Invariants of noncommutative projective schemes

TL;DR

This paper computes key algebraic invariants such as K-theory, cyclic homology, and topological Hochschild homology for noncommutative projective schemes linked to Koszul algebras of finite global dimension.

Contribution

It provides explicit calculations of invariants for a class of noncommutative schemes, advancing understanding in noncommutative algebraic geometry.

Findings

Computed algebraic K-theory for these schemes

Determined cyclic homology and topological Hochschild homology invariants

Established methods applicable to Koszul algebras of finite global dimension

Abstract

In this note we compute several invariants (e.g. algebraic K-theory, cyclic homology and topological Hochschild homology) of the noncommutative projective schemes associated to Koszul algebras of finite global dimension.

Equations

$$0⟶\mathcal{A}⟶\mathcal{B}⟶\mathcal{C}⟶0$$

$$E_m^o({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{qgr}(A)))\simeq R[t]/⟨h_A^{\prime }(t{)}^{-1}⟩{\otimes }_R{E}_m^o(k)$$

$$I\text{mspace}-6.0mu{K}_m(\mathrm{qgr}(A))\simeq \mathbb{Z}[t]/⟨h_A(t{)}^{-1}⟩{\otimes }_{\mathbb{Z}}I\text{mspace}-6.0mu{K}_m(k)$$

$$HH(\mathrm{qgr}(A))\simeq HH(k{)}^{\oplus d^{\prime }}$$

$$HP(\mathrm{qgr}(A))\simeq HP(k{)}^{\oplus d^{\prime }}$$

$$THH_{m}(\mathrm{qgr}(A))\simeq\left\{\begin{array}[]{ll}(\mathbb{F}_{p})^{\oplus d^{\prime}}&m\geq 0\,\,\,\mathrm{even}\\ 0&\text{otherwise}\,.\end{array}\right.$$

$$A:=k⟨x_1,\dots ,x_d⟩/⟨x_j{x}_i-q_{ij}{x}_i{x}_j|1\le i<j\le d⟩,$$

$$x_1{x}_2=q{x}_2{x}_1$$

$$x_2{x}_3=x_3{x}_2$$

$$R:=H^0(C\times C,(\mathcal{L}⊠\mathcal{L})(-{\Gamma }_{\sigma }))\subset H^0(C\times C,\mathcal{L}⊠\mathcal{L})=V\otimes V,$$

$$A:=T(\mathfrak{g}\oplus kz)/⟨\{z\otimes x-x\otimes z|x\in \mathfrak{g}\}\cup \{x\otimes y-y\otimes x-[x,y]\otimes z|x,y\in \mathfrak{g}\}⟩,$$

$$f_1:=x_1(\Theta x_1-x_3)+x_3(x_1-\Theta x_3)$$

$$f_3:=x_2(\Theta x_1-x_3)+x_4(x_1-\Theta x_3)$$

$$f_5:=x_1(\Delta x_1-x_2)+x_4(x_1-\Delta x_2)$$

$${\iota }_1:{\mathcal{E}}^{\prime }⟶\mathrm{Ex}({\mathcal{E}}^{\prime })a\mapsto (a=a\to 0)$$

$${\pi }_1:\mathrm{Ex}({\mathcal{E}}^{\prime })\stackrel{\epsilon \mapsto a}{⟶}{\mathcal{E}}^{\prime }$$

$$0⟶{\mathcal{D}}_{\mathrm{dg}}^b({\mathcal{E}}^{\prime })\stackrel{{\iota }_1}{⟶}{\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{Ex}({\mathcal{E}}^{\prime }))\stackrel{{\pi }_3}{⟶}{\mathcal{D}}_{\mathrm{dg}}^b({\mathcal{E}}^{\prime })⟶0$$

$$E({\mathcal{D}}_{\mathrm{dg}}^b({\mathcal{E}}^{\prime }))\stackrel{E({\iota }_1)}{⟶}E({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{Ex}({\mathcal{E}}^{\prime })))\stackrel{E({\pi }_3)}{⟶}E({\mathcal{D}}_{\mathrm{dg}}^b({\mathcal{E}}^{\prime }))\stackrel{\partial }{⟶}\Sigma E({\mathcal{D}}_{\mathrm{dg}}^b({\mathcal{E}}^{\prime })).$$

$$[E({\iota }_1)E({\iota }_2)]:E({\mathcal{D}}_{\mathrm{dg}}^b({\mathcal{E}}^{\prime }))\oplus E({\mathcal{D}}_{\mathrm{dg}}^b({\mathcal{E}}^{\prime }))\stackrel{\simeq }{⟶}E({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{Ex}({\mathcal{E}}^{\prime }))).$$

$$\phi :\mathrm{grproj}(B_0)⟶\mathrm{grproj}(B)$$

$$\psi :\mathrm{grproj}(B)⟶\mathrm{grproj}(B_0)$$

$$\mathrm{grproj}(B_0)\stackrel{\simeq }{⟶}{⨿}_{n\in \mathbb{Z}}\mathrm{proj}(B_0)$$

$$E(\phi \circ \psi ):E({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{grproj}(B)))⟶E({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{grproj}(B))).$$

$$E(\phi \circ \psi )=\sum _{n\in \mathbb{Z}}E(\psi (-{)}_n{\otimes }_{B_0}B(-n)).$$

$${\mathrm{grproj}}_0(B)\subset {\mathrm{grproj}}_1(B)\subset \cdots \subset {\mathrm{grproj}}_q(B)\subset \cdots \subset \mathrm{grproj}(B).$$

$$\sum _{n\in \mathbb{Z}}E(\psi (-{)}_n{\otimes }_{B_0}B(-n))=\sum _{n\in \mathbb{Z}}E(F_n(-)/F_{n-1}(-)).$$

$$\sum _{n\in \mathbb{Z}}E(F_n(-)/F_{n-1}(-))=\mathrm{id}.$$

$$0⟶{\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{tors}(A))⟶{\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{gr}(A))⟶{\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{qgr}(A))⟶0.$$

$${\mathcal{D}}^b(\mathrm{tors}(A))⟶{\mathcal{D}}^b(\mathrm{gr}(A))⟶{\mathcal{D}}^b(\mathrm{qgr}(A))$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{tors}(A)))⟶E({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{gr}(A)))⟶E({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{qgr}(A)))\stackrel{\partial }{⟶}\Sigma E({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{tors}(A))).$$

$${\oplus }_{-\infty }^{+\infty }E(k)\simeq E({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{grproj}(A)))\stackrel{\simeq }{⟶}E({\mathcal{D}}_{\mathrm{dg}}^b(\mathrm{gr}(A))).$$

