1 Introduction↩︎

This paper is part of our effort to understand the geometries attached to a vertex algebra. Corresponding to each vertex algebra \(V\), there is a Poisson algebra \(R(V)\) which defines a Poisson scheme \(X_V=\operatorname{\sf spec}{R(V)}\). In the rational \(C_2\)-cofinite case, \(X_V\) has only one point. In [1], we defined a cohomological dual \(X_{V,x}^!=\operatorname{\sf spec}((\operatorname{Ext}_{R(V)}^*(\mathsf{k}_x, \mathsf{k}_x))^{ab})\) for each closed point \(x\in X_V\) and call it the cohomological variety of \(V\) at \(x\). While it defines an invariant of the vertex algebra \(V\), it is difficult to compute in most cases, in particular when \(x \in X_V\) is not a local complete intersection. In [1], we estimated a lower bound on the dimension of this variety for many rational \(C_2\)-cofinite vertex algebras. Our goal is to establish a cohomological support variety theory ([2] for restricted Lie algebras in positive characteristic, [3] for Lie superalgebras) for representations of vertex algebras in terms of quasicoherent sheaves. In the present paper, we explore a model different from \(X_{V,x}^!\). The Yoneda algebra \(\operatorname{Ext}_{R(V)}^*(\mathsf{k}_x, \mathsf{k}_x)\) is actually a Hopf algebra, which is the universal enveloping algebra of a graded Lie algebra, called the homotopy Lie algebra of \(R(V)\), in the symmetric monoidal category of graded vector spaces (with Koszul braiding) when \(R(V)\) is Noetherian. Hence the idea of associated varieties of representations of Lie algebras can be applied [4]–[7]. To prepare for this, we first need to explore this Lie algebra from the differential graded context. Since the algebra \(R(V)\) is in general not Noetherian and not complete intersection, we give a more functorial construction of the homotopy Lie algebra as well as its dg enhancement, depending on the construction of the Koszul-Tate resolution, preparing for applications to dg vertex algebras which have been established in [8]–[10]. One of our objectives is to construct a cohomological duality for vertex algebras, by establishing a Koszul-Tate type of resolutions of vertex algebras.

The Koszul-Tate resolution requires a divided power structure (PD structure) [11]–[15] on the resolution algebra, which is called semi-free in some literature. The free algebras with PD structures have been studied with concrete constructions by Roby in [11]–[13]. One of the approaches is to use the functorial construction of the subalgebra \(\operatorname{\sf TS}_A(V)\) of symmetric tensors in the shuffle algebra (\(T_A(V)\) with shuffle product) over a free \(A\)-module \(V\). This construction has an apparent functorial advantage, automatically making \(\operatorname{\sf TS}_A(V)\) a cocommutative commutative Hopf algebra with PD structure. In this paper, we extend this construction to the symmetric monoidal category of differential complexes over general commutative rings. This is done in Section 3 with elementary setup in Section 2. More precisely, in Section 3 we first extend the construction of free PD algebras by Roby in [13] using the shuffle algebra of symmetric tensors \(\operatorname{\sf TS}_{A}(M)\) of a free \(A\)-module \(M\) to the symmetric tensor category \(\operatorname{\sf Cplx}(A)^{free}\) of differential complexes of free \(A\)-modules. This defines a functor \(\operatorname{\sf TS}_{A}^*(-):\operatorname{\sf Cplx}(A)^{free} \to \operatorname{\sf grPDAlg}(A)\) to the category of graded PD algebras over \(A\) (see [14] for graded PD algebras), sending direct sums to tensor products. Thus there is a PBW type basis for each given basis \(\mathcal{B}\) of \(M\) with a given total order. This functor automatically defines a graded Hopf algebra with PD structure on \(\operatorname{\sf TS}_A(M)\). The freeness then makes it possible to extend differentials if \(A\) is a PD dg algebra. We use this functor \(\operatorname{\sf TS}_A(-)\) to construct a Koszul-Tate resolution for an arbitrary commutative ring \(A\) with an ideal \(I\) and quotient ring \(A/I\) as an \(A\)-algebra.

In most constructions in the literature, we find that either \(A\) is assumed to be Noetherian (most in the context of Noetherian local rings such as in [16]–[19]) or to be a \(\mathbb{Q}\)-algebra, from rational homotopy theory [20] (thus the PD structure becomes automatic and unique) such as in [21]–[23]. In Tate’s construction [19], the approach consists in killing one cocycle at a time, which is good enough to apply when \(A\) is Noetherian. Northcott [24] extended Tate’s construction by still working over generators and taking limits. This extension is then used in Gulliksen-Levin’s lecture notes [18] as well as Stacks-Project ([15]) and others.

We take a generator free approach and apply the functor \(\operatorname{\sf TS}_{A}(-)\) to kill the entire (co)homology at a degree at once. This is a Sullivan type construction from homotopy theory [20], [25] where we attach all cells of a given dimension at once. We will rely heavily on the use of the PD structure to extend PD dg algebra homomorphisms as well as derivations. The Koszul-Tate resolution \(P^*=\varinjlim_{n} P_{n}^*\) of \(A/I\) as an \(A\)-algebra depends on the choice of the free \(A\)-module cover \(\phi_{n+1}: F_{n+1}\to H^{-n}(P^*_n)\) that kills degree \(-n\) cohomology of earlier stage \(P_n^*\) and the next stage is obtained as \(P_{n+1}^*=P_{n}^*\otimes_A \operatorname{\sf TS}_{A}(F_{n+1}[n+1])\). The differential on \(P^*_{n+1}\) is the unique extension of \(d_{P_{n}^*}\) and \(\phi_{n+1}\), making \(P_{n+1}^*\) a PD dg algebra and the natural embedding \(P_{n}^*\to P_{n+1}^*\) a homomorphism of PD dg algebras. We remark that, by construction, each \(P_n^*\) is a graded PD Hopf algebra which is both graded commutative and graded cocommutative. Thus, forgetting the differential, \(P^*\) is a connected graded cocommutative PD Hopf algebra over \(A\). However it is not a PD dg Hopf algebra (see [14]) as the comultiplication will fail to be a chain map, as one might expect.

