1.1 Conventions on algebras

The letter $\mathbf{k}$ will denote an algebraically closed field. All our algebras $A$ will be $\mathbf{k}$ -algebras, and whenever we speak of a graded algebra we mean a graded $\mathbf{k}$ -algebra $A=\bigoplus _{i\in \mathbb{Z}}A^{i}$ .

For a (graded) $\mathbf{k}$ -algebra $A$ , we denote by the (graded) tensor product $A^{e}:=A\otimes _{\mathbf{k}}A^{\text{op}}$ its enveloping algebra. Whenever we speak in the following about ideals or modules over some (not necessarily commutative) algebra, we refer to finitely generated left ideals or modules. The formalism of enveloping algebras allows us to speak of $A\,-A$ -bimodules as left $A^{e}$ -modules.

1.2 Conventions on triangulated categories

All triangulated categories are assumed to be $\mathbf{k}$ -linear, and subcategories thereof to be triangulated and full. The shift functor will be denoted by $[1]$ . All triangles are meant to be distinguished and denoted by $A\rightarrow B\rightarrow C$ , hiding the morphism $C\rightarrow A[1]$ . We write $\mathsf{Hom}^{\ast }(A,B)=\bigoplus _{i\in \mathbb{Z}}\mathsf{Hom}(A,B[i])[-i]$ for the derived homomorphisms in a triangulated category; this is a complex equipped with the zero differential. In contrast, $\mathsf{Hom}^{\bullet }(A,B)$ will be the homomorphism complex if $A$ and $B$ are objects of a dg-category, usually an enhancement of a triangulated category. In the literature this is sometimes also denoted by $\operatorname{RHom}(A,B)$ .

All functors between triangulated categories are meant to be exact. In particular, we will abuse notation and write $\otimes$ for the derived functor $\otimes ^{L}$ , using the same symbol as for the functor between abelian categories.

1.3 dg-algebras and Hochschild cohomology

Definition 1.1. An dg-algebra $A$ (over $\mathbf{k}$ ) consists of a graded $\mathbf{k}$ -vector space

$$\begin{eqnarray}A=\bigoplus _{i\in \mathbb{Z}}A^{i}\end{eqnarray}$$

and graded $\mathbf{k}$ -linear maps

1. – $d:A\rightarrow A$ of degree $1$ and

2. – $m:A\otimes A\rightarrow A$ of degree $0$

satisfying the following compatibilities:

1. – $d^{2}=0$ , so $d$ is a differential;

2. – $m(m\otimes \operatorname{id})=m(\operatorname{id}\otimes m)$ , so $m$ is an associative multiplication;

3. – $d(m(a,b))=m(da,b)+(-1)^{\mathsf{deg}\,a}m(a,db)$ for homogeneous elements (the Leibniz rule).

A morphism $\unicode[STIX]{x1D719}:A\rightarrow B$ of dg-algebras is a $\mathbf{k}$ -linear map compatible with differential and multiplication, that is, $\unicode[STIX]{x1D719}(d_{A}a)=d_{B}(\unicode[STIX]{x1D719}(a))$ and $\unicode[STIX]{x1D719}(m_{A}(a,b))=m_{B}(\unicode[STIX]{x1D719}(a),\unicode[STIX]{x1D719}(b))$ . Note that $\unicode[STIX]{x1D719}$ induces a map on cohomology $\mathsf{H}(\unicode[STIX]{x1D719}):\mathsf{H}(A)\rightarrow \mathsf{H}(B)$ .

Recall that we denote by $A^{e}=A\otimes _{\mathbf{k}}A^{\text{op}}$ the enveloping algebra of $A$ . Note that $A$ has a natural $A^{e}$ -module structure by multiplication from left and right. Given $q\in \mathbb{Z}$ and two graded $A^{e}$ -modules $N$ and $M$ , we write $\mathsf{Hom}_{A^{e}}^{q}(N,M)$ for the $A^{e}$ -module homomorphisms which are homogeneous of degree $q$ .

Definition 1.5. The complex

$$\begin{eqnarray}B^{\bullet }:\cdots \rightarrow A^{\otimes (q+2)}\xrightarrow[{}]{d^{-q}}A^{\otimes (q+1)}\rightarrow \cdots \rightarrow A^{\otimes 2}\rightarrow 0,\end{eqnarray}$$

with $A^{\otimes (q+2)}$ in degree $q$ , is called the Bar resolution of $A$ as an $A^{e}$ -module, where $d=d^{-q}$ is given by

$$\begin{eqnarray}d(a_{1}\otimes \cdots \otimes a_{q+2})=\mathop{\sum }_{i}\pm a_{1}\otimes \cdots \otimes a_{i}\cdot a_{i+1}\otimes \cdots \otimes a_{q+2}.\end{eqnarray}$$

Note that the differentials are of degree zero.

Definition 1.6. Let $A$ be a graded algebra and $M$ a graded $A^{e}$ -module. The Hochschild cohomology of $A$ with values in $M$ is given by

$$\begin{eqnarray}\mathsf{HH}^{p,q}(A,M):=\mathsf{H}^{p}(\mathsf{Hom}_{A^{e}}^{q}(B^{\bullet },M))\quad \text{for }p,q\in \mathbb{Z}.\end{eqnarray}$$

Our main tool for proving intrinsic formality of certain graded algebras will be the following result due to Kadeishvili.

Remark 1.9. Note that $\mathsf{Hom}_{A^{e}}^{q}(A,A)=\mathsf{Hom}_{A^{e}}^{0}(A,A(q))$ . Moreover, graded $A^{e}$ -modules, with $\mathsf{Hom}_{A^{e}}^{0}$ as morphisms, form an abelian category $\text{gr}A^{e}$ . There, the Bar resolution is a projective resolution of the $A^{e}$ -module $A$ . It follows that $\mathsf{HH}^{p,q}(A,M)=\mathsf{Ext}_{\text{gr}A^{e}}^{p}(A,M(q))$ . Hence, for any other projective resolution $P^{\bullet }$ of $A$ as a graded $A^{e}$ -module, we also have

$$\begin{eqnarray}\mathsf{HH}^{p,q}(A,M)=\mathsf{H}^{p}(\mathsf{Hom}_{A^{e}}^{q}(P^{\bullet },M))=\mathsf{H}^{p}(\mathsf{Hom}_{A^{e}}^{0}(P^{\bullet },M(q))).\end{eqnarray}$$

1.4 Derived categories and dg-categories

In this paper we will encounter two types of derived categories. First, for an abelian category ${\mathcal{A}}$ there is the category $\mathsf{D}({\mathcal{A}})$ which is the localisation of the homotopy category of complexes with values in ${\mathcal{A}}$ at the class of quasi-isomorphisms. In our examples, the abelian category ${\mathcal{A}}$ will be a category of (coherent or quasi-coherent) ${\mathcal{O}}_{X}$ -modules over a variety or manifold $X$ . For details on $\mathsf{D}({\mathcal{A}})$ see, for example, [Reference HuybrechtsHuy06, ch. 2].

Let us very quickly recall some facts and fix notation concerning dg-categories and enhancements; see, for example, [Reference Kuznetsov and LuntsKL15, §3] for details. A dg-category is a $\mathbf{k}$ -linear category ${\mathcal{E}}$ whose Hom-spaces are dg- $\mathbf{k}$ -modules and the compositions are compatible with the dg-structure. The homotopy category $\mathsf{H}^{0}({\mathcal{E}})$ is defined to have the same objects as ${\mathcal{E}}$ and morphisms $\mathsf{Hom}_{\mathsf{H}^{0}({\mathcal{E}})}(E,F):=\mathsf{H}^{0}(\mathsf{Hom}_{{\mathcal{E}}}(E,F))$ . The category $\mathsf{dg-Mod}({\mathcal{E}})$ of (right) dg-modules over ${\mathcal{E}}$ is defined as the category of dg-functors from ${\mathcal{E}}^{\text{op}}$ to the category of dg-modules over $\mathbf{k}$ . Its homotopy category $\mathsf{H}^{0}(\mathsf{dg-Mod}({\mathcal{E}}))$ carries the structure of a triangulated category. The dg-category ${\mathcal{E}}$ is called pretriangulated if the image of the Yoneda embedding $\mathsf{H}^{0}({\mathcal{E}}){\hookrightarrow}\mathsf{H}^{0}(\mathsf{dg-Mod}({\mathcal{E}}))$ is a triangulated subcategory.