Full text

The author was partially supported by a NSF CAREER Award

1. Introduction

Noncommutative projective schemes

Let $k$ be a field and $A=\bigoplus_{n\geq 0}A_{n}$ a $\mathbb{N}$-graded Noetherian $k$-algebra. Throughout the note, we will always assume that $A$ is connected, i.e. $A_{0}=k$, and locally finite-dimensional, i.e. $\mathrm{dim}_{k}(A_{n})<\infty$ for every $n$. Following Manin [12], Gabriel [6], Artin-Zhang [1], and others, the noncommutative projective scheme $\mathrm{qgr}(A)$ associated to $A$ is defined as the quotient category $\mathrm{gr}(A)/\mathrm{tors}(A)$, where $\mathrm{gr}(A)$ stands for the abelian category of finitely generated $\mathbb{Z}$-graded (right) $A$-modules and $\mathrm{tors}(A)$ for the Serre subcategory of torsion $A$-modules. This definition was motivated by Serre’s celebrated result [19, Prop. 7.8], which asserts that in the particular case where $A$ is commutative and generated by elements of degree $1$ the quotient category $\mathrm{qgr}(A)$ is equivalent to the abelian category of coherent ${\mathcal{O}}_{\mathrm{Proj}(A)}$-modules $\mathrm{coh}(\mathrm{Proj}(A))$. For example, when $A$ is the polynomial $k$-algebra $k[x_{1},\ldots,x_{d}]$, with $\mathrm{deg}(x_{i})=1$, we have the following equivalence $\mathrm{qgr}(k[x_{1},\ldots,x_{d}])\simeq\mathrm{coh}(\mathbb{P}^{d-1})$.

Invariants of dg categories

A dg category ${\mathcal{A}}$ is a category enriched over complexes of $k$-vector spaces; consult Keller’s survey [9]. Every (dg) $k$-algebra $B$ gives naturally rise to a dg category with a single object. Another source of examples is provided by exact categories since the bounded derived category ${\mathcal{D}}^{b}({\mathcal{E}})$ of every exact category ${\mathcal{E}}$ admits a canonical dg enhancement ${\mathcal{D}}^{b}_{\mathrm{dg}}({\mathcal{E}})$; see [9, §4.4]. In what follows, we will denote by $\mathrm{dgcat}(k)$ the category of dg categories and dg functors. A functor $E\colon\mathrm{dgcat}(k)\to{\mathcal{T}}$, with values in a triangulated category, is called:

• (i)

Morita invariant if it inverts the Morita equivalences; see [9, §4.6].

• (ii)

Localizing if it sends short exact sequences of dg categories, in the sense of Drinfeld/Keller (see [3][9, §4.6]), to distinguished triangles:

[TABLE]

• (iii)

Co-continuous if it preserves sequential (homotopy) colimits.

Examples of functors satisfying the conditions (i)-(iii) include nonconnective algebraic $K$-theory $I\mspace{-6.0mu}K$, homotopy $K$-theory $KH$, étale $K$-theory $K^{\mathrm{et}}$, the mixed complex $C$, Hochschild homology $HH$, cyclic homology $HC$, and topological Hochschild homology $THH$; see [22, §8.2]. Some other functors such as periodic cyclic homology $HP$ and negative cyclic homology $HN$ only satisfy conditions (i)-(ii). When applied to $B$, resp. to ${\mathcal{D}}^{b}_{\mathrm{dg}}({\mathcal{E}})$, all the preceding invariants of dg categories reduce to the corresponding invariants of the (dg) $k$-algebra $B$, resp. of the exact category ${\mathcal{E}}$.

Notation 1.1*.*

Given a functor $E\colon\mathrm{dgcat}(k)\to{\mathcal{T}}$, an object $o\in{\mathcal{T}}$, an integer $m\in\mathbb{Z}$, and a dg category ${\mathcal{A}}$, let us write $E_{m}^{o}({\mathcal{A}}):=\mathrm{Hom}_{\mathcal{T}}(\Sigma^{m}(o),E({\mathcal{A}}))$. Whenever ${\mathcal{T}}$ is symmetric monoidal with $\otimes$-unit ${\bf 1}$, we will write $E_{m}({\mathcal{A}})$ instead of $E_{m}^{{\bf 1}}({\mathcal{A}})$.

Statement of results

Let $k$ be a field and $A=\bigoplus_{n\geq 0}A_{n}$ a $\mathbb{N}$-graded Noetherian $k$-algebra. Assume that $A$ is Koszul and has finite global dimension $d$. Under these assumptions, the Hilbert series $h_{A}(t):=\sum_{n\geq 0}\mathrm{dim}_{k}(A_{n})t^{n}\in\mathbb{Z}[\![t]\!]$ is invertible and its inverse $h_{A}(t)^{-1}$ is a polynomial $1-\beta_{1}t+\beta_{2}t^{2}-\cdots+(-1)^{d}\beta_{d}t^{d}$ of degree $d$, where $\beta_{i}$ stands for the dimension of the $k$-vector space $\mathrm{Tor}^{A}_{i}(k,k)$ (or $\mathrm{Ext}_{A}^{i}(k,k)$). In what follows, we write $\beta:=\beta_{d}$. Our main result is the following computation:

Theorem 1.2**.**

Let $A$ be a $k$-algebra as above and $E\colon\mathrm{dgcat}(k)\to{\mathcal{T}}$ a functor satisfying conditions (i)-(iii). Assume that ${\mathcal{T}}$ is $R$-linear for a commutative ring $R$.