It is well-known that the Yoneda algebra \(\operatorname{\sf Ext}_A^*(A/I,A/I)\) is isomorphic to \(H^*(\mathcal{H}om_{A}^*(P^*,P^*))\), not only as graded \(A\)-modules, but also as graded algebras with the composition product on \(H^*(\mathcal{H}om_{A}^*(P^*,P^*))\) up to a Koszul sign. In Section 4, we consider the subcomplex \(\mathcal{D}er^{*,\operatorname{pd}}_A(P^*, P^*)\subseteq \mathcal{H}om_{A}^*(P^*,P^*)\) of the PD derivations. The complex \(\mathcal{D}er^{*,\operatorname{pd}}_A(P^*, P^*)\) is naturally a dg Lie algebra, which is a dg Lie subalgebra of \(\mathcal{D}er^{*}_A(P^*, P^*)\). The graded Lie algebra \(H^*(\mathcal{D}er^{*,\operatorname{pd}}_A(P^*, P^*))\) (in the symmetric tensor category of graded \(A\)-modules) naturally maps to a graded Lie subalgebra in \(H^*( \mathcal{H}om_{A}^*(P^*,P^*))\). Its image is defined to be the Homotopy Lie algebra. In the case where \(A\) is Noetherian, \(I\) is maximal, and the resolution \(P^*\) is \(I\)-minimal (i.e., \(d(P^{-n}) \subseteq IP^{-n+1}\)), the Yoneda algebra \(\operatorname{\sf Ext}_{A}^*(A/I,A/I)\) has a connected Hopf algebra structure whose homogeneous primitive elements form a graded Lie algebra, which is classically defined as the homotopy Lie algebra [16]. Moreover, the homotopy Lie algebra has a restricted structure, and the Yoneda algebra is the restricted enveloping algebra of the homotopy Lie algebra.

We could not prove that the graded Lie algebra \(H^*(\mathcal{D}er^{*,\operatorname{pd}}_A(P^*, P^*))\) does not depend on the resolution \(P^*\) for the \(A\)-algebra \(A/I\). We also will not discuss the model category structure on the category of PD dg algebras so that the Koszul-Tate resolution \(P^*\) constructed is a cofibrant object.

The concept of Homotopy Lie algebra was first defined by the homotopy groups of \(H\)-spaces in [26] and later interpreted for commutative local Noetherian rings (see [16]–[18]). We choose the definition using the dg Lie algebra \(\mathcal{D}er^{*, \operatorname{pd}}_A(P^*, P^*)\), which has a natural restricted structure given by \(D \mapsto D \circ D\). This Lie algebra plays the role of tangent complex in the context of rational homotopy theory and model category, or derived algebraic geometry. We will not interpret the cotangent complex in terms of the PD dg algebra \(P^*\). Cotangent complexes mostly appear in the simplicial context [27]. There is a dg approach to cotangent complexes in [28]. In the paper [29], Richter has proved that divided power structures on the chain complexes will naturally arise from the simplicial commutative algebra structure via normalized Moore complex. There is a different notion of homotopy Lie algebra operadic approach in [30] using Lie operad in the homotopy category of differential complexes. In case \(\mathbb{Q}\subseteq A\), the homotopy Lie algebra is closely related to the Homotopy Lie algebra described in this paper.

We have not yet considered the case where \(A\) is a Poisson \(\mathsf{k}\)-algebra and \(I\) is a Poisson ideal, but we expect that there is a Koszul-Tate resolution \(P^*\) which has a Poisson dg algebra with PD structure so that the augmentation \(P^*\to A/I\) [31] is a Poisson algebra homomorphism. Due to the size of the paper, we also do not mention the algebra of PD dg differential operators on the PD dg algebra \(P^*\).

In Section 5, we consider the special case when \(A\) is a complete intersection as well as several examples arising from simple singularities. In this case, \(P^*\) can constructed as \(I\)-minimal and the homotopy Lie algebra can be computed as a certain graded Lie algebra concentrated on degree 1 and 2. Conversely, any such graded Lie algebra (with a restricted structure) can be seen as the homotopy Lie algebra of some complete intersection algebra. In this section, we also consider the finite generation question of Yoneda algebras. This is a classical question regarding finite generation of cohomology rings of groups, restricted Lie algebras, and Hopf algebras (see [2], [32]–[36]). In contrast to finite groups or finite dimensional restricted Lie algebras, the Yoneda algebra is finitely generated if and only if there is a resolution \(P^*\) that is finitely generated as a PD dg algebra, which is equivalent to the homotopy Lie algebra having finite total dimension or to \(A\) being a complete intersection ring.

Although we are focusing on the case where \(A\) is a commutative ring and \(I\) is an ideal in this paper, the Yoneda algebra has the following geometric formulation. Let \(X\) be any \(\mathsf{k}\)-scheme and \(Z\subseteq X\) be a closed subscheme. Let \(\mathcal{I} \subseteq \mathcal{O}_X\) be the ideal sheaf (on \(X\)) defining \(Z\). The structure sheaf of \(Z\), \(\mathcal{O}_Z=\mathcal{O}_X/\mathcal{I}\), is an \(\mathcal{O}_X\)-module supported over \(Z\). We then want to compute the graded sheaf of \(\mathcal{O}_Z\)-modules \({\mathcal{E}xt}^*_{\mathcal{O}_X}(\mathcal{O}_Z, \mathcal{O}_Z)\) in the category of \(\mathcal{O}_X\)-modules on \(X\). More generally, when \(Z'\) is another closed subscheme of \(X\) with the defining ideal \(\mathcal{J}\), we want to compute the graded sheaf \({\mathcal{E}xt}^*_{\mathcal{O}_X}(\mathcal{O}_Z, \mathcal{O}_{Z'})\). The sheaves \({\mathcal{E}xt}^*_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\) for certain vector bundles \(\mathcal{F}\) on \(Z\) and \(\mathcal{G}\) on \(Z'\) are expected to have physical interpretations [37]. We note that this sheaf is supported over the scheme intersection \(Z\times_X Z'\), which is a subscheme of \(X\) defined by the ideal \(\mathcal{I}+\mathcal{J}\) with structure sheaf \(\mathcal{O}_Z\otimes_{\mathcal{O}_X}\mathcal{O}_{Z'}\). The sheaves \(\mathcal{T}or^{\mathcal{O}_X}_*(\mathcal{O}_Z, \mathcal{O}_{Z'})\) measure the derived intersection of \(Z\) and \(Z'\) [38]. When \(x\in Z\cap Z'\), by considering the localisations of \(\mathcal{O}_X\), \(\mathcal{O}_Z\) and \(\mathcal{O}_{Z'}\) at \(x\), the modules \(\mathcal{T}or^{\mathcal{O}_{X, x}}_i(\mathcal{O}_{Z,x}, \mathcal{O}_{Z', x})\) provide the intersection multiplicity of \(Z\) and \(Z'\) at \(x\) as Serre’s homological characterisation of intersection theory.