Given a triangulated category ${\mathcal{T}}$ , a dg-enhancement of ${\mathcal{T}}$ is a pretriangulated dg-category ${\mathcal{E}}$ together with an exact equivalence $\unicode[STIX]{x1D6F7}:\mathsf{H}^{0}({\mathcal{E}})\stackrel{{\sim}}{\rightarrow }{\mathcal{T}}$ .

We can consider every dg-algebra $A$ as a dg-category with one object. Then the homotopy category $\mathsf{H}^{0}(\mathsf{dg-Mod}({\mathcal{E}}))$ of dg-modules over that category agrees with the usual notion of the category of dg-modules over the algebra $A$ . The derived category of the a dg-algebra $A$ is defined as $\mathsf{D}(A):=\mathsf{H}^{0}(\mathsf{dg-Mod}(A))[\mathsf{qis}^{-1}]$ , the category $\mathsf{dg-Mod}(A)$ of right dg-modules over $A$ localised at the class of quasi-isomorphisms; for details see, for example, [Reference Keller, Angeleri Hügel, Happel and KrauseKel07, §8]. If the dg-algebra $A$ is concentrated in degree 0 (i.e. it is an ordinary algebra), then $\mathsf{D}(A)=\mathsf{D}(\mathsf{Mod}(A))$ where the latter is the derived category of the abelian category $\mathsf{Mod}(A)$ in the sense explained above.

1.5 Formality and triangulated categories

The following results are well known to experts and can be found in essence or in part in, for example, the survey [Reference Keller, Sanz-Solé, Soria, Varona and VerderaKel06a] by Keller, the lecture notes [Reference Toën and CortiñasToë11] by Toën or the book [Reference Bernstein and LuntsBL94, §10] by Bernstein and Lunts.

We recall terminology. Let ${\mathcal{T}}$ be a triangulated category. The category ${\mathcal{T}}$ is called cocomplete if arbitrary direct summands exist. It is called idempotent complete if every idempotent endomorphism splits. An object $T\in {\mathcal{T}}$ is called compact if for every set $\{Y_{i}\}$ of objects in ${\mathcal{T}}$ the natural morphism

$$\begin{eqnarray}\bigoplus _{i}\mathsf{Hom}(T,Y_{i})\rightarrow \mathsf{Hom}\biggl(T,\bigoplus _{i}Y_{i}\biggr)\end{eqnarray}$$

is an isomorphism. We write ${\mathcal{T}}^{c}\subset {\mathcal{T}}$ for the full subcategory of compact objects. It is a thick (i.e. closed under direct summands) triangulated subcategory of ${\mathcal{T}}$ .

For an object $T\in {\mathcal{T}}$ , we write $\langle \!\langle T\rangle \!\rangle$ for the smallest cocomplete triangulated subcategory of ${\mathcal{T}}$ containing $T$ . We write $\langle T\rangle$ for the smallest thick triangulated subcategory of ${\mathcal{T}}$ containing $T$ . If $T$ is compact, then $\langle T\rangle \subset {\mathcal{T}}^{c}$ . There is an inclusion $\langle T\rangle \subset \langle \!\langle T\rangle \!\rangle$ , since every cocomplete triangulated category is thick; see [Reference NeemanNee01, Proposition 1.6.8].

The compact objects in the derived category $\mathsf{D}(\operatorname{QCoh}(X))$ of coherent sheaves on a separated scheme of finite type over $\mathbf{k}$ coincide with the perfect objects, that is, objects locally quasi-isomorphic to bounded complexes of locally free sheaves of finite rank.

The following statement can be found in a similar form in [Reference KellerKel94, §4.2] or [Reference Lunts and OrlovLO10, Propositions 1.16 and 1.17].

Corollary 1.12. Let ${\mathcal{T}}$ be a cocomplete dg-enhanced triangulated category and $T\in {\mathcal{T}}^{c}$ a compact object. If the graded algebra $A=\mathsf{End}^{\ast }(T)$ is intrinsically formal, then

$$\begin{eqnarray}\langle \!\langle T\rangle \!\rangle \cong \mathsf{D}(A)\quad \text{and}\quad \langle T\rangle \cong \mathsf{D}(A)^{c}.\end{eqnarray}$$

2.1 Definition and basic examples

$\mathbb{P}^{1}[k]$ -objects are well-known as spherical objects; see [Reference Seidel and ThomasST01]. Without the Calabi–Yau property they are called spherelike objects and studied in [Reference Hochenegger, Kalck and PloogHKP16, Reference Hochenegger, Kalck and PloogHKP19] by Kalck, Ploog and the first author.

$\mathbb{P}^{n}[2]$ -objects are known as $\mathbb{P}^{n}$ -objects and studied in [Reference Huybrechts and ThomasHT06] by Huybrechts and Thomas. The focus there is on hyperkähler manifolds, whose structure sheaves are $\mathbb{P}^{n}$ -objects. Another standard example of a $\mathbb{P}^{n}$ -object is the structure sheaf ${\mathcal{O}}_{Z}\in \mathsf{D}(X)$ of the centre ${\mathcal{O}}_{\mathbb{P}^{n}}\cong Z\subset X$ of a Mukai flop of a variety of dimension $\dim X=2n$ .

The terminology $\mathbb{P}^{n}[k]$ was introduced by the second author in [Reference KrugKru18], where examples of varieties are also given, whose structure sheaves are $\mathbb{P}^{n}[k]$ -objects.

2.2 Associated $\mathbb{P}$ -twists

In this subsection we assume that ${\mathcal{T}}$ is a $\mathbf{k}$ -linear triangulated category that admits a dg-enhancement and that the $\mathbb{P}^{n}[k]$ -object $P\in {\mathcal{T}}$ is proper, that is, $\mathsf{Hom}^{\ast }(P,F)$ is a finite-dimensional graded vector space for all $F\in {\mathcal{T}}$ . Under these assumptions there is an autoequivalence $\mathsf{P}\!_{P}:{\mathcal{T}}\stackrel{{\sim}}{\rightarrow }{\mathcal{T}}$ , the $\mathbb{P}$ -twist along $P$ , whose construction, due to [Reference Huybrechts and ThomasHT06], we sketch in the following. Whenever we speak about the $\mathbb{P}$ -twist associated to a $\mathbb{P}$ -like object in later sections, we will tacitly assume that these assumptions are met.

So, let $P$ be a $\mathbb{P}^{n}[k]$ -object and $t$ be a non-zero element of $\mathsf{Ext}^{k}(P,P)$ . Using this generator, one can define the upper triangle for any $F\in {\mathcal{T}}$ :

As the composition $\text{ev}\circ H=0$ , the arrow $\operatorname{Cone}(H){\dashrightarrow}F$ exists. Completing this arrow to a triangle gives the double cone which we denote by $\mathsf{P}\!_{P}(F)$ .

2.3 Formality of single $\mathbb{P}$ -objects

The following proposition is Theorem A in the introduction.

Proposition 2.7. Let $P$ be a $\mathbb{P}^{n}[k]$ -like object in a cocomplete $\mathbf{k}$ -linear dg-enhanced triangulated category with $n,k$ positive integers. Then there are equivalences

$$\begin{eqnarray}\langle \!\langle P\rangle \!\rangle \cong \mathsf{D}(\mathsf{End}^{\ast }(P))\quad \text{and}\quad \langle P\rangle \cong \mathsf{D}(\mathsf{End}^{\ast }(P))^{c}.\end{eqnarray}$$

Proof. In order to apply the criterion of Proposition 1.7 for intrinsic formality, we have to show the vanishing of the Hochschild cohomology groups $\mathsf{HH}^{q,2-q}(A,A)$ for $q>2$ .