• (i)

For every compact object $o\in{\mathcal{T}}$, we have $R$-module isomorphisms

[TABLE]

where $h^{\prime}_{A}(t)^{-1}=1-\beta^{\prime}_{1}t+\beta^{\prime}_{2}t^{2}-\cdots+(-1)^{d^{\prime}}\beta^{\prime}_{d^{\prime}}t^{d^{\prime}}$ stands for the image of the polynomial $h_{A}(t)^{-1}$ in $R[t]$. In what follows, we write $\beta^{\prime}:=\beta^{\prime}_{d^{\prime}}$.

• (ii)

Assume moreover that $1/\beta^{\prime}\in R$ and that ${\mathcal{T}}$ is compactly generated. Under these assumptions, we have an isomorphism $E({\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{qgr}(A)))\simeq E(k)^{\oplus d^{\prime}}$.

Remark 1.4*.*

• (i)

If $\beta=1$, then $\beta^{\prime}=\beta$ and $d^{\prime}=d$. As proved in [21, Cor. 0.2], in the particular case where $d=3$, we always have $h_{A}(t)^{-1}=(1-t)^{3}$.

• (ii)

If $R$ is a field, then $1/\beta^{\prime}\in R$. Moreover, $\beta^{\prime}=\beta$ and $d^{\prime}=d$ if and only if the characteristic of $R$ does not divides $\beta$.

Corollary 1.5**.**

Let $A$ be a $k$-algebra as above and $E\colon\mathrm{dgcat}(k)\to{\mathcal{T}}$ a functor satisfying conditions (i)-(iii). Assume moreover that ${\mathcal{T}}$ is compactly generated. Under these assumptions, we have an isomorphism $E({\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{qgr}(A)))_{1/\beta^{\prime}}\simeq E(k)_{1/\beta^{\prime}}^{\oplus d^{\prime}}$ in the $\mathbb{Z}[1/\beta^{\prime}]$-linearized triangulated category111Let ${\mathcal{G}}$ be a set of compact generators of ${\mathcal{T}}$. Recall that ${\mathcal{T}}_{1/\beta^{\prime}}$ may be defined as the Verdier quotient of ${\mathcal{T}}$ by the smallest localizing (=closed under arbitrary direct sums) triangulated subcategory containing the objects $\{\mathrm{cone}(\beta\cdot\operatorname{id}_{o})\,|\,o\in{\mathcal{G}}\}$. ${\mathcal{T}}_{1/\beta^{\prime}}$.

Proof.

By construction, the triangulated category ${\mathcal{T}}_{1/\beta^{\prime}}$ is compactly generated and $R[1/\beta^{\prime}]$-linear. Moreover, the $\mathbb{Z}[1/\beta^{\prime}]$-linearization functor $(-)_{1/\beta^{\prime}}\colon{\mathcal{T}}\to{\mathcal{T}}_{1/\beta^{\prime}}$ is triangulated and preserves arbitrary direct sums. Therefore, the proof follows from Theorem 1.2(ii) applied to $E=E(-)_{1/\beta^{\prime}}$ (with $R=R[1/\beta^{\prime}]$). ∎

Example 1.6* (Algebraic $K$-theory).*

Nonconnective algebraic $K$-theory gives rise to a functor $I\mspace{-6.0mu}K\colon\mathrm{dgcat}(k)\to\mathrm{Ho}(\mathrm{Spt})$, with values in the homotopy category of spectra, satisfying conditions (i)-(iii); see [22, §8.2.1]. Therefore, by applying Theorem 1.2(i) to $E=I\mspace{-6.0mu}K$ (with $R=\mathbb{Z}$) and to the sphere spectrum $o=\mathbb{S}$, we obtain isomorphisms222In the particular case where $m=0$, the isomorphism (1.7) was originally established by Mori-Smith in [14, Thm. 2.3].

[TABLE]

Moreover, since the triangulated category $\mathrm{Ho}(\mathrm{Spt})$ is compactly generated, Corollary 1.5 implies that $I\mspace{-6.0mu}K(\mathrm{qgr}(A))_{1/\beta}\simeq I\mspace{-6.0mu}K(k)^{\oplus d}_{1/\beta}$. All the above holds mutatis mutandis with $I\mspace{-6.0mu}K$ replaced by $KH$ or $K^{\mathrm{et}}$.

Example 1.8* (Mixed complex).*

Following Kassel [8], a mixed complex is a (right) dg module over the $k$-algebra of dual numbers $\Lambda:=k[\epsilon]/\epsilon^{2}$ with $\mathrm{deg}(\epsilon)=-1$ and $d(\epsilon)=0$. The mixed complex gives rise to a functor $C\colon\mathrm{dgcat}(k)\to{\mathcal{D}}(\Lambda)$, with values in the derived category of $\Lambda$, satisfying conditions (i)-(iii); see [22, §8.2.4]. Therefore, since the category ${\mathcal{D}}(\Lambda)$ is compactly generated, by applying Theorem 1.2(ii) to $E=C$ (with $R=k$), we obtain an isomorphism $C(\mathrm{qgr}(A))\simeq C(k)^{\oplus d^{\prime}}$.