If \(Z\subseteq X\) is a closed subvariety then \(\operatorname{Ext}^1_{\mathcal{O}_X}(\mathcal{O}_Z, \mathcal{O}_Z)\) is the tangent space of the moduli space of subvarieties at the point \([Z]\). The (skew)-commutativity of the Yoneda product can be used to construct symplectic structure on the moduli space (see [39]). If \([Z]\) is a smooth point of the moduli space, then the Yoneda product is skew commutative. When \(Z=\{z\}\) is a closed point, \(\operatorname{Ext}^{1}_{\mathcal{O}_X }(\mathsf{k}_z, \mathsf{k}_z)\) is the tangent space \(T_zX\).

Acknowledgements. The authors would like to thank the anonymous referee for a detailed report containing thoughtful comments that greatly helped in improving the present paper. This work started with substantial discussions with Cuipo Jiang on the cohomological varieties for vertex algebras and both authors appreciate her contributions. The second author also wants to thank Amnon Yekutieli for sending his preliminary version of [28].

2 The category of PD dg algebras over a PD dg ring↩︎

2.1 Complexes and graded modules↩︎

Fix a commutative ring \(A\) and consider the categories \(\operatorname{\sf Cplx}(A)\) of differential (cochain) complexes of \(A\)-modules, and \(A\operatorname{\sf -mod}^{\mathbb{Z}}\) the category of \(\mathbb{Z}\)-graded \(A\)-modules. Note that \(A\operatorname{\sf -mod}^{\mathbb{Z}}\) is a symmetric monoidal category with tensor product \(\otimes = \otimes_A\) where \[(M^* \otimes N^*)^n=\bigoplus_{i+j=n}M^i \otimes N^j\] and with braiding \[\begin{array}{cccc} b_{M^*,N^*}:&M^i \otimes N^j & \to & N^j \otimes M^i \\ & m \otimes n & \mapsto & (-1)^{ij}n \otimes m. \end{array}\] Then \(\operatorname{\sf Cplx}(A)\) is also a symmetric monoidal category with tensor product of differential complexes with a similar braiding. We label the cochain complexes \[\cdots \to M^{i}\stackrel{d^{i}}{\to} M^{i+1}\to \cdots\] if the differential map is of degree \(1\). By simply relabeling \(M_i=M^{-i}\), then each differential cochain complex becomes a differential chain complex with differential \(d^{i}=d_{-i}\) of degree \(-1\). We say that \((M^*,d^*)\) is \(A\)-free if \(M^i\) is a free \(A\)-module for all \(i\), and the category of these complexes is written \(\operatorname{\sf Cplx}(A)^{free}\).

Finally we have a forgetful functor \[\operatorname{\sf Cplx}(A) \to A\operatorname{\sf -mod}^{\mathbb{Z}}\] sending \((M^*,d^*)\) to \(M^*=\bigoplus_{n \in \mathbb{Z}}M^n\). For simplicity reasons, we will later on write \(M^*\) for either the complex or the resulting graded \(A\)-module, depending on the context.

We will also consider the full subcategory \(\operatorname{\sf Cplx}(A)^+\) of the cochain complexes \((M^* , d^*)\) with \(M^i=0\) for all \(i>0\).

2.2 PD dg rings↩︎

A strictly graded commutative dg ring refers to a graded commutative dg ring satisfying \(xx=0\) for all odd degree elements \(x\).

Let \((R^*, d^*)\) be a strictly graded commutative dg ring with \(I \subseteq R\) a dg ideal. We write \(I_{ev}=\bigoplus_{n \in \mathbb{Z}}I^{2n}\) and \(I_{odd}=\bigoplus_{n \in \mathbb{Z}}I^{2n+1}\) for the even and odd components of \(I\). We define \(R_{ev}\) and \(R_{odd}\) similarly. A divided power (PD) structure on \(I\) is a sequence of maps \(\gamma_n: I_{ev}\to I_{ev}\) (\(n \in \mathbb{Z}_{+}\)) and \(\gamma_0:I_{ev} \to R^*\) given by \(\gamma_0(x)=1\) satisfying the following conditions:

1. \(\gamma_1(x)=x\) for all \(x \in I\);

2. \(\gamma_n(x)\gamma_m(x)=\binom{n+m}{m} \gamma_{n+m}(x)\) for all \(x\in I_{ev}\);

3. \(\gamma_n(ax)=a^n\gamma_n(x)\) for all \(a\in R_{ev}\) and \(x\in I_{ev}\);

4. \(\gamma_n(x+y)=\sum_{i=0}^n\gamma_i(x)\gamma_{n-i}(y)\) for all \(x, y\in I_{ev}\);

5. \(\gamma_p( \gamma_q(x))=\frac{(pq)!}{p!(q!)^p}\gamma_{pq}(x)\) for all \(x \in I_{ev}\);

6. \(\gamma_n(I_{2r})\subseteq I_{2rn}\) for \(n>0\);

7. \(\gamma_n(xy)=0 \text{ for all } x \in R_{odd}, y\in I_{odd} \text{ homogeneous of odd degrees and } n\geq 2\);

8. \(d(\gamma_n(x))=\gamma_{n-1}(x)d(x)\).

A PD dg ring is a triple \((R^*,I,\gamma)\) where \(R^*\) is a dg ring, \(I\) is a dg ideal of \(R^*\), and \(\gamma\) is a PD structure on \(I\). We will call \(I\) the PD ideal of \((R^*,I,\gamma)\). When the ideal \(I\) and the PD structure \(\gamma\) are clear, we will simply write \(R^*\) for the PD dg ring.

The condition [axiom957] on the odd degree was imposed by André in [14]. We recall that the PD structure is heavily dependent on the ideal \(I\). The use of the term “PD” comes from the French “puissances divisées”, the term under which the notion was introduced (see [12]). It follows from the definition that:

1. Given any PD dg ring \(R^*\), for any \(x \in I_{ev}\), then \(n! \gamma_n(x)=x^n\) for all \(n \geq 0\).

2. If \(R\) is a graded commutative dg \(\mathbb{Q}\)-algebra, i.e., \(\mathbb{Q}\subset R_0\), then for any dg ideal \(I \subset R\) there exists a unique PD dg structure on \(R\) given by \(\gamma_n(x)=\frac{x^n}{n!}\) for \(x \in I_{ev}\).

3. Any strictly graded commutative dg ring \(R^*\) is a PD dg ring with \(I=\{0\}\). In particular, this is true when \(R^*\) is concentrated in degree \(0\), which is a commutative ring.