There is the well-known 2-periodic free resolution

$$\begin{eqnarray}\cdots \rightarrow \text{}\underline{A}^{e}\xrightarrow[{}]{v}\text{}\underline{A}^{e}\xrightarrow[{}]{u}\text{}\underline{A}^{e}\xrightarrow[{}]{v}\text{}\underline{A}^{e}\xrightarrow[{}]{u}\text{}\underline{A}^{e}\xrightarrow[{}]{m}\text{}\underline{A}\rightarrow 0\end{eqnarray}$$

of the underlying non-graded algebra $\text{}\underline{A}=\mathbf{k}[t]/t^{n+1}$ considered as the diagonal bimodule over itself. Here, $m$ is multiplication in $A$ , $u$ is multiplication by $t\otimes 1-1\otimes t$ , and $v$ is multiplication by $t^{n}\otimes 1+t^{n-1}\otimes t+\cdots +1\otimes t^{n}$ ; see [Reference WeibelWei94, Example 9.1.4]. Now one can check easily that this becomes a graded free resolution

$$\begin{eqnarray}\cdots \xrightarrow[{}]{v}A^{e}(-(n+2)k)\xrightarrow[{}]{u}A^{e}(-(n+1)k)\xrightarrow[{}]{v}A^{e}(-k)\xrightarrow[{}]{u}A^{e}\xrightarrow[{}]{m}A\rightarrow 0.\end{eqnarray}$$

So we obtain a graded free resolution $F^{\bullet }$ of the $A^{e}$ -module $A$ where

$$\begin{eqnarray}F^{q}=\left\{\begin{array}{@{}ll@{}}A^{e}(-i(n+1)k)\quad & \text{for }q=2i\text{ even,}\\ A^{e}(-(i(n+1)+1)k)\quad & \text{for }q=2i+1\text{ odd.}\end{array}\right.\end{eqnarray}$$

By Remark 1.9, $\mathsf{HH}^{q,2-q}(A,A)$ is a subquotient of $\mathsf{Hom}_{A^{e}}^{0}(F^{q},A(2-q))$ so it is sufficient to show that the latter vanishes for $q>2$ . For $q=2i$ even,

$$\begin{eqnarray}\mathsf{Hom}_{A^{e}}^{0}(F^{q},A(2-q))=\mathsf{Hom}_{A^{e}}^{0}(A^{e},A(2-2i+i(n+1)k))=A^{2-2i+i(n+1)k}.\end{eqnarray}$$

We have $2-2i+i(n+1)k=2+i(nk+k-2)>nk$ for $i\geqslant 2$ . But $A$ is concentrated in degrees between $0$ and $nk$ , so we get

$$\begin{eqnarray}\mathsf{Hom}_{A^{e}}^{0}(F^{2i},A(2-2i))=A^{2-2i+i(n+1)k}=0.\end{eqnarray}$$

The verification that $\mathsf{Hom}_{A^{e}}^{0}(F^{q},A(2-q))=0$ for $q>2$ odd is similar.◻

Proof. This is proved in [Reference SegalSeg18, Proposition 4.2] under the assumption that $\mathsf{End}^{\bullet }(P)$ is formal. By Lemma 2.8, this assumption is always satisfied.

To be precise, Segal’s assumption is that the dg-algebra $\mathsf{End}^{\bullet }(P)$ is formal as a dg-module over $B$ , so we have to show that this is implied by its formality as a dg-algebra. The $B$ -module structure is given by choosing an isomorphism $\mathsf{End}^{\ast }(P)\cong \mathbf{k}[s]/s^{n+1}$ and an element $u\in \mathsf{End}^{\bullet }(P)$ whose cohomology class is mapped to $s$ under this isomorphism. Then $t^{l}$ acts on $\mathsf{End}^{\bullet }(P)$ by multiplication by $u^{l}$ . Now, we know that $\mathsf{End}^{\bullet }(P)$ is formal as a dg-algebra. Hence there is a roof

where $f$ and $g$ are quasi-isomorphisms of dg-algebras and $f$ is surjective. Indeed one can take $f:W^{\bullet }\rightarrow \mathsf{End}^{\bullet }(P)$ to be a cofibrant replacement with respect to the structure of a model category on the category of augmented dg-algebras as described in [Reference Keller, de la Peña and BautistaKel06b, §4.2].

Let $v\in W^{\bullet }$ be a preimage of $u$ under $f$ . Then we can equip $W^{\bullet }$ with the structure of a $B$ -algebra by letting $t$ act by $v$ so that $f$ becomes an quasi-isomorphism of $B$ -modules. Furthermore, the cohomology class of $v$ is non-zero, hence $g(v)$ is a non-zero multiple of $s$ . Therefore, $g$ is a quasi-isomorphism of $B$ -modules, too.◻

3.1 Separably augmented algebras and resolutions of diagonal bimodules

We recall that a $\mathbf{k}$ -algebra $R$ is separable if and only if there is an element $p=\sum x_{i}\otimes y_{i}\in R^{e}$ (called separability idempotent), such that $ap=pa$ for all $a\in R$ and $\sum x_{i}y_{i}=1$ in $R$ . For general facts on separable algebras see the textbook by Weibel [Reference WeibelWei94, §9.2].

We denote by $\mathbb{N}$ the semigroup of non-negative integers. A separably augmented $\mathbf{k}$ -algebra is an $\mathbb{N}$ -graded $\mathbf{k}$ -algebra such that $R=A^{0}$ is a separable $\mathbf{k}$ -algebra.

For a separable $\mathbf{k}$ -algebra $R$ and an $R\,-R$ -bimodule $V$ , we denote by $T(V)=R\,\oplus \,V\,\oplus \,(V\otimes _{R}V)\oplus \cdots \,$ its (free) tensor algebra over $R$ . Moreover, let $J\subset T(V)$ be the two-sided ideal generated by $V$ . If $V$ carries an $\mathbb{N}_{+}$ -grading, where $\mathbb{N}_{+}$ denotes the positive integers, the tensor algebra inherits a canonical $\mathbb{N}$ -grading given by

$$\begin{eqnarray}\mathsf{deg}(v_{1}\otimes v_{2}\otimes \cdots \otimes v_{n})=\mathsf{deg}(v_{1})+\mathsf{deg}(v_{2})+\cdots +\mathsf{deg}(v_{n}).\end{eqnarray}$$

Then $T(V)^{0}=R$ , so that $T(V)$ is a separably augmented algebra with $T(V)^{+}=J$ . Conversely, every separably augmented algebra with $A^{0}=R$ has a graded surjection $T(V){\twoheadrightarrow}A$ for some $R\,-R$ -bimodule $V$ , for example $V=A^{+}$ . Given a separably augmented algebra $A$ and a graded surjection $T(V){\twoheadrightarrow}A$ with kernel $I$ , we call the induced isomorphism $A\cong T(V)/I$ a tensor presentation of $A$ .

Replacing $V$ by $V/(V\cap I)$ if necessary, we may always choose a presentation such that $I\subset J^{2}$ . In the following, we will also assume that the inclusion $J^{n}\subset I$ holds for some $n\geqslant 2$ . This is automatic if $A$ is of finite dimension over $\mathbf{k}$ , as it will be in the applications.

Proposition 3.2 (Cf. [Reference Butler and KingBK99, Proposition 2.4]).

Let $A$ be a separably augmented algebra with $A^{0}=R$ . A minimal resolution $P^{\bullet }$ of $A$ as a graded $A^{e}$ -module has terms $P^{m}=A\otimes _{R}\mathsf{Tor}_{m}^{A}(R,R)\otimes _{R}A$ .

Moreover, suppose that $A\cong T(V)/I$ is a tensor presentation with $J^{n}\subset I\subset J^{2}$ for some $n\geqslant 2$ . Then there are isomorphisms of graded $R$ -algebras

$$\begin{eqnarray}\mathsf{Tor}_{2p}^{A}(R,R)=\frac{I^{p}\cap JI^{p-1}J}{JI^{p}+I^{p}J}\quad \text{and}\quad \mathsf{Tor}_{2p+1}^{A}(R,R)=\frac{JI^{p}\cap I^{p}J}{I^{p+1}+JI^{p}J},\end{eqnarray}$$

where the grading on the left-hand side is induced by the grading on $A$ and the one on the right-hand side is induced by the grading on $T(V)$ .