Example 1.9* (Cyclic homology and its variants).*

As explained by Keller in [11, §2.2], Hochschild homology $HH$, cyclic homology $HC$, periodic cyclic homology $HP$, and negative cyclic homology $HN$, can be recovered from the mixed complex $C$. Therefore, making use of Example 1.8, we conclude that

[TABLE]

Example 1.10* (Topological Hochschild homology).*

Topological Hochschild homology gives rise to a (lax symmetric monoidal) functor $THH\colon\mathrm{dgcat}(k)\to\mathrm{Ho}(\mathrm{Spt})$ satisfying conditions (i)-(iii); see [22, §8.2.8]. Since the “inclusion of the $0^{\mathrm{th}}$ skeleton” yields a ring homomorphism $k\to THH_{0}(k)$, the abelian groups $THH_{\ast}$ are then naturally equipped with a $k$-linear structure. Therefore, using the fact that the triangulated category $\mathrm{Ho}(\mathrm{Spt})$ is (compactly) generated by the sphere spectrum $\mathbb{S}$, an argument similar to the one used in the proof of Theorem 1.2(ii) allows us to conclude that $THH(\mathrm{qgr}(A))\simeq THH(k)^{\oplus d^{\prime}}$. For example, in the particular where $k=\mathbb{F}_{p}$, with $p$ a prime number, we have the following isomorphisms:

[TABLE]

Intuitively speaking, Theorem 1.2 (as well as Corollary 1.5 and Examples 1.6-1.10) shows that all the different invariants of a noncommutative projective scheme $\mathrm{qgr}(A)$ are completely determined by the Hilbert series $h_{A}(t)$ of $A$. Theorem 1.2 (as well as Corollary 1.5) may be applied to the following algebras:

Example 1.11* (Quantum polynomial algebras).*

Choose constant elements $q_{ij}\in k^{\times}$ with $1\leq i<j\leq d$. The following $\mathbb{N}$-graded Noetherian $k$-algebra

[TABLE]

with $\mathrm{deg}(x_{i})=1$, is called the quantum polynomial algebra associated to $q_{ij}$. This algebra is Koszul, has global dimension $d$, and $h_{A}(t)^{-1}=(1-t)^{d}$; see [13, §1].

Example 1.12* (Quantum matrix algebras).*

Choose a $q\in k^{\times}$. The $\mathbb{N}$-graded Noetherian $k$-algebra $A$ defined as the quotient of $k\langle x_{1},x_{2},x_{3},x_{4}\rangle$ by the relations

[TABLE]

with $\mathrm{deg}(x_{i})=1$, is called the quantum matrix algebra associated to $q$. This algebra is Koszul, has global dimension $4$, and $h_{A}(t)^{-1}=(1-t)^{4}$; see [13, §1].

Example 1.13* (Sklyanin algebras).*

Let $C$ be a smooth elliptic curve, $\sigma\in\mathrm{Aut}(C)$ an automorphism given by translation under the group law, and ${\mathcal{L}}$ a line bundle on $C$ of degree $d\geq 3$. We write $\Gamma_{\sigma}\subset C\times C$ for the graph of $\sigma$ and $V$ for the $d$-dimensional $k$-vector space $H^{0}(C,{\mathcal{L}})$. The $\mathbb{N}$-graded Noetherian $k$-algebra $A:=T(V)/R$, where

[TABLE]

is called the Sklyanin algebra associated to the triple $(C,\sigma,{\mathcal{L}})$. This algebra is Koszul, has global dimension $d$, and $h_{A}(t)^{-1}=(1-t)^{d}$; see [4][24, §1].

Example 1.14* (Homogenized enveloping algebras).*

Let $\mathfrak{g}$ be a finite dimensional Lie algebra. The following $\mathbb{N}$-graded Noetherian $k$-algebra ($z$ is a new variable)

[TABLE]

is called the homogenized enveloping algebra of $\mathfrak{g}$. This algebra is Koszul, has global dimension $d:=\mathrm{dim}(\mathfrak{g})+1$, and $h_{A}(t)^{-1}=(1-t)^{d}$; see [20, §12].

Example 1.15*.*

Let $k$ be an uncountable algebraically closed field. Choose a pair of elements $(\theta,\rho)$ of $k^{\times}$ which are algebraically independent over the prime field of $k$ and write $\Theta:=\frac{\theta-1}{\theta+1}$ and $\Delta:=\frac{\rho-1}{\rho+1}$. Under these assumptions and notations, the $\mathbb{N}$-graded Noetherian $k$-algebra $A:=k\langle x_{1},x_{2},x_{3},x_{4}\rangle/\langle f_{1},\ldots,f_{6}\rangle$, where

[TABLE]

is Koszul, has global dimension $4$, and $h_{A}(t)^{-1}=(1-t)^{4}$; see [18, Thm. 3.5].

Gorenstein algebras

Recall that a $\mathbb{N}$-graded Noetherian $k$-algebra $A=\bigoplus_{n\geq 0}A_{n}$ is called Gorenstein, with Gorenstein parameter $l$, if it has finite injective dimension $m$ and ${\bf R}\mathrm{Hom}_{A}(k,A)\simeq\Sigma^{-m}k(l)$, where $k(l)$ stands for the $\mathbb{Z}$-graded (right) $A$-module $k(l)_{n}:=k_{n+l}$. Let us assume moreover that $A$ has finite global dimension $d$; this implies that $d=m$. Under these assumptions, a remarkable result of Orlov (see [15, Cor. 2.7]) asserts that the bounded derived category ${\mathcal{D}}^{b}(\mathrm{qgr}(A))$ admits a full exceptional collection of length $l$. This leads naturally to the following result:

Theorem 1.16**.**

Let $A$ be a $\mathbb{N}$-graded Noetherian $k$-algebra and $E$ a functor satisfying conditions333More generally, condition (ii) can be replaced by additivity in the sense of [22, Def. 2.1]. (i)-(ii). Assume that $A$ is Gorenstein, with Gorenstein parameter $l$, and has finite global dimension $d$. Under these assumptions, we have an isomorphism $E({\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{qgr}(A)))\simeq E(k)^{\oplus l}$.