Let \((R^*, d^*)\) be a dg ring in \(\operatorname{\sf Cplx}(A)\). Set \(B^*(R^*)\subseteq Z^*(R^*)\subseteq R^*\) the coboundary and cocycle subcomplexes of \(R^*\). Then \(Z^*(R^*)Z^*(R^*)\subseteq Z^*(R^*)\) by the Leibniz rule, rule that also implies \(d(1)=0\). Thus \(Z^*(R^*)\) is a dg subring of \(R^*\) (with trivial differential) and \(B^*(R^*)\) is a two-sided ideal of \(Z^*(R^*)\). Therefore the cohomology \(H^*(R^*)\) is dg ring (with trivial differential).

Note that \(Z^*(I)=I\cap Z^*(R^*)\) is a dg ideal of \(Z^*(R^*)\). It follows from [axiom958] in the definition of the PD structure that \(\gamma_n(Z^*(I)_{ev})\subseteq Z^*(I)_{ev}\). Thus \((Z^*(R^*), Z^*(I), \gamma)\) is a PD dg ring. Set \(I(H^*(R^*))\) the image of \(Z^*(I)\) in \(H^*(R^*)\), which is clearly an ideal. We want to assume that the PD structure on \((Z^*(R^*), Z^*(I), \gamma)\) induces a PD structure \((H^*(R^*), I(H^*(R^*)), \bar \gamma)\). This requires \[\begin{align} \gamma_n(z+b)=\gamma_n(z)+\sum_{i=1}^n\gamma_{n-i}(z)\gamma_{i}(b)\in \gamma_n(z)+I_{ev}\cap B^*(R^*) \end{align}\] for \(z\in I_{ev}\cap Z^*(R^*)\) and \(b\in I_{ev}\cap B^*(R^*)\), i.e., \(B^*(R^*)\) should be a PD dg ideal of \(Z^*(R^*)\). Since \(n\gamma_{n}(d(r))=\gamma_{n-1}(d(r))d(r)\) by Axioms [axiom951] and [axiom952], then \(B^*(R^*)\) is a PD dg ideal of \(Z^*(R^*)\) if \(R^*\) is a \(\mathbb{Q}\)-algebra.

Given two PD dg rings \((R', I',\gamma')\) and \((R'', I'',\gamma'')\), with the ideal \(I=\operatorname{\sf Im}(R'\otimes_{\mathbb{Z}} I''+I'\otimes_{\mathbb{Z}} R'')\subseteq R=R'\otimes_{\mathbb{Z}} R''\), there exists (cf. [18]) a unique PD structure \((R,I,\gamma)\) such that the obvious dg ring homomorphisms \(\iota':(R',I',\gamma')\to(R,I, \gamma)\) and \(\iota'': (R'',I'',\gamma'')\to(R,I, \gamma)\) are PD dg ring homomorphisms. Using Axiom [axiom953], we can make \(\gamma_n\) explicit. If \(x'\in R'\) and \(x''\in R''\) are homogeneous, then \((x'\otimes x'')=(x'\otimes 1)(1\otimes x'')=(-1)^{|x'||x''|}(1\otimes x'')(x'\otimes 1)\). For \(x''\in I''_{ev}\) and \(x' \in R'_{ev}\), we have \[\begin{array}{rcl} \gamma_n(x'\otimes x'') & = & (x'\otimes 1)^{n}\gamma_n(1\otimes x'') \\[5pt] & = & (x'\otimes 1)^{n}\gamma_n \circ \iota''(x'') \\[5pt] & = & (x'\otimes 1)^{n} \iota'' \circ \gamma_n''(x'') \\[5pt] & = & (x'\otimes 1)^{n}(1\otimes \gamma''_n(x'')) \\[5pt] & = & x'^{n}\otimes \gamma''_n(x'') \end{array}\] because \(\iota''\) is a PD homomorphism. Likewise, for \(x'\in I'_{ev}\) and \(x'' \in R''_{ev}\) we have \[\begin{align} \gamma_n(x'\otimes x'')=&(-1)^{n|x'||x''|}(1\otimes x'')^{n}( \gamma'_n(x')\otimes 1) \notag \\[5pt] =&(-1)^{n|x'||x''|}(-1)^{n|x'|n|x''|}\gamma'_n(x')\otimes x''^{n}\notag \\[5pt] =&\gamma'_n(x')\otimes x''^{n}. \label{pd95tensor95first} \end{align}\tag{1}\] Moreover, \(\gamma_n(R'_{odd} \otimes I''_{odd}+I'_{odd} \otimes R''_{odd})=0\) for \(n >1\) because of Axiom [axiom957].

Therefore the category of PD dg rings is a tensor category. Let \(F\) be the forgetful functor from the category of PD dg rings to the category of dg rings. This makes \(F\) into a strictly monoidal functor. However, the tensor product is not the coproduct (cf. [15]). In fact, there is a homomorphism of PD dg rings \[(R_1^*,I_1^*, \gamma)\coprod_{(R_0^*,I_0^*, \gamma)}(R_2^*,I_2^*, \gamma) \to (R_1^*,I_1^*, \gamma)\otimes_{(R_0^*,I_0^*, \gamma)}(R_2^*,I_2^*, \gamma)\] and there is a natural homomorphism of dg rings \[F((R_1^*,I_1^*, \gamma))\otimes_{F((R_0^*,I_0^*, \gamma))}F((R_2^*,I_2^*, \gamma))\to F\left((R_1^*,I_1^*, \gamma)\coprod_{(R_0^*,I_0^*, \gamma)}(R_2^*,I_2^*, \gamma)\right)\] which is not an isomorphism ([15]).

The example in ([15]) suggests that this failure can be avoided if one requires a certain flatness condition, in particular, freeness condition. Thus working on resolutions becomes necessary in many applications.

We note that the braiding in the tensor category of complexes of abelian groups induces a symmetric tensor category structure on the category of commutative dg rings. The same braiding defines a symmetric tensor structure on the category of PD dg rings. We will need this symmetric tensor category structure later to work on Hopf algebras.

The following argument will become useful later on. Let \(A'\) be a commutative \(A\)-algebra. We can regard \(A'\) as a PD dg \(A\)-algebra \((A',\{0\}, 0)\). Let \(\operatorname{\sf PDdgAlg}(A)\) be the category of PD dg rings with PD dg ring homomorphisms \((A, \{0\},0) \to (R^*, I^*, \gamma)\). Morphisms of \(\operatorname{\sf PDdgAlg}(A)\) are morphisms of PD dg rings which are \(A\)-linear. In particular, \(\operatorname{\sf PDdgAlg}(A)\) contains the category of commutative \(A\)-algebras as a full subcategory.

