathscr R}(\mathsf E,\mathsf F)=\sum_l(-1)^l \dim_{\Bbbk} \operatorname{Hom}_{ \operatorname{perf}\!{-} \mathscr R}(\mathsf E,\mathsf F[l]),
\end{equation}
\tag{13}
$$
which defines the so-called Euler bilinear form on the abelian group $K_0( \operatorname{perf}\!{-} \mathscr R)$. Consider the case where the semisimple part $S$ of the underlying algebra $R$ is isomorphic to $S_N\cong{\Bbbk\times\cdots\times \Bbbk}$. Denote by $\{e_1,\dots,e_N\}$ the complete set of primitive idempotents of the algebra $R$. Moreover, assume also that the DG algebra $\mathscr R$ is $S$-split. This means that there are morphisms of DG algebras $\epsilon\colon S\to \mathscr R$ and $\pi\colon \mathscr R\to S$ such that $\pi\circ \epsilon=\operatorname{id}_S$. In particular, $d_{\mathscr R}(e_i)=0$ for any $i=1,\dots,N$ and $d(J)\subseteq J$, where $J$ is the radical. If $\mathscr R$ is smooth, then $K_0( \operatorname{perf}\!{-} \mathscr R)\cong{\mathbb Z}^N$, and the classes of the simple modules $\mathsf S_i$ form a basis in $K_0( \operatorname{perf}\!{-} \mathscr R)$. Denote by $\mathsf P_{i}=e_i\mathscr R$, $i=1,\dots,N$, the semi-projective DG modules. Their classes also form a basis of $K_0( \operatorname{perf}\!{-} \mathscr R)$, and we have $\chi_{\mathscr R}(\mathsf P_i,\mathsf S_j)=\delta_{ij}$. Let $\mathtt X_{\mathscr R}$ be the matrix of the bilinear form $\chi_{\mathscr R}$ in the basis $\mathsf P_{i}$. A direct calculation gives us the following equalities:
$$
\begin{equation}
\begin{aligned} \, \notag (\mathtt X_{\mathscr R})_{ij}&=\chi_{\mathscr R}(\mathsf P_i,\mathsf P_j)= \sum_l (-1)^l \dim_{\Bbbk}\operatorname{Hom}_{ \operatorname{perf}\!{-} \mathscr R}(\mathsf P_i,\mathsf P_j[l]) \\ &=\sum_l (-1)^l \dim_{\Bbbk} H^l(e_j\mathscr R e_i)= \sum_l (-1)^l \dim_{\Bbbk} e_j\mathscr R^l e_i, \end{aligned}
\end{equation}
\tag{14}
$$
where $\mathscr R^l$ is the degree $l$ component of $\mathscr R$. Proposition 5.1. Let $\mathscr A$ and $\mathscr B$ be two smooth finite-dimensional $S$-split DG algebras as above, where $S\cong{\Bbbk\times\cdots\times \Bbbk}$. Let $\mathscr C^{\nabla}=\mathscr A\otimes_S^{\nabla,\tau}\mathscr B$ be a DG twisted tensor product over $S$. Then, for any $\tau$ and $\nabla$, the DG algebra $\mathscr C^{\nabla}$ is also $S$-split and smooth, and $\mathtt X_{\mathscr C^{\nabla}}=\mathtt X_{\mathscr B}\cdot \mathtt X_{\mathscr A}$. Proof. As $\mathscr A$ is $S$-split, we have $(\mathsf J_{\mathscr A})_-=(\mathsf J_{\mathscr A})_+=J_{A}$. By Remark 2.11 the twisting map $\tau$ takes $B\otimes_S J_{A}$ to $J_{A}\otimes_S B$. This means that the two-sided ideal $J_{A}\otimes_S B\subset \mathscr C^{\nabla}$ is nilpotent and therefore contained in the external radical $(\mathsf J_{\mathscr C^{\nabla}})_{+}$. Thus, by Theorem 3.13 the DG algebra $\mathscr C^{\nabla}$ is also smooth. Moreover, the morphisms of DG algebras $i_A\colon \mathscr A\to \mathscr C^{\nabla}$ and $p_B\colon \mathscr C^{\nabla}\to \mathscr B$ induce morphisms $\epsilon_C\colon S\to \mathscr C^{\nabla}$ and $\pi_C\colon \mathscr C^{\nabla}\to S$ such that $\pi_C\circ \epsilon_C=\operatorname{id}_S$. Therefore, the DG algebra $\mathscr C^{\nabla}$ is $S$-split too. Let us apply equality (14) to the DG algebra $\mathscr C^{\nabla}$:
$$
\begin{equation*}
\begin{aligned} \, (\mathtt X_{\mathscr C^{\nabla}})_{ij}&=\!\sum_l (-1)^l \dim_{\Bbbk} e_j(\mathscr C^{\nabla})^l e_i=\!\sum_l (-1)^l \dim_{\Bbbk} \biggl(\,\bigoplus_{s+t=l} \bigoplus_{r=1}^{N} e_j\mathscr A^s e_r\otimes_{\Bbbk} e_r \mathscr B^t e_i\biggr) \\ &=\!\sum_{s,t}(-1)^{s+t}\dim_{\Bbbk}\biggl(\,\bigoplus_{r=1}^{N} e_j\mathscr A^s e_r\otimes_{\Bbbk} e_r \mathscr B^t e_i\biggr) \\ &=\!\sum_{s,t} (-1)^{s+t}\sum_{r=1}^N \dim_{\Bbbk}(e_j\mathscr A^s e_r) \cdot\dim_{\Bbbk}(e_r \mathscr B^t e_i). \end{aligned}
\end{equation*}
\notag
$$
Changing the order of summation leads to the following equations:
$$
\begin{equation*}
\begin{aligned} \, (\mathtt X_{\mathscr C^{\nabla}})_{ij}&=\!\sum_{r=1}^N \biggl(\,\sum_{s}(-1)^s \dim_{\Bbbk}e_j\mathscr A^s e_r\biggr)\cdot \biggl(\sum_{t}(-1)^t \dim_{\Bbbk} e_r\mathscr B^t e_i\biggr) \\ &=\!\sum_{r=1}^N(\mathtt X_{\mathscr B})_{ir}\cdot(\mathtt X_{\mathscr A})_{rj}. \end{aligned}
\end{equation*}
\notag
$$
Thus, we obtain the required matrix equality $\mathtt X_{\mathscr C^{\nabla}}= \mathtt X_{\mathscr B}\cdot\mathtt X_{\mathscr A}$. The proposition is proved. In the previous section (see Proposition 4.4) we introduced generalized Green DG algebras. Recall that we considered DG algebras $K_{ij}[d]$, $1\leqslant i\ne j\leqslant N$, that have only one arrow from $i$ to $j$ of degree $d$ in the Jacobson radical, that is, $J=e_jJe_i\cong\Bbbk$. Generalized Green DG algebras were defined as iterated twisted tensor products over $S=S_N$ of such DG algebras with twisting map given by formula (5) from Construction 2.12. The matrix $\mathtt X_{K_{ij}[d]}$ is equal to ${\mathtt E}_{ij}^{\epsilon}$, where ${\mathtt E}_{ij}$ is the elementary matrix and $\epsilon=(-1)^d$. Since the elementary matrices generate the group $\operatorname{SL}(n,{\mathbb Z})$, we obtain the following corollary. Corollary 5.2. For any matrix $\mathtt X\in\operatorname{SL}(n,{\mathbb Z})$ there is a smooth DG algebra $\mathscr R$ such that $\mathtt X_{\mathscr R}=\mathtt X$. Moreover, for any $\mathtt X\in\operatorname{SL}(n,{\mathbb Z})$ there is a generalized Green DG algebra $\mathscr R$ with $\mathtt X_{\mathscr R}=\mathtt X$. 5.2. Grothendieck groups of algebras $R_{\mathfrak{F}}$Now consider the algebras $R_{\mathfrak{F}}$ constructed in subsection 4.3. We fix a family ${\mathfrak{F}}$ such that the algebra $R_{\mathfrak{F}}$ is smooth. The Grothendieck group $K_0( \operatorname{perf}\!{-} R_{\mathfrak{F}})$ is isomorphic to ${\mathbb Z}^2$, and the classes of the projective modules $P_1$ and $P_2$ form a basis. Denote by $\mathtt X_{R_{\mathfrak{F}}}$ the matrix of the bilinear form $\chi_{R_{\mathfrak{F}}}$ in this basis. Since the algebra $R_{\mathfrak{F}}$ is smooth, the classes of the simple modules $S_1$ and $S_2$ also form a basis of the Grothendieck group, and the matrix of $\chi_{R_{\mathfrak{F}}}$ is equal to $(\mathtt X_{R_{\mathfrak{F}}}^{-1})^\top$ in this basis. A direct calculation gives us the following matrices:
$$
\begin{equation*}
\begin{aligned} \, \mathtt X_{R_{\mathfrak{F}}}&=\begin{pmatrix} m(n-k) +1 & n+mk(n-k) \\ m & mk+1 \end{pmatrix}, \\ (\mathtt X_{R_{\mathfrak{F}}}^{-1})^\top&= \begin{pmatrix} mk +1 & -m \\ -n- mk(n-k) & m(n-k)+1 \end{pmatrix}. \end{aligned}
\end{equation*}
\notag
$$
Let us denote by $q_{{\mathfrak{F}}}(\mathsf E)$ the Euler quadratic form $\chi_{R_{\mathfrak{F}}}(\mathsf E,\mathsf E)$. It is an integral binary quadratic form that is equal to
$$
\begin{equation*}
q_{{\mathfrak{F}}}(x,y)=(m(n-k)+1)x^2+(mk(n-k)+m+n)xy+(mk+1)y^2
\end{equation*}
\notag
$$
in the basis of the classes of the projective modules. Proposition 5.3. Let $q_{{\mathfrak{F}}}$ be the Euler quadratic form for the algebra $R_{\mathfrak{F}}$. Then the following assertions hold. Proof. Statements from parts 1), 2), and 3) follow by direct calculations.
Let us discuss part 4). The case $n=2$ is evident, so we assume that $n\geqslant 3$. For brevity we denote the quadratic form $ax^2+bxy+cy^2$ by $(a,b,c)$. An indefinite form $(a,b,c)$ of discriminant $D > 0$ is called reduced if $0 < b <\sqrt{D}$ and $\sqrt{D}- b < 2|a|<\sqrt{D}+b$. It is well known that any indefinite form is equivalent to a reduced form of the same discriminant and the number of reduced forms of a fixed discriminant is finite (see, for example, [2; Proposition 3.3]). Two reduced forms $(a,b,c)$ and $(a',b',c')$ are called adjacent if $a'=c$ and $b+b' \equiv 0\ (\operatorname{mod}2c)$. It can be shown that there is a unique reduced form adjacent to the right of any given reduced form. Thus, the set of reduced forms of a fixed discriminant can be partitioned into cycles of adjacent forms (see [2; Proposition 3.4]). Finally, two reduced forms are equivalent if and only if they are in the same cycle (see [2; Theorem 3.5]). In our case the principal form, that is, a reduced form of type $(1,b,c)$, with discriminant $D=F^2-4$, looks like $(1,F-2,2-F)$. The principal cycle of adjacent forms consists of two forms, $(1,F-2,2-F)$ and $ (2-F,F-2,1)$. Consider the quadratic form $q'_{{\mathfrak{F}}}=(mk+1,F-2k,1-k(n-k))$ from 3). It is reduced and equivalent to the form $q_{{\mathfrak{F}}}$, but it does not belong to the principal cycle. Therefore, our form $q_{\mathfrak{F}}$ does not represent $1$ for any $m\geqslant 1$ and $n>k\geqslant 1$. The proposition is proved. Corollary 5.4. For any $m\geqslant 1$ and $n>k\geqslant 1$ the triangulated category $ \operatorname{perf}\!{-} R_{\mathfrak{F}}$ does not have exceptional objects. 5.3. Final remarksConsider the DG algebra $\mathscr R=\Bbbk[\varepsilon]/\varepsilon^2$ with $\deg (\varepsilon)=\delta$. It can be shown that $\mathscr R$ has many different smooth realization with Grothendieck groups of rank $2$. Suppose $F\colon \operatorname{perf}\!{-} \mathscr R\hookrightarrow \operatorname{perf}\!{-} \mathscr E$ is a full embedding, where $\mathscr E$ is a smooth proper DG algebra, and the right adjoint $G\colon \operatorname{perf}\!