Proof.

As explained in [22, §2.4.2 and §8.4.5], every functor $E$ satisfying conditions (i)-(ii) sends a full exceptional collections of length $l$ to the direct sum $E(k)^{\oplus l}$. ∎

Remark 1.17*.*

• (i)

Since $A$ is connected and has finite global dimension, the Hilbert series $h_{A}(t)$ is invertible and its inverse $h_{A}(t)^{-1}$ is a polynomial. Moreover, the Gorenstein condition implies that $h_{A}(t)^{-1}$ is monic and has degree $l$.

• (ii)

As proved in [16, Chap. 2  Thm. 2.5], $A$ is moreover Koszul if and only if $d=l$.

Note that Theorem 1.2 does not follows from Theorem 1.16 because, in general, Koszulness does not implies444In the particular case where $d=3$, Koszulness indeed implies Gorensteiness; see [21, Cor. 0.2]. Gorensteiness. For instance, the algebras $A$ of Example 1.15 are Koszul but not Gorenstein; see [18, Thm. 3.5]. In this latter example, we have moreover $\mathrm{dim}_{k}(\mathrm{Ext}^{i}_{A}(k,A))=\infty$ for $i=2,3,4$; see [18, Prop. 5.11]. Consequently, the $k$-linear triangulated categories ${\mathcal{D}}^{b}(\mathrm{qgr}(A))$ are not even Ext-finite.

2. Proof of Theorem 1.2

Recall from Quillen [17, §2] that an exact category ${\mathcal{E}}$ is an additive category equipped with a family of short exact sequences satisfying some standard conditions. In order to simplify the exposition, given an exact functor $F\colon{\mathcal{E}}\to{\mathcal{E}}^{\prime}$, we will still denote by $F\colon{\mathcal{D}}^{b}_{\mathrm{dg}}({\mathcal{E}})\to{\mathcal{D}}^{b}_{\mathrm{dg}}({\mathcal{E}}^{\prime})$ the induced dg functor. We start with the following general result of independent interest:

Proposition 2.1**.**

Let $0\to F_{1}\to F_{2}\to F_{3}\to 0$ be a short exact sequence of exact functors $F_{1},F_{2},F_{3}\colon{\mathcal{E}}\to{\mathcal{E}}^{\prime}$. Given any localizing functor $E\colon\mathrm{dgcat}(k)\to{\mathcal{T}}$, we have the following equality $E(F_{2})=E(F_{1})+E(F_{3})$.

Proof.

Let $\mathrm{Ex}({\mathcal{E}}^{\prime})$ be the category of short exact sequences $\varepsilon=(a\to b\to c)$ in ${\mathcal{E}}^{\prime}$; this is also an exact category with short exact sequence defined componentwise. By construction, $\mathrm{Ex}({\mathcal{E}}^{\prime})$ comes equipped with the following exact functors

[TABLE]

[TABLE]

satisfying the equalities $\pi_{1}\circ\iota_{1}=\pi_{2}\circ\iota_{1}=\operatorname{id}$, $\pi_{3}\circ\iota_{1}=\pi_{1}\circ\iota_{2}=0$, and $\pi_{2}\circ\iota_{2}=\pi_{3}\circ\iota_{2}=\operatorname{id}$. Moreover, we have the following short exact sequence of dg categories

[TABLE]

and consequently the following distinguished triangle

[TABLE]

Since $\pi_{3}\circ\iota_{2}=\operatorname{id}$, the preceding triangle splits an induces an isomorphism

[TABLE]

Note that a short exact sequence of exact functors $0\to F_{1}\to F_{2}\to F_{3}\to 0$ is the same data as an exact functor $F\colon{\mathcal{E}}\to\mathrm{Ex}({\mathcal{E}}^{\prime})$. Therefore, by combining the equalities $E(\pi_{2})\circ[E(\iota_{1})\,\,E(\iota_{2})]=[\operatorname{id}\,\,\operatorname{id}]$ and $(E(\pi_{1})+E(\pi_{3}))\circ[E(\iota_{1})\,\,E(\iota_{2})]=[\operatorname{id}\,\,\operatorname{id}]$ with the fact that (2.2) is an isomorphism, we conclude that $E(\pi_{2})=E(\pi_{1})+E(\pi_{3})$. The proof follows now from the equalities $\pi_{1}\circ F=F_{1}$, $\pi_{2}\circ F=F_{2}$, and $\pi_{3}\circ F=F_{3}$. ∎

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $\mathbb{N}$-graded $k$-algebra and $\mathrm{grproj}(B)$ the exact category of finitely generated projective $\mathbb{Z}$-graded (right) $B$-modules. The following general computation is also of independent interest:

Proposition 2.3**.**

We have an isomorphism $E({\mathcal{D}}_{\mathrm{dg}}^{b}(\mathrm{grproj}(B)))\simeq\oplus_{-\infty}^{+\infty}E(B_{0})$.

Proof.

Consider $B_{0}$ as an $\mathbb{N}$-graded $k$-algebra concentrated in degree zero. The canonical inclusion $B_{0}\to B$ and projection $B\to B_{0}$ of $\mathbb{N}$-graded $k$-algebras give rise to the following base-change exact functors:

[TABLE]

Since $\psi\circ\varphi=\operatorname{id}$, it follows from Lemma 2.5 below that $\varphi$ and $\psi$ give rise to inverse isomorphisms between $E({\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{grproj}(B)))$ and $E({\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{grproj}(B_{0})))$. Now, note that we have the following canonical equivalence of exact categories

[TABLE]

where $\mathrm{proj}(B_{0})$ stands for the exact category of finitely generated projective (right) $B_{0}$-modules. Since the dg category ${\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{proj}(B_{0}))$ is Morita equivalent to the $k$-algebra $B_{0}$ and the functor $E$ is co-continuous, we then conclude from the equivalence (2.4) that $E({\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{grproj}(B_{0})))\simeq\oplus^{+\infty}_{-\infty}E(B_{0})$. This finishes the proof. ∎

Lemma 2.5**.**

The following endomorphism is equal to the identity

[TABLE]

Proof.