If \(A\to A'\) is a ring homomorphism and \((A^*,I,\gamma)\) is a PD dg \(A\)-algebra, we define the base extension \(A^*_{A'}=A^* \otimes_{A} A'\). It has an ideal \(I_{A'}=\operatorname{\sf Im}(I\otimes A') \subset A^*_{A'}\). Moreover, \(\gamma:I_{ev} \to I_{ev}\) extends to \(\gamma_{A'}:(I_{A'})_{ev} \to (I_{A'})_{ev}\) by \((\gamma_{A'})_n(x \otimes \alpha)=\gamma_n(x) \otimes \alpha^n\) for all \(x \in I_{ev}, \alpha \in A'\).

3 Koszul-Tate Resolutions for arbitrary commutative rings↩︎

3.1 Symmetric tensors of differential complexes↩︎

Let \(A\) be a commutative ring. We briefly recall the symmetric strict monoidal category \(\operatorname{\sf Cplx}(A)\) of differential (cochain) complexes of \(A\)-modules with morphisms being chain maps. The tensor product over \(A\) of two complexes \((X^*, d^*)\otimes (Y^*, d^*)\) is defined by \[((X^*, d^*)\otimes(Y^*, d^*))^n=\bigoplus _{i+j=n}X^i\otimes Y^j\] with differential \(d^{X\otimes Y}(x\otimes y)=d(x)\otimes y+(-1)^{|x|}x\otimes d(y)\) for all homogeneous \(x\in X^*\) and \(y\in Y^*\). In this section we will use \(\otimes=\otimes_{A}\) unless there is a confusion with other tensor products.

The braiding \(b_{X, Y}: X^*\otimes Y^*\to Y^*\otimes X^*\) is defined by \[b_{X,Y}(x\otimes y)=(-1)^{|x||y|}y\otimes x\] for all homogeneous \(x\in X^*\) and \(y\in Y^*\). The map \(b_{X,Y}\) is clearly a chain map of chain complexes: \[\begin{align} b_{X,Y}d^{X\otimes Y}(x\otimes y)&=(-1)^{(|x|+1)|y|}y\otimes d(x) +(-1)^{|x|}(-1)^{|x|(|y|+1)} d(y)\otimes x\\ &=(-1)^{|x||y|}(d(y)\otimes x+(-1)^{|y|}y\otimes d(x))=d^{Y\otimes X}b_{X,Y}(x\otimes y). \end{align}\] An associative algebra object \((A^*, m, 1)\) (with multiplication \(m: A^*\otimes A^*\to\) being a chain map) in \(\operatorname{\sf Cplx}(A)\) is called a dg algebra over \(A\). We use \(\operatorname{\sf DGA}(A)\) to denote the category of all dg algebras over \(A\). A dg algebra \((A^*, m, 1)\) is called strictly graded commutative if \[\begin{align} \label{diag:commutativity} \xymatrix{A^*\otimes A^*\ar[dr]_m\ar[rr]^{b_{A, A}}&& A^*\otimes A^*\ar[dl]^{m}\\ &A^*& } \end{align}\tag{2}\] is a commutative diagram and if \(m(x, x)=0\) for all homogeneous \(x\in X^{odd}\). Let \(\operatorname{\sf scDGA}(A)\) denote the category of strictly graded commutative dg algebras in \(\operatorname{\sf Cplx}(A)\). If \((A^*, d^A)\) and \((B^*, d^B)\) are two strictly graded commutative dg algebras, then \(A^*\otimes B^*\) is also a strictly graded commutative dg algebra. We recall the multiplication \(m_{A\otimes B}\) is defined as the following: \[(A^*\otimes B^*)\otimes (A^*\otimes B^*)\xrightarrow{1\otimes b_{B, A}\otimes 1}(A^*\otimes A^*)\otimes (B^*\otimes B^*)\xrightarrow{m_A\otimes m_B}A^*\otimes B^*.\]

If \(\operatorname{\sf cAlg}(A)\) is the category of commutative \(A\)-algebras, then there are embeddings \(\operatorname{\sf cAlg}(A) \hookrightarrow \operatorname{\sf scDGA}(A) \hookrightarrow \operatorname{\sf PDdgA}(A)\) as full subcategories. This is clear from [note95ring95dg] in Section 2.2.

Let \((X^*, d^*)\) be an object in \(\operatorname{\sf Cplx}(A)\). Then \(T_A^n(X^*)=X^*\otimes\cdots\otimes X^*\) is a differential complex. The symmetric group \(\mathfrak{S}_n\) acts on \(T_A^n(X^*)\) as automorphisms of chain complexes as follows: \[\begin{align} \label{action95sigma} \sigma(x_1\otimes\cdots\otimes x_n)=(-1)^{\ell(\sigma; (x_1, \dots, x_n))}(x_{\sigma^{-1}(1)}\otimes \cdots \otimes x_{\sigma^{-1}(n)}). \end{align}\tag{3}\] Here \(\ell(\sigma;(x_1, \dots, x_n))=\sum_{(i, j)\in\operatorname{\sf Inv}(\sigma^{-1})}|x_i||x_j|\) and \(\operatorname{\sf Inv}(\sigma)=\{ (i< j) \;|\; \sigma(i)> \sigma (j)\}\) is the set of inversions of \(\sigma\). This group action is generated by the braiding \((i, i+1)\mapsto b_{i, i+1}=1^{\otimes (i-1)}\otimes b_{X,X}\otimes 1^{\otimes(n-i-1)}\). As \(b_{X,X}\) is a chain map, then so is \(\sigma: T_A^n(X^*)\to T_A^n(X^*)\).