Proof. This follows by setting $L=T(V)^{e}$ in [Reference Butler and KingBK99, Proposition 2.4]. Unfortunately, Butler and King assume that the grading of $A$ is induced by the natural grading of $T(V)$ (i.e. the elements of $V$ have degree $1$ ), which will never be the case in our applications.

However, one can check that every step of the proof of [Reference Butler and KingBK99, Proposition 2.4] works in our general graded set-up. Indeed, the proof of the equality $P^{m}=A\,\otimes _{R}\mathsf{Tor}_{m}^{A}(R,R)\,\otimes _{R}A$ mainly refers to Eilenberg [Reference EilenbergEil56], who works in the general graded setting of separably augmented algebras throughout; compare Remark 3.1. The arguments in [Reference Butler and KingBK99] needed for this equality can all be turned into arguments that also work in our graded set-up using the fact that an object in the category of graded $A$ -modules is projective if and only if the underlying non-graded $\text{}\underline{A}$ -module is projective; see, for example, [Reference EilenbergEil56, §1].

The computation of $\mathsf{Tor}_{m}^{A}(R,R)$ in terms of the ideals $I$ and $J$ is done using the projective resolution

$$\begin{eqnarray}\cdots \rightarrow \frac{JI^{n}}{JI^{n+1}}\rightarrow \frac{I^{n}}{I^{n+1}}\rightarrow \frac{JI^{n-1}}{JI^{n}}\rightarrow \cdots \rightarrow \frac{JI}{I}\rightarrow A\rightarrow R\rightarrow 0;\end{eqnarray}$$

see also [Reference BongartzBon83] for details. Its differentials are induced by the inclusions of the homogeneous ideals $I$ and $J$ , hence are graded homomorphisms.◻

Remark 3.3. Let $P^{\bullet }\rightarrow A$ be the minimal resolution of $A$ as in Proposition 3.2. Note that there is a natural isomorphism

$$\begin{eqnarray}\mathsf{Hom}_{A^{e}}(P^{q},\hspace{2.22198pt}\cdot \hspace{2.22198pt})=\mathsf{Hom}_{R^{e}}(\mathsf{Tor}_{q}^{A}(R,R),\hspace{2.22198pt}\cdot \hspace{2.22198pt}).\end{eqnarray}$$

3.2 Degree criterion for intrinsic formality

Definition 3.4. For a graded module $M$ we define the maximal degree of $M$ as

$$\begin{eqnarray}\mathsf{maxdeg}(M):=\mathsf{max}\{\mathsf{deg}(m)\mid \text{non-zero homogeneous }m\in M\}\end{eqnarray}$$

and analogously the minimal degree $\mathsf{mindeg}(M)$ .

Proposition 3.5. Let $A$ be a separably augmented algebra and let $q\in \mathbb{N}$ . If

$$\begin{eqnarray}\mathsf{maxdeg}(A)+q-2<\mathsf{mindeg}(\mathsf{Tor}_{q}^{A}(R,R))\end{eqnarray}$$

then the Hochschild cohomology $\mathsf{HH}^{q,2-q}(A,A)$ vanishes. In particular, if this inequality holds for all $q>2$ , then $A$ is intrinsically formal.

Proof. By Remarks 1.9 and 3.3, the Hochschild cohomology $\mathsf{HH}^{q,2-q}(A,A)$ is a subquotient of

$$\begin{eqnarray}\mathsf{Hom}_{R^{e}}(\mathsf{Tor}_{q}^{A}(R,R),A(2-q)).\end{eqnarray}$$

Hence, it is sufficient to show the vanishing of this $\mathsf{Hom}$ -space. But there cannot be any non-zero homomorphism of degree zero, since the minimal degree of the source is smaller than the maximal degree of the target.

Recall that $\mathsf{HH}^{q,2-q}(A,A)=0$ for $q>2$ implies intrinsic formality of $A$ by Proposition 1.7.◻

We will use Proposition 3.5 to prove intrinsic formality of a given separably augmented algebra using a suitable tensor representation, namely in the case of endomorphism algebras of configurations of $\mathbb{P}$ -objects.

Remark 3.6. Let $A$ be an $\mathbb{N}$ -graded $\mathbf{k}$ -algebra, $P$ a graded $A$ -module, and $M,N\subset P$ graded submodules. Then the following rules hold:

1. – $\mathsf{mindeg}(M+N)=\mathsf{min}\{\mathsf{mindeg}(M),\mathsf{mindeg}(N)\}$ ; hence $\mathsf{mindeg}(M)=\mathsf{min}\{\mathsf{deg}(m_{i})\}$ for $M=_{A}\langle m_{1},\ldots ,m_{l}\rangle$ with $m_{i}$ homogeneous;

2. – $\mathsf{mindeg}(M\cap N)\geqslant \mathsf{max}\{\mathsf{mindeg}(M),\mathsf{mindeg}(N)\}$ ;

3. – $\mathsf{mindeg}(P/M)\geqslant \mathsf{mindeg}(P)$ , with equality if $\mathsf{mindeg}(M)>\mathsf{mindeg}(P)$ .

If $I,J\subset A$ are ideals then we have additionally:

1. – $\mathsf{mindeg}(I\cdot J)\geqslant \mathsf{mindeg}(I)+\mathsf{mindeg}(J)$ .

Let $A$ be a separably augmented algebra with tensor representation $A\cong T(V)/I$ and let $J=A^{+}$ so that $\mathsf{mindeg}(J)=\mathsf{mindeg}(V)$ . By Proposition 3.2 and the above rules, we get

$$\begin{eqnarray}\displaystyle \mathsf{mindeg}\,\mathsf{Tor}_{2p}^{A}(R,R) & {\geqslant} & \displaystyle \mathsf{mindeg}(I^{p}\cap JI^{p-1}J)\nonumber\\ \displaystyle & {\geqslant} & \displaystyle \mathsf{max}\{p\,\mathsf{mindeg}(I),2\,\mathsf{mindeg}(J)+(p-1)\mathsf{mindeg}(I)\},\nonumber\\ \displaystyle \mathsf{mindeg}\,\mathsf{Tor}_{2p+1}^{A}(R,R) & {\geqslant} & \displaystyle \mathsf{mindeg}(JI^{p}\cap I^{p}J)\geqslant p\,\mathsf{mindeg}(I)+\mathsf{mindeg}(J).\nonumber\end{eqnarray}$$

4.1 Formality of configurations of $\mathbb{P}$ -objects

Proposition 4.3. Let $Q$ be a graph and let $P_{1},\ldots ,P_{m}$ be a $Q$ -configuration consisting of $\mathbb{P}^{n}[k]$ -like objects in a $\mathbf{k}$ -linear triangulated category with $n,k$ integers with $n,k\geqslant 2$ . Assume that there exists an integer $h$ with $nk/2\leqslant h\leqslant nk$ and $\gcd (k,h)>1$ such that

$$\begin{eqnarray}\mathsf{Hom}^{\ast }(P_{i},P_{j})=\mathbf{k}[-h]\quad \text{for all adjacent }P_{i}\text{ and }P_{j}.\end{eqnarray}$$

Then $A=\mathsf{End}^{\ast }(\bigoplus _{i=1}^{m}P_{i})$ is intrinsically formal.

Proof. We will use the tensor presentation of $A=T(V)/I$ as in Remark 4.2 together with the notation $e_{i}=\operatorname{id}_{P_{i}}$ , $\mathsf{End}^{k}(P_{i})=\mathbf{k}\cdot t_{i}$ and $\mathsf{Ext}^{h}(P_{i},P_{j})=\mathbf{k}\cdot a_{ij}$ for adjacent $P_{i}$ and $P_{j}$ . Here $V$ is the graded vector space spanned by all $t_{i}$ and $a_{ij}$ . Recall that $J=T(V)^{+}$ .