Let $P\in\mathrm{grproj}(B)$. Note first that the exact endofunctor $\varphi\circ\psi$ of $\mathrm{grproj}(B)$ is given by $P\mapsto\bigoplus_{n\in\mathbb{Z}}\psi(P)_{n}\otimes_{B_{0}}B(-n)$. Since the functor $E$ is co-continuous, this yields the following equality

[TABLE]

Given a finitely generated projective $\mathbb{Z}$-graded (right) $B$-module $P$ and an integer $m\in\mathbb{Z}$, let us write $F_{m}(P)$ for the $\mathbb{Z}$-graded (right) $B$-submodule of $P$ generated by the elements $\bigcup_{n\leq m}P_{n}$. In the same vein, given an integer $q\geq 0$, let us denote by $\mathrm{grproj}_{q}(B)$ the full subcategory of $\mathrm{grproj}(B)$ consisting of those $\mathbb{Z}$-graded (right) $B$-module $P$ such that $F_{-(q+1)}(P)=0$ and $F_{q}(P)=P$. Note that by definition we have an exhaustive increasing filtration

[TABLE]

As explained by Quillen in [17, pages 99-100], for every $m\in\mathbb{Z}$, the assignment $P\mapsto F_{m}(P)/F_{m-1}(P)$ is an exact endofunctor of $\mathrm{grproj}(B)$. Moreover, we have a canonical isomorphism of exact functors between $\psi(-)_{m}\otimes_{B_{0}}B(-m)$ and $F_{m}(-)/F_{m-1}(-)$. Consequently, we obtain the following equality

[TABLE]

Now, note that every $\mathbb{Z}$-graded (right) $B$-module $P\in\mathrm{grproj}_{q}(B)$ admits a canonical filtration $0=F_{-(q+1)}(P)\subset\cdots\subset F_{q}(P)=P$. This yields a sequence $0=F_{-(q+1)}(-)\to\cdots\to F_{q}(-)=\operatorname{id}$ of exact endofunctors of $\mathrm{grproj}_{q}(B)$. Consequently, an inductive argument using the above general Proposition 2.1 implies that the sum $\sum_{n=-q}^{q}E(F_{n}(-)/F_{n-1}(-))$ is equal to the identity of $E({\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{grproj}_{q}(B)))$. Finally, using the fact that the above filtration (2.7) of $\mathrm{grproj}(B)$ is exhaustive and that the functor $E$ is co-continuous, we hence conclude that

[TABLE]

The proof follows now from the combination of (2.6) with (2.8)-(2.9). ∎

Recall that $A$ is a (connected and locally finite-dimensional) $\mathbb{N}$-graded Noetherian $k$-algebra, which we assume to be Koszul and of finite global dimension $d$.

Proposition 2.10**.**

We have a short exact sequence of dg categories

[TABLE]

Proof.

As explained by Keller in [9, Thm. 4.11], (2.11) is a short exact sequence of dg categories if and only if the associated sequence of triangulated categories

[TABLE]

is exact sequence in the sense of Verdier. By definition, we have a short exact sequence of abelian categories $0\to\mathrm{tors}(A)\to\mathrm{gr}(A)\to\mathrm{qgr}(A)\to 0$. Therefore, thanks to [10, Lem. 1.15] (consult also [7]), in order to show that (2.12) is exact in the sense of Verdier, it suffices to prove the following condition: given a short exact sequence $0\to L\to M\to N\to 0$ in the abelian category $\mathrm{gr}(A)$, with $L\in\mathrm{tors}(A)$, there exists a morphism of short exact sequences

[TABLE]

with $L^{\prime}$ and $L^{\prime\prime}$ belonging to $\mathrm{tors}(A)$. Recall that the category $\mathrm{tors}(A)$ of torsion $A$-modules is defined as the full subcategory of $\mathrm{gr}(A)$ consisting of those $\mathbb{Z}$-graded (right) $A$-modules which are (globally) finite-dimensional over $k$. Given a $\mathbb{Z}$-graded (right) $A$-module $M$ and an integer $m\in\mathbb{Z}$, let us write $M_{\geq m}$ for the (right) $A$-submodule $\bigoplus_{n\geq m}M_{n}$ of $M$. Since by assumption $L$ is torsion and $M$ is finitely generated, there exists an integer $m\gg 0$ such that $L\cap M_{\geq m}=0$. Consequently, we can construct the following morphism of short exact sequences

[TABLE]

The proof follows now from the fact that, by construction, the $\mathbb{Z}$-graded (right) $A$-modules $M/M_{\geq m}$ and $M/\langle M_{\geq m}+L\rangle$ belong to $\mathrm{tors}(A)$. ∎

Remark 2.13*.*

By assumption, the functor $E$ is localizing. Therefore, the short exact exact sequence of dg categories (2.11) gives rise to a distinguished triangle:

[TABLE]

Since $A$ has finite global dimension, the inclusion of categories $\mathrm{grproj}(A)\subset\mathrm{gr}(A)$ induces a Morita equivalence ${\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{grproj}(A))\to{\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{gr}(A))$. Therefore, by first using the general Proposition 2.3 (with $B=A$) and then by applying the functor $E$ to the preceding Morita equivalence, we obtain an induced isomorphism

[TABLE]

Proposition 2.15**.**

We have a Morita equivalence

[TABLE]

where $A^{!}$ stands for the Koszul dual $k$-algebra of $A$.

Proof.