Let \(\operatorname{\sf TS}_A^n(X^*)=T_A^n(X^*)^{\mathfrak{S}_n}\) be the \(A\)-submodule of \(\mathfrak{S}_n\)-fixed points. Thus \(\operatorname{\sf TS}_A^n(X^*)\) is closed under differentials since the \(\mathfrak{S}_n\)-action commutes with the differentials, making \(\operatorname{\sf TS}_A^n(X^*)\) a subcomplex of \(T_A^n(X^*)\). It is obvious that \(\operatorname{\sf TS}_A^1(X^*)=T_A^1(X^*)=X^*.\)

Using induction, one can check that, for any sequence of homogeneous elements \((x_1,\dots, x_n)\) in \(X^*\) and \(\sigma, \tau\in \mathfrak{S}_n\), we have \[\ell(\sigma\tau; (x_1, \cdots, x_n))\equiv \ell(\sigma; (x_1, \cdots, x_n))+\ell(\tau; (x_{\sigma^{-1}(1)}, \cdots, x_{\sigma^{-1}(n)})) \pmod 2.\]

For \(m, n \in \mathbb{Z}_{>0}\), we set \(\mathfrak{S}(m,n)=\mathfrak{S}_{m+n}/(\mathfrak{S}_m\times \mathfrak{S}_n)\). The relative trace map \(\operatorname{\sf Tr}_{\operatorname{}(m,n)}:\operatorname{\sf TS}_A^{m}(X^*) \otimes \operatorname{\sf TS}_A^{n}(X^*) \to \operatorname{\sf TS}_A^{m+n}(X^*)\) (see [40]) is defined by \[\operatorname{\sf Tr}_{(m,n)}(f \otimes g)=\sum_{\sigma \in \mathfrak{S}(m,n)}\sigma (f \otimes g).\] Note that, for any \((\sigma_m, \sigma_n) \in \mathfrak{S}_m\times \mathfrak{S}_n\subseteq \mathfrak{S}_{m+n}\), \[\sigma \circ(\sigma_m, \sigma_n) (f\otimes g)=\sigma(f\otimes g)\] as \(f\) and \(g\) are symmetric tensors themselves. Hence \(\operatorname{\sf Tr}_{\mathfrak{S}(m,n)}(f \otimes g) \in \operatorname{\sf TS}_A^{m+n}(X^*)\). With this we define a multiplication (shuffle product) \[\begin{align} \label{TS-star-product} \star: \operatorname{\sf TS}_A^m(X^*)\otimes\operatorname{\sf TS}_A^n(X^*)\to \operatorname{\sf TS}_A^{n+m}(X^*) \end{align}\tag{4}\] as follows: for \(f\in \operatorname{\sf TS}_A^m(X^*)\) and \(g\in \operatorname{\sf TS}_A^n(X^*)\), \[\begin{align} \label{shuffle-product} f\star g=\operatorname{\sf Tr}_{\mathfrak{S}(m,n)}(f \otimes g). \end{align}\tag{5}\]

We note that the definition of \(\star\)-product in 5 does not apply to elements in \(T_A(X^*)\) directly. However, this product can be extended to all elements in \(T_A(X^*)\) as follows. There is a canonical choice of minimal coset representatives \(\sigma\in \mathfrak{S}(m,n)\), called \((m,n)\)-shuffles (see [40]), such that \[\sigma(1)<\cdots<\sigma(m)\; \text{ and } \; \sigma(m+1)<\cdots<\sigma(m+n).\] Let \(\operatorname{Sh}(m,n)\subseteq \mathfrak{S}_{m+n}\) be the set of all \((m,n)\)-shuffles. Then \[\begin{align} f \star g=\sum_{\sigma \in \operatorname{Sh}(m,n)}\sigma (f \otimes g) \end{align}\] for all \(f\in T_A^m(X^*)\) and \(g\in T_A^n(X^*)\). We will also refer to this product on \(T_A(X^*)\) as shuffle product. It is a standard argument that \((T_A(X^*), \star)\) is an associative dg algebra. In fact, the argument applies to any strict symmetric monoidal category in place of \(\operatorname{\sf Cplx}\) by using the obvious bijective maps \(\operatorname{Sh}(l+m, n)\times \operatorname{Sh}(l,m)\to \operatorname{Sh}(l,m,n)\) and \(\operatorname{Sh}(l, m+n)\times \operatorname{Sh}(m,n)\to \operatorname{Sh}(l,m,n)\). To distinguish with standard tensor multiplication, we use \(T_A(X^*)_{Sh}=(T_A(X^*), \star)\) to denote the dg algebra with respect to the \(\star\)-product following the notation of Roby [13] (in fact a slightly modified notation to avoid confusion with the base ring \(A\)).

We see that \(\operatorname{\sf TS}_A^+(X^*)=\bigoplus_{n>0}\operatorname{\sf TS}_A^n(X^*)\) is a dg ideal of the dg algebra \(\operatorname{\sf TS}_A(X^*)\) and is the kernel of the augmentation map \(\operatorname{\sf TS}_A(X^*)\twoheadrightarrow A\), which is also a chain map.

3.2 PD algebra structure↩︎

Let \((K')_{n}^p\) be the set of elements \(\sigma \in \mathfrak{S}_{np}\) that preserve the relative order of the terms within each of the \(p\) blocks of size \(n\) and such that \(\sigma (n)<\sigma(2n)<\cdots <\sigma(pn)\). They form a set of complete coset representatives in \(\mathfrak{S}_{pn}\) modulo the subgroup \(\mathfrak{S}_p\ltimes (\mathfrak{S}_n\times\cdots\times \mathfrak{S}_n)\) as in [13]. Let us state the following well-known lemma which will be used later on.

By using Theorem 5, we see that, as graded algebras, \[\operatorname{\sf TS}_A(X^*)=\bigotimes_{i\in \mathbb{Z}}\operatorname{\sf TS}_A(X^i[-i])\] where the \(A\)-modules \(X^i\) are regarded as complexes concentrated in degree \(0\). This also gives a PBW type basis for the algebra \(\operatorname{\sf TS}_A(X^*)\) if each \(X^i\) is a free \(A\)-module with basis \(B_n\) (with an argument similar to that of [40]). Set \(\mathbb{N}_s=\mathbb{N} \coprod \mathbb{Z}/2\mathbb{Z}\) and let \(f: B=\coprod_{n}B_n\to \mathbb{N}_s\) be a function with finite support where \(f(B_n) \subset \mathbb{N}\) if \(n\) is even and \(f(B_n) \subset \mathbb{Z}/2\mathbb{Z}\) if \(n\) is odd. We use \(\mathbb{N}_s^{(B)}\) to denote the set of all such maps \(f\). For each \(f \in \mathbb{N}_s^{(B)}\), set \[\begin{align} \label{PBW95TS} x^{(f)}=\prod^\star_{b\in B} b^{(f(b))} \end{align}\tag{8}\] where \(b^{(f(b))}=\gamma_{f(b)}(b)\). Here \(\prod^\star\) means a finite \(\star\)-product with a fixed ordering of the set \(B\). Note that \(x^{(f)}\in \operatorname{\sf TS}_A^{\sum_{b\in B}f(b)}(X^*)\) and its dg degree is \[|x^{(f)}|=\sum_{b\in B}f(b)|b|.\] In the following, we consider the differential \(d\). As \(T_A(X^*)_{Sh}\) is a dg algebra (cf. Proposition 1), it follows that for each element \(f\) we have: \[\begin{align} \label{eq:differential} d(x^{(f)})=\sum_{b\in B}\left ( \prod^\star_{b'<b}(-1)^{f(b')|b'|} {b'}^{(f(b'))}\right ) \star d(b^{(f(b))}) \star \left (\prod^\star_{b'>b} {b'}^{(f(b'))}\right ). \end{align}\tag{9}\]

Given two complexes \(X^*\) and \(Y^*\), we have PD dg algebra homomorphisms \(\iota_{\operatorname{\sf TS}_A(X)}:\operatorname{\sf TS}_A(X) \to \operatorname{\sf TS}_A(X) \otimes \operatorname{\sf TS}_A(Y)\) and \(\iota_{\operatorname{\sf TS}_A(Y)}:\operatorname{\sf TS}_A(Y) \to \operatorname{\sf TS}_A(X) \otimes \operatorname{\sf TS}_A(Y)\) (see Section 2).