Note that the homogeneous elements $1,t_{i},\ldots ,t_{i}^{n},a_{ij}$ constitute a basis of $A$ as a $\mathbf{k}$ -vector space. Hence,

( A ) $$\begin{eqnarray}\mathsf{maxdeg}(A)=nk\leqslant 2h.\end{eqnarray}$$

The only elements of $I$ not involving an $a_{ij}$ lie in the ideal generated by the elements $t_{i}^{n+1}$ . Indeed, $t_{i}^{l}\neq 0$ for $l<n+1$ by the definition of a $\mathbb{P}^{n}[k]$ -object. Furthermore, for $i\neq j$ , the tensor product $t_{i}\otimes t_{j}$ already vanishes as an element of $V\otimes _{R}V\subset T(V)$ as $t_{i}t_{j}=t_{i}\otimes t_{j}=(t_{i}e_{i})\otimes t_{j}=t_{i}\otimes (e_{i}t_{j})=0$ . Hence, the minimal degrees of $I$ and $J$ are

$$\begin{eqnarray}\displaystyle \mathsf{mindeg}(I) & {\geqslant} & \displaystyle \mathsf{min}\{\mathsf{deg}(t_{i}^{n+1}),\mathsf{deg}(a_{ij}a_{jl}),\mathsf{deg}(a_{ij}t_{j})\}=h+k,\nonumber\\ \displaystyle \mathsf{mindeg}(J) & = & \displaystyle \mathsf{min}\{\mathsf{deg}(t_{i}),\mathsf{deg}(a_{ij})\}=k,\nonumber\end{eqnarray}$$

where the assumption that $n\geqslant 2$ , hence $h\geqslant k$ , is used. Hence, by Remark 3.6, we get

(e) $$\begin{eqnarray}\mathsf{mindeg}\,\mathsf{Tor}_{2p}^{A}(R,R)\geqslant \mathsf{max}\{p(h+k),2k+(p-1)(h+k)\}=p(h+k),\end{eqnarray}$$

(o) $$\begin{eqnarray}\mathsf{mindeg}\,\mathsf{Tor}_{2p+1}^{A}(R,R)\geqslant p(h+k)+k.\end{eqnarray}$$

We can now confirm that the assumptions of Proposition 3.5 are satisfied for $q\geqslant 4$ . For $q=2p$ with $p\geqslant 2$ , we have

$$\begin{eqnarray}\mathsf{maxdeg}(A)+q-2\overset{(A)}{{\leqslant}}2h+2p-2\overset{(2\leqslant p)}{{<}}ph+2p\overset{(2\leqslant k)}{{\leqslant}}ph+pk\overset{(\text{e})}{{\leqslant}}\mathsf{mindeg}\,\mathsf{Tor}_{2p}^{A}(R,R).\end{eqnarray}$$

Similarly, for $q=2p+1$ with $p\geqslant 2$ ,

$$\begin{eqnarray}\mathsf{maxdeg}(A)+q-2\overset{(A)}{=}2h+2p-1\overset{(2\leqslant p)}{{<}}ph+2p\underset{(2\leqslant k)}{{<}}ph+pk+k\underset{(\text{o})}{{\leqslant}}\mathsf{mindeg}\,\mathsf{Tor}_{2p+1}^{A}(R,R).\end{eqnarray}$$

Hence, $\mathsf{HH}^{q,2-q}(A,A)$ vanishes for $q\geqslant 4$ . In order to apply Kadeishvili’s criterion for intrinsic formality (see Proposition 1.7) all that is left is to show that $\mathsf{HH}^{3,-1}(A,A)=0$ . This can be done using the Bar resolution $B^{\bullet }$ ; see §1.3. Indeed, $A$ is concentrated in degrees divisible by $\gcd (k,nk/2)>1$ . Hence, the same holds for $B^{3}=A^{\otimes 5}$ . Thus, there is no non-trivial degree-zero homomorphism $A^{\otimes 5}\rightarrow A(-1)$ .◻

The previous proposition does not cover the interesting case of configurations of spherical objects, which we treat in the following proposition.

Proposition 4.4. Let $Q$ be a graph and let $P_{1},\ldots ,P_{m}$ be a $Q$ -configuration consisting of $k$ -spherelike objects in a $\mathbf{k}$ -linear triangulated category with $k\geqslant 4$ . Moreover, assume that, for adjacent $P_{i}$ and $P_{j}$ ,

$$\begin{eqnarray}\mathsf{Hom}^{\ast }(P_{i},P_{j})=\mathbf{k}[-h_{ij}]\quad \text{with }\left\lfloor \frac{k}{2}\right\rfloor \leqslant h_{ij}\leqslant k.\end{eqnarray}$$

Then $A=\mathsf{End}^{\ast }(\bigoplus _{i=1}^{m}P_{i})$ is intrinsically formal.

Proof. This can be shown along the same lines as the proof of Proposition 4.3, but some of the estimates change. The minimal degrees of the ideals $I$ and $J$ become

$$\begin{eqnarray}\displaystyle \mathsf{mindeg}(I) & {\geqslant} & \displaystyle \mathsf{min}\{\mathsf{deg}(t_{i}^{2}),\mathsf{deg}(a_{ij}a_{jl}),\mathsf{deg}(a_{ij}t_{l})\}\geqslant 2h,\nonumber\\ \displaystyle \mathsf{mindeg}(J) & = & \displaystyle \mathsf{min}\{\mathsf{deg}(t_{i}),\mathsf{deg}(a_{ij})\}\geqslant h,\nonumber\end{eqnarray}$$

where we abbreviate $h:=\left\lfloor k/2\right\rfloor$ . Note that $h\geqslant 2$ by the assumption on $k$ . Hence, the minimal degrees in the minimal projective resolution are now

(e) $$\begin{eqnarray}\mathsf{mindeg}\,\mathsf{Tor}_{2p}^{A}(R,R)\geqslant \mathsf{max}\{2ph,2h+2(p-1)h\}=2ph,\end{eqnarray}$$

(o) $$\begin{eqnarray}\mathsf{mindeg}\,\mathsf{Tor}_{2p+1}^{A}(R,R)\geqslant 2ph+h.\end{eqnarray}$$

Furthermore, note that

( A ) $$\begin{eqnarray}\mathsf{maxdeg}(A)=k\leqslant 2h+1.\end{eqnarray}$$

We will check that the assumptions of Proposition 3.5 are satisfied for $q>2$ , which concludes the proof. Indeed, for $q=2p$ with $p\geqslant 2$ ,

$$\begin{eqnarray}\mathsf{maxdeg}(A)+q-2\overset{(A)}{{\leqslant}}2h+2p-1<2h+2p\leqslant 2ph\overset{(\text{e})}{{\leqslant}}\mathsf{mindeg}\,\mathsf{Tor}_{2p}^{A}(R,R).\end{eqnarray}$$

To see this, we still need $h+p\leqslant ph$ . This inequality is equivalent to $h/(h-1)\leqslant p$ , as $h\geqslant 2$ , and holds as $p\geqslant 2$ . Similarly, for $q=2p+1$ with $p\geqslant 1$ , we have

$$\begin{eqnarray}\mathsf{maxdeg}(A)+q-2\overset{(A)}{{\leqslant}}2h+2p<2ph+h\overset{(\text{e})}{{\leqslant}}\mathsf{mindeg}\,\mathsf{Tor}_{2p+1}^{A}(R,R).\end{eqnarray}$$

Here, the middle $2h+2p<2ph+h$ is equivalent to $h/(h-1)<2p$ , as $h\geqslant 2$ , hence the inequality holds due to $p\geqslant 1$ .◻

The following corollary, in combination with Theorem 1.10, gives Theorem B in the introduction.