Given a $\mathbb{N}$-graded $k$-algebra $B=\bigoplus_{n\geq 0}B_{n}$, let us denote by $\mathrm{Gr}(B)$ the category of all $\mathbb{Z}$-graded (right) $B$-modules and by ${\mathcal{D}}(\mathrm{Gr}(B))$ the associated (unbounded) derived category. Following Beilinson-Ginzburg-Soergel [2, §2.12], let ${\mathcal{D}}^{\downarrow}(\mathrm{Gr}(B))$, resp. ${\mathcal{D}}^{\uparrow}(\mathrm{Gr}(B))$, be the full subcategory of ${\mathcal{D}}(\mathrm{Gr}(B))$ consisting of those cochain complexes of $\mathbb{Z}$-graded (right) $B$-modules $M$ such that for some integer $m\gg 0$ we have $M^{q}_{n}\neq 0\Rightarrow(q\geq-m\,\,\mathrm{or}\,\,q+n\leq m)$, resp. $M^{q}_{n}\neq 0\Rightarrow(q\leq-m\,\,\mathrm{or}\,\,q+n\geq-m)$. These categories admit canonical dg enhancements ${\mathcal{D}}_{\mathrm{dg}}(\mathrm{Gr}(B))$, ${\mathcal{D}}_{\mathrm{dg}}^{\downarrow}(\mathrm{Gr}(B))$, and ${\mathcal{D}}_{\mathrm{dg}}^{\uparrow}(\mathrm{Gr}(B))$. Now, recall from [2, Thm. 2.12.1] (consult also [5, §2]) the construction of the Koszul duality dg functor ${\mathcal{D}}_{\mathrm{dg}}(\mathrm{Gr}(A))\to{\mathcal{D}}_{\mathrm{dg}}(\mathrm{Gr}(A^{!}))$. As proved in loc. cit., this dg functor restricts to a Morita equivalence

[TABLE]

which sends the $\mathbb{Z}$-graded (right) $A$-modules $k(i),i\in\mathbb{Z}$, to the $\mathbb{Z}$-graded (right) $A^{!}$-modules $\Sigma^{-i}A^{!}(i),i\in\mathbb{Z}$. Therefore, making use of the general Lemma 2.18 below (with $B=A$ and $B=A^{!}$), we conclude that (2.17) restricts furthermore to the above Morita equivalence (2.16). ∎

Lemma 2.18**.**

*Let $B=\bigoplus_{n\geq 0}B_{n}$ be a (connected and locally finite-dimensional) $\mathbb{N}$-graded Noetherian $k$-algebra. The smallest thick triangulated subcategory of ${\mathcal{D}}^{b}(\mathrm{gr}(B))$ containing the $\mathbb{Z}$-graded (right) $B$-modules $\{k(i)\,|\,i\in\mathbb{Z}\}$, resp. $\{B(i)\,|\,i\in\mathbb{Z}\}$, agrees with ${\mathcal{D}}^{b}(\mathrm{tors}(B))$, resp. ${\mathcal{D}}^{b}(\mathrm{grproj}(B))$. *

Proof.

Consult the proof of [15, Lem. 2.3]. ∎

Recall that since $A$ is connected, its Koszul dual $k$-algebra $A^{!}$ is also connected. Therefore, by first applying the functor $E$ to (2.16) and then by using the above general Proposition 2.3 (with $B=A^{!}$), we obtain an induced isomorphism

[TABLE]

Since $A$ is Koszul and of finite global dimension $d$, we have a linear free resolution

[TABLE]

of the $\mathbb{Z}$-graded (right) $A$-module $k$. As mentioned in §1, the integer $\beta_{i}$ agrees with the dimension of the $k$-vector space $\mathrm{Tor}^{A}_{i}(k,k)$ (or $\mathrm{Ext}_{A}^{i}(k,k)$).

Proposition 2.21**.**

Under the above isomorphisms (2.14) and (2.19), the distinguished triangle of Remark 2.13 identifies with

[TABLE]

*where $\mathrm{M}^{\prime}$ stands for the (infinite) matrix $\mathrm{M}^{\prime}_{ij}:=(-1)^{j}(-1)^{(i-j)}\beta^{\prime}_{i-j}$. *

Proof.

Let $\operatorname{NMot}(k)$ be the category of noncommutative motives constructed in [22, §8.2]; denoted by $\operatorname{NMot}(k)_{\mathrm{loc}}$ in loc. cit. By construction, this triangulated category comes equipped with a functor $U\colon\mathrm{dgcat}(k)\to\operatorname{NMot}(k)$ which is initial among all the functors satisfying conditions (i)-(iii). Concretely, given a functor $E\colon\mathrm{dgcat}(k)\to{\mathcal{T}}$ satisfying conditions (i)-(iii), there exists a (unique) triangulated functor $\overline{E}\colon\operatorname{NMot}(k)\to{\mathcal{T}}$ such that $\overline{E}\circ U\simeq E$. Moreover, $\overline{E}$ preserves arbitrary direct sums; see [22, Thm. 8.5]. This implies that in order to prove Theorem 2.21, it suffices to show that the triangle of Remark 2.13 (with $E=U$) identifies with

[TABLE]

where $\mathrm{M}$ stands for the (infinite) matrix $\mathrm{M}_{ij}:=(-1)^{j}(-1)^{(i-j)}\beta_{i-j}$. Recall from [22, §8.6] that, for every dg category ${\mathcal{A}}$, we have a natural isomorphism

[TABLE]

Moreover, $U(k)$ is a compact object of the triangulated category $\operatorname{NMot}(k)$. Therefore, since $K_{0}(k)\simeq\mathbb{Z}$, an endomorphism of $\oplus_{-\infty}^{+\infty}U(k)$ corresponds to an infinite matrix with integer coefficients in which every column has solely a finite number of non-zero entries. Let us denote by $\mathrm{M}$ the matrix corresponding to $U({\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{tors}(A)))\to U({\mathcal{D}}^{b}_{\mathrm{dg}}(\mathrm{gr}(A)))$ under the isomorphisms (2.14) and (2.19) (with $E=U$). By applying the functor $\mathrm{Hom}_{\operatorname{NMot}(k)}(U(k),-)$ to the isomorphisms (2.14) and (2.19) (with $E=U$), we obtain induced abelian group isomorphisms