3.3 Hopf algebra structures↩︎

Consider \(X^* \in \operatorname{\sf Cplx}(A)^{free}\). By using the isomorphism \(h\) in Theorem 6, the maps \(+: X^*\oplus X^*\to X^*\), and \(\delta: X^*\to X^*\oplus X^*\) with \(\delta(x)=(x, x)\) induce homomorphisms \(\mu=\operatorname{\sf TS}_A(+) \circ h: \operatorname{\sf TS}_A(X^*)\otimes \operatorname{\sf TS}_A(X^*)\to \operatorname{\sf TS}_A(X^*)\) and \(\Delta=h^{-1} \circ \operatorname{\sf TS}_A(\delta) \;:\operatorname{\sf TS}_A(X^*)\to \operatorname{\sf TS}_A(X^*)\otimes \operatorname{\sf TS}_A(X^*)\) in \(\operatorname{\sf PDdgAlg}(A)\). In fact, \(\mu\) is exactly the \(\star\)-product in \(\operatorname{\sf TS}_A(X^*)\). Indeed, for \(x,x' \in \operatorname{\sf TS}_A(X^*)\), we have \[\begin{array}{rcl} \mu(x \otimes x') & = & \operatorname{\sf TS}_A(+) \circ h(x \otimes x') \\ & = & \operatorname{\sf TS}_A(+)\big(\operatorname{\sf TS}_A(\iota_1)(x) \star \operatorname{\sf TS}_A(\iota_2)(x') \big) \\ & = & \operatorname{\sf TS}_A(+)(\operatorname{\sf TS}_A(\iota_1)(x)) \star \operatorname{\sf TS}_A(+)(\operatorname{\sf TS}_A(\iota_2)(x')) \text{ as } \operatorname{\sf TS}_A(+) \text{ is a morphism} \\ & & \text{of algebras} \\ & = & x \star x' \text{ as } +\! \circ \iota_i=\operatorname{id}_X \text{ for } i=1,2. \end{array}\] As \(\operatorname{\sf TS}_A(\{0\})=A\), the maps \(\{0\}\to X^*\) and \(X^*\to \{0\}\) define morphisms \(u:A \to \operatorname{\sf TS}_A(X^*)\) and \(\varepsilon: \operatorname{\sf TS}_A(X^*)\to A\), making \(\operatorname{\sf TS}_A(X^*)\) into a commutative and cocommutative bialgebra with \(u\) and \(\epsilon\) the unit and counit respectively (see [40]). In fact, the map \(\operatorname{\sf neg}: X^*\to X^*\) sending \(\operatorname{\sf neg}(x)=-x\) defines the antipode \(S: \operatorname{\sf TS}_A(X^*)\to \operatorname{\sf TS}_A(X^*)\) making \(\operatorname{\sf TS}_A(X^*)\) into a connected commutative and cocommutative Hopf algebra. The antipode property follows easily from the functoriality of \(\operatorname{\sf TS}_A(-)\) and the following commutative diagrams \[\xymatrix{ X^*\oplus X^*\oplus X^* \ar[rr]^{(+, \operatorname{\sf Id})}\ar[d]_{( \operatorname{\sf Id}, +)}&&X^*\oplus X^*\ar[d]^{+} \\ X^*\oplus X^* \ar[rr]_{+}&& X^*} \quad \quad \xymatrix{ X^*\oplus X^*\oplus X^* &&X^*\oplus X^*\ar[ll]_{(\operatorname{\sf Id},\delta)} \\ X^*\oplus X^* \ar[u]^{(\delta, \operatorname{\sf Id})}&& \ar[ll]^{\delta}X^*\ar[u]_{\delta}}\] \[\xymatrix{ X^*\ar[dr]^{\varepsilon}\ar[rr]^{\delta}\ar[d]_{\delta}&& X^*\oplus X^* \ar[d]^{(\operatorname{\sf Id}, \operatorname{\sf neg})}\\ X^*\oplus X^*\ar[d]_{( \operatorname{\sf neg},\operatorname{\sf Id})}&\{0\}\ar[dr]^{u} &X^*\oplus X^*\ar[d]^{+}\\ X^*\oplus X^* \ar[rr]_{+}&& X^* }.\]

3.4 Freeness of PD dg algebra \(\operatorname{\sf TS}_A(X^*)\)↩︎

We now show that the functor \(\operatorname{\sf TS}_A(-): \operatorname{\sf Cplx}(A) \to \operatorname{\sf PDdgAlg}(A)\) is the left adjoint of the forgetful functor \(\operatorname{\sf PDdgAlg}(A)\to \operatorname{\sf Cplx}(A)\) in the sense of the following theorem:

We remark that \(\operatorname{\sf TS}_A(X^*)\) is not free as strictly commutative dg algebra in general unless \(\mathbb{Q}\subseteq A\). In there are definitions of semi-free resolutions in the literature, in which each step is assumed to be free commutative algebra. In that sense, the resolution we are going to construct will not be semi-free. However, one could understand that in the PD dg sense, the construction we will get is indeed semi-free.