Proof. Replacing the objects $P_{i}$ by appropriate shifts $P_{i}[n_{i}]$ , we may assume that they satisfy the assumptions on $\mathsf{Hom}^{\ast }(P_{i},P_{j})$ of Propositions 4.3 and 4.4, respectively. For the restrictions on $n$ and $k$ see Remark 4.5. Denote by $Q$ the underlying tree. We may start with some edge $i\in Q$ and set $n_{i}=0$ . By definition of a $Q$ -configuration of objects, for adjacent $i$ and $j$ , we have

$$\begin{eqnarray}\mathsf{Hom}^{\ast }(P_{i},P_{j})=\mathbf{k}[-a],\quad \mathsf{Hom}^{\ast }(P_{j},P_{i})=\mathbf{k}[-b]\end{eqnarray}$$

for some $a,b\in \mathbb{Z}$ . Note that we assume the objects to be $\mathbb{P}$ -objects (not just $\mathbb{P}$ -like). Hence, Serre duality gives $a+b=nk$ . Hence, in the case where $nk$ is even, we may set $n_{j}=a-h$ , so after replacing $P_{j}$ by $P_{j}[a-h]$ we get

$$\begin{eqnarray}\mathsf{Hom}^{\ast }(P_{i},P_{j})=\mathbf{k}[-h]=\mathsf{Hom}^{\ast }(P_{j},P_{i}).\end{eqnarray}$$

Since, by assumption, $Q$ has no loops, there is no obstruction to extending this procedure to the whole of $Q$ . The case where $n=1$ and $k\geqslant 5$ is odd works similarly; compare [Reference Seidel and ThomasST01, §4c].

Now we can use Propositions 4.3 and 4.4 together with Corollary 1.12 to conclude that $\langle \{P_{i}\}\rangle \cong \mathsf{D}(A)^{c}$ where $A=\mathsf{End}(\bigoplus _{i}P_{i})$ .◻

Remark 4.8. There are results, analogous to those of this subsection, on the formality of the endomorphism algebras of configurations of $\mathbb{P}^{n}[k]$ -objects for $k$ negative. To see this, one can use that a non-positively graded algebra is intrinsically formal as soon as

$$\begin{eqnarray}\mathsf{mindeg}(A)+q-2>\mathsf{maxdeg}(\mathsf{Tor}_{q}^{A}(R,R))\quad \text{for }q\geqslant 3,\end{eqnarray}$$

which is analogous to Proposition 3.5.

We chose to concentrate on $\mathbb{P}^{n}[k]$ -objects with $k$ positive, since those with negative $k$ are rare in practice, for example, negative Calabi–Yau objects cannot appear in derived categories of smooth varieties; compare [Reference Hochenegger, Kalck and PloogHKP19, Lemma 1.7].

4.2 Actions induced by $A_{m}$ -configurations of $\mathbb{P}$ -objects

Seidel and Thomas [Reference Seidel and ThomasST01] used the formality of $\mathsf{End}(\bigoplus _{i}P_{i})$ , where $P_{1},\ldots ,P_{m}$ is an $A_{m}$ -configuration of $k$ -spherical objects in a cocomplete dg-enhanced triangulated category ${\mathcal{T}}$ , in order to prove that the induced action of the braid group $B_{m+1}$ on ${\mathcal{T}}$ is faithful. This means that the subgroup $\langle \mathsf{T}\!_{P_{1}},\ldots ,\mathsf{T}\!_{P_{m}}\rangle \subset \operatorname{Aut}({\mathcal{T}})$ generated by the spherical twists is isomorphic to $B_{m+1}$ . Consider the $P_{i}$ as $\mathbb{P}^{1}[k]$ -objects and the associated $\mathbb{P}$ -twists are then the square of the spherical twists: $\mathsf{P}\!_{P_{i}}\cong \mathsf{T}\!_{P_{i}}^{\,2}$ ; compare Remark 2.6. It follows from the description of the group spanned by the squares of the standard generators of the braid group [Reference CollinsCol94] that the only relations between the $\mathbb{P}$ -twists are the commutativity relations

$$\begin{eqnarray}\mathsf{P}_{i}\mathsf{P}_{j}=\mathsf{P}_{j}\mathsf{P}_{i}\quad \text{for }|i-j|>1.\end{eqnarray}$$

Hence, it makes sense to conjecture the following more general faithfulness result.

Conjecture. Let $P_{1},\ldots ,P_{m}$ be an $A_{m}$ -configuration of $\mathbb{P}^{n}[k]$ -objects with $k\geqslant 2$ . Then the only relations between the associated $\mathbb{P}$ -twists $\mathsf{P}_{i}:=\mathsf{P}_{E_{i}}\in \operatorname{Aut}({\mathcal{T}})$ are the commutativity relations

$$\begin{eqnarray}\mathsf{P}_{i}\mathsf{P}_{j}=\mathsf{P}_{j}\mathsf{P}_{i}\quad \text{for }|i-j|>1.\end{eqnarray}$$

It is easy to see that, for two $\mathbb{P}^{n}[k]$ -objects with vanishing graded Hom-space between them, the associated $\mathbb{P}$ -twists commute; see [Reference KrugKru15, Corollary 2.5]. Hence, the unknown and probably difficult part of the conjecture is that there are no further relations between the twists associated to an $A_{m}$ -configuration of $\mathbb{P}$ -objects.

By Corollary 4.7, it would be sufficient to consider one particular example of an $A_{m}$ -configuration of $\mathbb{P}^{n}[k]$ -objects in order to prove (or disprove) the conjecture for a fixed value of $m$ , $n$ and $k$ with $nk$ even and $\gcd (k,nk/2)>1$ .

6.1 Trees of $\mathbb{P}$ -like objects on symmetric quotient stacks

We recall a construction of $\mathbb{P}^{n}[k]$ -like objects from $k$ -spherelike objects, which is essentially due to Ploog and Sosna in [Reference Ploog and SosnaPS14]. Let $X$ be a smooth projective variety and $E$ be a $k$ -spherelike object in $\mathsf{D}^{b}(X)$ . Consider the $n$ -fold cartesian product $X^{n}$ with its projections $\unicode[STIX]{x1D70B}_{i}:X^{n}\rightarrow X$ . Then we define

$$\begin{eqnarray}E^{\boxtimes n}=\unicode[STIX]{x1D70B}_{1}^{\ast }E\otimes \cdots \otimes \unicode[STIX]{x1D70B}_{n}^{\ast }E\in \mathsf{D}^{b}(X^{n}).\end{eqnarray}$$

There is a natural action on $X^{n}$ by the permutation group $\mathfrak{S}_{n}$ . Actually, we can turn $E^{\boxtimes n}$ into an object $E^{\{n\}}$ in the equivariant derived category $\mathsf{D}_{\mathfrak{S}_{n}}^{b}(X^{n})$ by equipping $E^{\boxtimes n}$ with the canonical linearisation given by permutation of the tensor factors. By $E^{-\{n\}}\in \mathsf{D}_{\mathfrak{S}_{n}}^{b}(X^{n})$ , we denote the object $E^{\boxtimes n}$ equipped with the linearisation which differs from the canonical one by the non-trivial character (also known as sign or alternating representation) $\mathfrak{a}$ of $\mathfrak{S}_{n}$ . Let $[X^{n}/\mathfrak{S}_{n}]$ be the quotient stack of $X^{n}$ by the permutation action of $\mathfrak{S}_{n}$ . By the definition of sheaves on a quotient stack, there is an equivalence $\mathsf{D}_{\mathfrak{S}_{n}}^{b}(X^{n})\cong \mathsf{D}^{b}([X^{n}/\mathfrak{S}_{n}])$ .

Proposition 6.3 (Cf. [Reference Ploog and SosnaPS14, Lemma 4.2]).

Let $n,k\in \mathbb{N}$ with $n\geqslant 2$ and $k$ even. Let $X$ be a smooth projective variety and let $E$ be a $k$ -spherelike object in $\mathsf{D}^{b}(X)$ . Then $E^{\{n\}},E^{-\{n\}}\in \mathsf{D}_{\mathfrak{S}_{n}}^{b}(X^{n})$ are $\mathbb{P}^{n}[k]$ -like objects.

Moreover, if $E$ is a $k$ -spherical object (e.g. if $E$ is spherelike and $X$ is a Calabi–Yau variety of dimension $k$ ), then $E^{\{n\}}$ and $E^{-\{n\}}$ are $\mathbb{P}^{n}[k]$ -objects.