{-} \mathscr E\to {\mathcal D}_{\mathrm{cf}}(\mathscr R)$ is a Verdier quotient. The objects $F(\mathscr R)$ and $K\in \operatorname{perf}\!{-} \mathscr E$ such that $G(K)\cong\Bbbk$ give two elements $r,k\in K_0( \operatorname{perf}\!{-} \mathscr E)$ with $\chi_{\mathscr E}(r,r)=1+(-1)^{\delta}$ and $\chi_{\mathscr E}(r,k)=1$, which generate ${\mathbb Z}^2\subseteq K_0( \operatorname{perf}\!{-} \mathscr E)$. Hence the rank of $K_0( \operatorname{perf}\!{-} \mathscr E)$ is at least $2$. For any $p,q\geqslant 0$, the DG algebras
$$
\begin{equation*}
\mathscr E_{[p,q;\delta]}=K_{p}^{\circ}\otimes^{\mathbf{v}}_S K_{1} \otimes^{\mathbf{v}}_S K_{1}^{\circ}[\delta]\otimes^{\mathbf{v}}_S K_{q}
\end{equation*}
\notag
$$
provide different examples of smooth categories ${\mathcal T}= \operatorname{perf}\!{-} \mathscr E_{[p,q; \delta]}$ with $K_0({\mathcal T})\cong{\mathbb Z}^2$, for which there is a full embedding $ \operatorname{perf}\!{-} \mathscr R\hookrightarrow{\mathcal T}$. It works because
$$
\begin{equation*}
\mathsf{End}_{\mathscr E_{[p,q;\delta]}}(P_2)\cong \mathscr R.
\end{equation*}
\notag
$$
Consider two of them, $\mathscr E_{[0,0;\delta]}$ and $\mathscr E_{[0,1;\delta]}$. Informally, we feel that the first DG algebra is simpler than the second. But are there invariants allowing this to be strictly determined? Following [24], we can define the dimension of a triangulated category ${\mathcal T}$ as the least integer $d\geqslant 0$ for which there is a strong generator $E\in {\mathcal T}$ with $\langle E\rangle_{d+1}={\mathcal T}$. It can be shown that $\dim {\mathcal D}_{\mathrm{cf}}(\mathscr R)= \dim ( \operatorname{perf}\!{-} \mathscr E_{[0,0;\delta]})=1$. The dimension of the category $ \operatorname{perf}\!{-} \mathscr E_{[0,1;\delta]}$ is not so easy to calculate, but using methods from [6; § 6] one can show that it is also equal to $1$. We conjecture that the dimension of the category $ \operatorname{perf}\!{-} R_{\mathfrak{F}}$ is equal to $1$, at least in the case $n=2$. To see a difference between $\mathscr E_{[0,0;\delta]}$ and $\mathscr E_{[0,1;\delta]}$, we can try to consider dimension spectra. Recall that the dimension spectrum $\sigma({\mathcal T})\subset{\mathbb Z}$ consists of all $d\geqslant 0$, for which there is an object $E\in {\mathcal T}$ with $\langle E\rangle_{d+1}={\mathcal T}$ and $\langle E\rangle_{d}\ne{\mathcal T}$ (see [19]). It can be shown that $\sigma( \operatorname{perf}\!{-} \mathscr E_{[0,0;\delta]})=\{1,2\}$. On the other hand it is not very difficult to check that $\sigma( \operatorname{perf}\!{-} \mathscr E_{[0,1; \delta]})$ contains the integer $3$. The final remark is that the dimension of the category $ \operatorname{perf}\!{-} R_{\mathfrak{F}}$ does not exceed $3$ for any $\mathfrak{F}$, since the nilpotency index is equal to $4$, but at the same time, for an arbitrarily large odd integer, there exists $\mathfrak{F}$ such that the spectrum of the algebra $R_{\mathfrak{F}}$ contains this integer, since the spectrum contains the global dimension.
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