[TABLE]

[TABLE]

The element $1\in\mathbb{Z}$, placed at the $j^{\mathrm{th}}$ component of the direct sum $\oplus_{-\infty}^{+\infty}\mathbb{Z}$, corresponds under (2.25) to the Grothendieck class $[\Sigma^{-j}k(-j)]=(-1)^{j}[k(-j)]\in K_{0}({\mathcal{D}}^{b}(\mathrm{tors}(A)))$. In the same vein, the element $1\in\mathbb{Z}$, placed at the $i^{\mathrm{th}}$ component of the direct sum $\oplus_{-\infty}^{+\infty}\mathbb{Z}$, corresponds under (2.24) to the Grothendieck class $[A(-i)]\in K_{0}({\mathcal{D}}^{b}(\mathrm{gr}(A)))$. Thanks to the above linear free resolution (2.20), we have moreover the following equality $[k(-j)]=\sum_{i=0}^{d}(-1)\beta_{i}[A(-i-j)]$ in the Grothendieck group $K_{0}({\mathcal{D}}^{b}(\mathrm{gr}(A)))$. The above considerations allow us to conclude that the $(i,j)^{\mathrm{th}}$ entry of the matrix $\mathrm{M}$ is given by the integer $(-1)^{j}(-1)^{(i-j)}\beta_{i-j}$. This finishes the proof. ∎

We now have all the ingredients necessary for the conclusion of the proof of Theorem 1.2(i). Let $o\in{\mathcal{T}}$ be a compact object. By applying the functor $\mathrm{Hom}_{\mathcal{T}}(o,-)$ to the triangle (2.22), we obtain an induced long exact sequence of $R$-modules:

[TABLE]

Since $\mathrm{M}^{\prime}_{ij}=(-1)^{j}(-1)^{(i-j)}\beta^{\prime}_{i-j}$, with $\beta^{\prime}_{0}=1$ and $\beta^{\prime}_{r}=0$ whenever $r\notin\{0,\ldots,d^{\prime}\}$, a simple matrix computation shows that the preceding homomorphism $\mathrm{M}^{\prime}$ of $R$-modules is injective. Consequently, the long exact sequence breaks-up into short exact sequences of $R$-modules:

[TABLE]

Thanks to Lemma 2.28 below and to the definition of the homomorphism $\phi$ (see below), we also have the following short exact sequences of $R$-modules:

[TABLE]

Now, consider the Poincaré polynomial $p_{A}(t):=\sum^{d}_{i=0}(-1)^{i}\beta_{i}t^{i}$ (and $p^{\prime}_{A}(t):=\sum_{i=0}^{d^{\prime}}(-1)^{i}\beta^{\prime}_{i}t_{i}$). Thanks to the linear free resolution (2.20), we have $h_{A}(t)^{-1}=p_{A}(t)$ (and $h^{\prime}_{A}(t)^{-1}=p^{\prime}_{A}(t)$). This implies that under the canonical isomorphism between $\oplus_{-\infty}^{+\infty}E^{o}_{m}(k)$ and $R[t,t^{-1}]\otimes_{R}E^{o}_{m}(k)$, the matrix $\mathrm{M}^{\prime}$ corresponds to the homomorphism $\phi\otimes\operatorname{id}$. Consequently, we obtain induced $R$-module isomorphisms

[TABLE]

This concludes the proof of Theorem 1.2(i).

Lemma 2.28**.**

We have the following short exact sequence of $R$-modules

[TABLE]

where $\phi$ stands for the homomorphism $p(t)\mapsto p(-t)\cdot h^{\prime}_{A}(t)^{-1}$.

Proof.

Since $h^{\prime}_{A}(0)^{-1}=1$, the homomorphism $\phi$ is injective. Moreover, we have the following natural isomorphisms

[TABLE]

where $(a)$ follows from the fact that the homomorphisms $\phi$ and $-\cdot h^{\prime}_{A}(t)^{-1}$ have the same image, and $(b)$ from the fact that the polynomial $t$ is invertible in $R[t]/\langle h^{\prime}_{A}(t)^{-1}\rangle$ (this follows from the fact that $h^{\prime}_{A}(0)^{-1}=1$). This concludes the proof. ∎

We now have all the ingredients necessary for the conclusion of the proof of Theorem 1.2(ii). Consider the following composition

[TABLE]

By assumption, the triangulated category ${\mathcal{T}}$ is compactly generated. Therefore, the morphism (2.29) is invertible if and only if for every compact object $o\in{\mathcal{T}}$ the induced $R$-module homomorphisms

[TABLE]

are invertible. Under the canonical identification $\oplus_{n=0}^{d^{\prime}-1}R\otimes_{R}E^{o}_{m}(k)\simeq\oplus_{n=0}^{d^{\prime}-1}E^{o}_{m}(k)$, the composition of (2.30) with (2.27) corresponds to the $R$-module homomorphisms:

[TABLE]

By assumption, we have $1/\beta^{\prime}\in R$. Therefore, the factorization algorithm for polynomials applied to $R[t]$ allows us to conclude that the $R$-module homomorphism $(1,t,\ldots,t^{d^{\prime}-1})$ is invertible. This implies that the induced $R$-module homomorphisms (2.30) are also invertible, and so the proof of Theorem 1.2(ii) is finished. Acknowledgments: The author is grateful to Michael Artin for useful discussions concerning noncommutative projective schemes and also to Theo Raedschelders for important comments on a previous version of this note.