Finally, let $E,F\in \mathsf{D}^{b}(X)$ be objects with $\mathsf{Hom}^{\ast }(E,F)=\mathbf{k}[-m]$ for some $m\in \mathbb{N}$ . Then

$$\begin{eqnarray}\displaystyle \mathsf{Hom}^{\ast }(E^{\{n\}},F^{\{n\}})=\mathsf{Hom}^{\ast }(E^{-\{n\}},F^{-\{n\}})=\left\{\begin{array}{@{}ll@{}}\mathbf{k}[-nm]\quad & \text{if }m\text{ is even,}\\ 0\quad & \text{if }m\text{is odd,}\end{array}\right. & & \displaystyle \nonumber\\ \displaystyle \mathsf{Hom}^{\ast }(E^{-\{n\}},F^{\{n\}})=\mathsf{Hom}^{\ast }(E^{\{n\}},F^{-\{n\}})=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if }m\text{ is even,}\\ \mathbf{k}[-mn]\quad & \text{if }m\text{is odd.}\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$

Proof. All of this follows from the equivariant Künneth formula which says that

$$\begin{eqnarray}\displaystyle \mathsf{Hom}^{\ast }(E^{\{n\}},F^{\{n\}})=\mathsf{Hom}^{\ast }(E^{-\{n\}},F^{-\{n\}}) & = & \displaystyle S^{n}\mathsf{Hom}^{\ast }(E,F),\nonumber\\ \displaystyle \mathsf{Hom}^{\ast }(E^{-\{n\}},F^{\{n\}})=\mathsf{Hom}^{\ast }(E^{\{n\}},F^{-\{n\}}) & = & \displaystyle \bigwedge ^{n}\mathsf{Hom}^{\ast }(E,F).\nonumber\end{eqnarray}$$

Note that both the symmetric and the exterior product are formed in the graded sense. For example, if $m\in \mathbb{N}$ is odd, then

$$\begin{eqnarray}S^{n}(\mathbf{k}[-m])\cong \Bigl(\bigwedge ^{n}\mathbf{k}\Bigr)[-mn]=0.\Box\end{eqnarray}$$

6.2 A geometric example of a tree of $\mathbb{P}$ -objects

Let $Q$ be a tree and $X$ be a smooth quasi-projective surface together with a $Q$ -configuration of $(-2)$ -curves. This means that, for every vertex $i$ of the tree $Q$ , there is a $(-2)$ -curve $\mathbb{P}^{1}\cong C_{i}\subset X$ , $C_{i}$ and $C_{j}$ intersect in one point if there is an edge joining $i$ and $j$ and they do not intersect otherwise. Note that such a configuration might not exist for any given tree $Q$ ; see [Reference Hochenegger and PloogHP19, §6] for sufficient criteria.

The objects ${\mathcal{O}}_{C_{i}}\in \mathsf{D}^{b}(X)$ form a $Q$ -tree of $2$ -spherical objects with $\mathsf{Hom}^{\ast }({\mathcal{O}}_{C_{i}},{\mathcal{O}}_{C_{j}})=\mathbf{k}[-1]$ for adjacent $i$ and $j$ ; see [Reference Seidel and ThomasST01, Example 3.5]. By Corollary 6.4, there is an induced $Q$ -tree of $\mathbb{P}^{n}$ -objects of the form ${\mathcal{O}}_{C_{i}}^{\unicode[STIX]{x1D700}_{i}\{n\}}$ in $\mathsf{D}_{\mathfrak{S}_{n}}^{b}(X^{n})$ . Here we have to choose opposite signs $\unicode[STIX]{x1D700}_{i}=-\unicode[STIX]{x1D700}_{j}$ for adjacent $i$ and $j$ , as the graded Hom-space is concentrated in the odd degree 1.

We use the derived McKay correspondence as mentioned in Remark 6.2 (recall that we omit $R$ and $L$ in front of derived functors),

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}:=p_{\ast }\circ q^{\ast }:\mathsf{D}^{b}(X^{[n]})\stackrel{{\sim}}{\rightarrow }\mathsf{D}_{\mathfrak{S}_{n}}^{b}(X^{n}),\end{eqnarray}$$

to interpret this as a tree of $\mathbb{P}^{n}$ -objects on the Hilbert scheme $X^{[n]}$ . Here we denote by $q:{\mathcal{Z}}\rightarrow X^{[n]}$ and $p:{\mathcal{Z}}\rightarrow X^{n}$ the projections from the universal family of $\mathfrak{S}_{n}$ -clusters ${\mathcal{Z}}\subset X^{[n]}\times X^{n}$ . Hence there is a commutative diagram

where $X^{(n)}:=X^{n}/\mathfrak{S}_{n}$ is the symmetric product, $\unicode[STIX]{x1D70B}$ is the $\mathfrak{S}_{n}$ -quotient morphism, and $\unicode[STIX]{x1D707}$ is the Hilbert–Chow morphism. Furthermore, ${\mathcal{Z}}\cong (X^{[n]}\times _{X^{(n)}}X^{n})_{\mathsf{red}}$ is the reduced fibre product of this diagram. Note that every closed subscheme $C$ of $X$ induces a canonical closed embedding $C^{[n]}{\hookrightarrow}X^{[n]}$ .

Let $i$ and $j$ be two adjacent vertices of $Q$ . Then $C_{i}\cap C_{j}$ is a reduced point, hence cannot contain a subscheme of length $n\geqslant 2$ . Thus, $C_{i}^{[n]}$ and $C_{j}^{[n]}$ do not intersect inside $X^{[n]}$ . So we see geometrically that $\mathsf{Hom}^{\ast }({\mathcal{O}}_{C_{i}}^{^{\{n\}}},{\mathcal{O}}_{C_{j}}^{^{\{n\}}})=0$ for support reasons.

For $n=2$ , we give a concrete description of the image of ${\mathcal{O}}_{C}^{^{-\{n\}}}$ under the McKay correspondence, where $C=C_{i}$ is one of the rational curves. Denote by $\unicode[STIX]{x1D6FF}:X\rightarrow X^{(2)}$ the diagonal embedding into the symmetric product. Then $E:=\unicode[STIX]{x1D707}^{-1}\unicode[STIX]{x1D6FF}(X)$ is the exceptional divisor of $X^{[2]}\rightarrow X^{(2)}$ . There is a line bundle ${\mathcal{L}}\in \operatorname{Pic}(X^{[2]})$ such that ${\mathcal{L}}^{2}\cong {\mathcal{O}}(E)$ ; see [Reference LehnLeh99, Lemma 3.7].

We summarise this situation in the following diagram, consisting of a blow-up square on the right and its restriction to $C$ , so both are cartesian:

(□)

Note that the restrictions $\unicode[STIX]{x1D70C}:E\rightarrow X$ and $\unicode[STIX]{x1D6F4}_{C}\rightarrow C$ of $\unicode[STIX]{x1D707}$ are $\mathbb{P}^{1}$ -bundles (actually, $\unicode[STIX]{x1D6F4}_{C}$ is isomorphic to the Hirzebruch surface $\unicode[STIX]{x1D6F4}_{4}$ ).

Before we prove the proposition, we need to recall another feature of the Hilbert scheme of two points, namely that the functor

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}:=\unicode[STIX]{x1D704}_{\ast }\unicode[STIX]{x1D70C}^{\ast }:\mathsf{D}^{b}(X)\rightarrow \mathsf{D}^{b}(X^{[2]})\end{eqnarray}$$

is spherical; see [Reference KrugKru15, Remark 4.3] or [Reference Krug, Ploog and SosnaKPS18, Theorem 4.26(ii)]. This means, in particular, that the associated twist functor $\mathsf{T}\!_{\unicode[STIX]{x1D6E9}}$ , defined by the triangle of functors

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}\unicode[STIX]{x1D6E9}^{R}\xrightarrow[{}]{\unicode[STIX]{x1D700}}\operatorname{id}\rightarrow \mathsf{T}\!_{\unicode[STIX]{x1D6E9}},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}$ is the counit of adjunction, is an autoequivalence of $\mathsf{D}^{b}(X^{[n]})$ . The right adjoint $\unicode[STIX]{x1D6E9}^{R}$ of $\unicode[STIX]{x1D6E9}$ is given by

(*) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}^{R}\cong \unicode[STIX]{x1D70C}_{\ast }\unicode[STIX]{x1D704}^{!}\cong \unicode[STIX]{x1D70C}_{\ast }(\unicode[STIX]{x1D704}^{\ast }(\hspace{2.22198pt}\cdot \hspace{2.22198pt})\otimes {\mathcal{O}}_{E}(E))[-1].\end{eqnarray}$$

Proof. The second assertion implies the first one since transversality means that the scheme-theoretic intersection $E\cap C^{[2]}$ is reduced and of the expected dimension 1.

The composition $C^{[2]}{\hookrightarrow}X^{[2]}\xrightarrow[{}]{\unicode[STIX]{x1D70C}}X^{(2)}$ is a closed embedding with image $C^{(2)}\subset X^{(2)}$ . Using the right cartesian diagram in ( $\Box$ ), it follows that $\unicode[STIX]{x1D70C}$ maps $E\cap C^{[2]}$ isomorphically to the scheme-theoretic intersection $\unicode[STIX]{x1D6FF}(X)\cap C^{(2)}$ .

Hence, we only have to prove that $\unicode[STIX]{x1D6FF}(X)\cap C^{(2)}=\unicode[STIX]{x1D6FF}(C)$ . This question is local in the analytic topology so that we may assume that $X=\operatorname{Spec}\mathbb{C}[x_{1},x_{2}]$ and $C=\operatorname{Spec}\mathbb{C}[x_{1}]$ . We set $s_{i}=x_{i}+y_{i}$ and $t_{i}=x_{i}-y_{i}$ so that $X^{2}=\operatorname{Spec}\mathbb{C}[s_{1},s_{2},t_{1},t_{2}]$ with the natural action of $\mathfrak{S}_{2}=\langle \unicode[STIX]{x1D70F}\rangle$ given by $\unicode[STIX]{x1D70F}\cdot s_{i}=s_{i}$ and $\unicode[STIX]{x1D70F}\cdot t_{i}=-t_{i}$ . Therefore,

$$\begin{eqnarray}{\mathcal{O}}(X^{(2)})=\mathbb{C}[s_{1},s_{2},t_{1},t_{2}]^{\mathfrak{S}_{2}}=\mathbb{C}[s_{1},s_{2},t_{1}^{2},t_{1}t_{2},t_{2}^{2}].\end{eqnarray}$$

The ideal of $\unicode[STIX]{x1D6FF}(X)\subset X^{(2)}$ is given by $I=(t_{1},t_{2})^{\mathfrak{S}_{2}}=(t_{1}^{2},t_{1}t_{2},t_{2}^{2})$ and the ideal of $C^{(2)}\subset X^{(2)}$ is given by $J=(s_{2},t_{2})^{\mathfrak{S}_{2}}=(s_{2},t_{2}^{2})$ . Hence,

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}(X)\cap C^{(2)}=\operatorname{Spec}(\mathbb{C}[s_{1},s_{2},t_{1}^{2},t_{1}t_{2},t_{2}^{2}]/(I+J))=\operatorname{Spec}\mathbb{C}[s_{1}]=\unicode[STIX]{x1D6FF}(C).\Box\end{eqnarray}$$

Proof. By ${\mathcal{O}}_{C^{[2]}}(-E)$ we mean $\left.{\mathcal{O}}_{X^{[2]}}(-E)\right|_{C^{[2]}}$ . Using (*) and the previous lemma, we compute

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}^{R}({\mathcal{O}}_{C^{[2]}}(-E))\cong {\mathcal{O}}_{C}[-1].\end{eqnarray}$$

Note that $\unicode[STIX]{x1D6E9}({\mathcal{O}}_{C})\cong {\mathcal{O}}_{\unicode[STIX]{x1D6F4}_{C}}$ , so the twist triangle applied to ${\mathcal{O}}_{C^{[2]}}(-E)$ becomes, after shift,

$$\begin{eqnarray}{\mathcal{O}}_{C^{[2]}}(-E)\rightarrow \mathsf{T}\!_{\unicode[STIX]{x1D6E9}}({\mathcal{O}}_{C^{[2]}}(-E))\rightarrow {\mathcal{O}}_{\unicode[STIX]{x1D6F4}_{C}}.\end{eqnarray}$$

The long exact cohomology sequence shows that ${\mathcal{H}}^{i}(\mathsf{T}\!_{\unicode[STIX]{x1D6E9}}({\mathcal{O}}_{C^{[2]}}(-E)))=0$ for $i\neq 0$ . Hence, the triangle reduces to the short exact sequence

(1) $$\begin{eqnarray}0\rightarrow {\mathcal{O}}_{C^{[2]}}(-E\cap C^{[2]})\rightarrow {\mathcal{H}}^{0}(\mathsf{T}\!_{\unicode[STIX]{x1D6E9}}({\mathcal{O}}_{C^{[2]}}(-E)))\rightarrow {\mathcal{O}}_{\unicode[STIX]{x1D6F4}_{C}}\rightarrow 0.\end{eqnarray}$$

As $\unicode[STIX]{x1D6F4}_{C}\cap C^{[2]}=E\cap C^{[2]}$ , there is the canonical short exact sequence

(2) $$\begin{eqnarray}0\rightarrow {\mathcal{O}}_{C^{[2]}}(-E\cap C^{[2]})\rightarrow {\mathcal{O}}_{Y_{C}}\rightarrow {\mathcal{O}}_{\unicode[STIX]{x1D6F4}_{C}}\rightarrow 0.\end{eqnarray}$$

One can compute that $\mathsf{Ext}^{1}({\mathcal{O}}_{\unicode[STIX]{x1D6F4}_{C}},{\mathcal{O}}_{C^{[2]}}(-\unicode[STIX]{x1D6F4}_{C}\cap C^{[2]}))=\mathbb{C}$ , using for example [Reference Căldăraru, Katz and SharpeCKS03, Theorem A.1]. It follows that (1) and (2) coincide, so

$$\begin{eqnarray}\mathsf{T}\!_{\unicode[STIX]{x1D6E9}}({\mathcal{O}}_{C^{[2]}}(-E))\cong {\mathcal{H}}^{0}(\mathsf{T}\!_{\unicode[STIX]{x1D6E9}}({\mathcal{O}}_{C^{[2]}}(-E)))\cong {\mathcal{O}}_{Y_{C}}.\Box\end{eqnarray}$$

Proof of Proposition 6.7.

Combining the formulae of [Reference Krug, Ploog and SosnaKPS18, Theorem 4.26], we get an isomorphism of functors

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{-1}(\unicode[STIX]{x1D6F7}(\hspace{2.22198pt}\cdot \hspace{2.22198pt})\otimes \mathfrak{a})\cong {\mathcal{L}}\otimes \mathsf{T}\!_{\unicode[STIX]{x1D6E9}}(\hspace{2.22198pt}\cdot \hspace{2.22198pt}\otimes {\mathcal{L}}^{-2}),\end{eqnarray}$$

where ${\mathcal{L}}^{2}={\mathcal{O}}(E)$ . Combining this with Proposition 6.6 gives

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{-1}({\mathcal{O}}_{C}^{^{-\{2\}}})\cong {\mathcal{L}}\otimes \mathsf{T}\!_{\unicode[STIX]{x1D6E9}}({\mathcal{O}}_{C^{[2]}}\otimes {\mathcal{L}}^{-2}).\end{eqnarray}$$

Now the assertion follows by Lemma 6.9. ◻

For $C_{i},C_{j}\in X$ two $(-2)$ -curves which intersect in one point, $Y_{C_{i}}$ and $C_{j}^{[2]}$ intersect transversally in one point of $X^{[2]}$ . This confirms geometrically that

$$\begin{eqnarray}\mathsf{Hom}_{\mathsf{D}_{\mathfrak{ S}_{2}}^{b}(X^{2})}^{\ast }({\mathcal{O}}_{C_{i}}^{^{-\{2\}}},{\mathcal{O}}_{C_{j}}^{^{\{2\}}})\cong \mathsf{Hom}_{\mathsf{D}^{b}(X^{[2]})}^{\ast }({\mathcal{O}}_{Y_{C_{i}}}\otimes {\mathcal{L}},{\mathcal{O}}_{C_{j}^{[2]}})=\mathbb{C}[-2];\end{eqnarray}$$

compare Proposition 6.3.