3.5 Lifting properties↩︎

3.6 PD dg algebra extensions↩︎

Let \(F=V[-p]\) be a complex concentrated in degree \(p\) (\(p \leq 0\)) for a free \(A\)-module \(V\). Then we have \(\operatorname{Hom}_{\operatorname{\sf Cplx}}(F, X^*)=\operatorname{Hom}_{A\operatorname{\!-Mod}}(V, Z^p(X^*))\). Let \((A^*,d^A)\) be an object in \(\operatorname{\sf scDGA}(A)\) and \(\phi: F\to A^*\) be a chain map. We define a map \(d^\phi: \operatorname{\sf TS}_A(F[1])\otimes A^*\to \operatorname{\sf TS}_A(F[1])\otimes A^*\) as follows:

1. \(d^\phi(x \otimes 1)=1 \otimes \phi(x)\) for all \(x \in \operatorname{\sf TS}_A^1(F[1])\);

2. \(d^\phi(1 \otimes a)=1 \otimes d^A(a)\) for all \(a \in A^*\);

3. \(d^\phi(x^{(f)} \otimes a)=d(x^{(f)})\otimes a + (-1)^{|x^{(f)}|}x^{(f)} \otimes d^A(a)\) for all \(x^{(f)} \in \operatorname{\sf TS}_A(F[1])\) and \(a \in A^*\);

where \[\begin{align} d(x^{(f)}) \otimes a=\sum_{b\in B}\left ( \prod^\star_{b'<b}(-1)^{f(b')|b'|} {b'}^{(f(b'))}\right ) \star b^{(f(b)-1)} \star \left (\prod^\star_{b'>b} {b'}^{(f(b'))}\right) \otimes \phi(b)a . \end{align}\] Then \(d^\phi\) is derivation extending \(d^A\) such that \(\operatorname{\sf TS}_A(F[1])\otimes A^*\) is a dg algebra with derivation \(d^\phi\) and \(A^*\to \operatorname{\sf TS}_A(F[1])\otimes A^*\) is a homomorphism of dg algebras.

Let \(\theta: P^* \to Q^*\) and \(\tau: Q^* \to P^*\) be PD dg \(A\)-algebra homomorphisms such that \(\theta \circ \tau \simeq \operatorname{id}_{Q^*}\) and \(\tau \circ \theta \simeq \operatorname{id}_{P^*}\). Define \(\alpha: \mathcal{H}om_A^*(P^*, P^*) \to \mathcal{H}om_A^*(Q^*, Q^*)\) by \(\alpha(D)=\theta \circ D \circ \tau\). We note that \(\alpha\) is chain map. But \(\alpha\) does not preserve the composition multiplication unless \(\theta\) and \(\tau\) are inverse to each other on the chain level. However, we have the following:

4 Derivations of PD dg algebras↩︎

Let \(A\) be a commutative ring. A quick consequence of \(P^*\) in Theorem 12 being K-projective is that the functor \(\mathcal{H}om_A^*(P^*, -)\) sends quasi-isomorphisms to quasi-isomorphisms. To see this, consider a quasi-isomorphism \(f: M^*\to N^*\). We know that \(f\) is a quasi-isomorphism if and only if \(\operatorname{\sf Cone}(f)\) is acyclic [42]. It follows that \(\mathcal{H}om_A^*(P^*,\operatorname{\sf Cone}(f))\) is also acyclic. But one can show that \(\mathcal{H}om_A^*(P^*,\operatorname{\sf Cone}(f)) \cong \operatorname{\sf Cone}(\mathcal{H}om_A^*(P^*,f))\) ([43]), implying that \(\mathcal{H}om_A^*(P^*,f)\) is a quasi-isomorphism (using [42] once more). Finally, it can be shown that any complex of projective \(A\)-modules bounded above is K-projective (cf. [15]).

Set \(I \subset A\) an ideal and consider the \(A\)-algebra \(A/I\). In this section, we will rely on the \(A\)-algebra structure of \(A/I\) in order to make use of Theorem 12. Moreover, this structure naturally makes \(A/I\) into an \(A\)-module, and we will be able to consider the Yoneda algebra \(\operatorname{Ext}_A^*(A/I,A/I)\). Let \(\epsilon: P^*\to A/I\) be an \(A\)-projective resolution. Given an element \(\widetilde{\varphi} \in \mathcal{H}om_A^{n}(P^*,P^*)\), we write \(\widetilde{\varphi}_{-k} :P^{-k} \to P^{-k+n}\) for each homogeneous component. We say that \(\widetilde{\varphi}\) is a cocycle if \(d(\widetilde{\varphi})=0\), i.e, if \(d \circ \widetilde{\varphi}_{-k-1} -(-1)^{n}\widetilde{\varphi}_{-k} \circ d=0\) for all \(k\). Notice that \(|d(\widetilde{\varphi})|=|\widetilde{\varphi}|+1\), hence \(\mathcal{H}om_A^{*}(P^*,P^*)\) is a cochain complex.

Set \(\varphi:P^{-n} \to A/I\) a morphism of \(A\)-modules. We say that a morphism of complexes \(\widetilde{\varphi}:P^* \to P^*\) is a lift of \(\varphi\) if the following conditions are satisfied:

• \(|\widetilde{\varphi}|=n\);

• \(\widetilde{\varphi}\) is a cocycle;

• \(\epsilon \circ \widetilde{\varphi}_{-n}=\varphi\).

This can be summarized in the following commutative diagram:

The Yoneda algebra \(\operatorname{Ext}_A^*(A/I,A/I)\) is equipped with two distinct products, which we differentiate here to avoid sign problems. To define them, take \(\varphi:P^{-n} \to A/I\) and \(\psi:P^{-m} \to A/I\) be representatives of elements in \(\operatorname{Ext}_A^*(A/I,A/I)\). Using the lifts defined above, the composition product is given as \(\psi \circ \varphi=\epsilon \circ \widetilde{\psi} \circ \widetilde{\varphi}\). To compute the other product, called Yoneda product (see [44]), one needs a different lift from the one defined above. For a map \(\varphi:P^{-n} \to A/I\), we define \(\varphi_{-n-i} :P_{-n-i} \to P_{-i}\) (\(i \geq 0\)) as we did \(\widetilde{\varphi}_{-n-i}\) but the bottom row of the diagram is given by \(d\) and not \((-1)^nd\). The Yoneda product is then \(\psi \cdot \varphi=\psi \circ \varphi_{-m-n}\).

The \(A\)-projective resolution \(\epsilon: P^*\to A/I\) is a quasi-isomorphism of complexes of \(A\)-modules with \(A/I\) regarded as a complex concentrated in degree \(0\). Moreover, \(\mathcal{H}om_A^*(P^*, P^*)\) is a dg algebra over \(A\). Post-composition with \(\epsilon\) defines a chain map \(\mathcal{H}om_A^*(P^*,\epsilon): \mathcal{H}om_A^*(P^*, P^*)\to \mathcal{H}om_A^*(P^*, A/I)\), which is surjective because of the projectivity of \(P^*\). As each summand of \(P^*\) is projective and they are bounded above, it follows that \(P^*\) is \(K\)-projective. Then, following the discussion at the beginning of the section, \(\mathcal{H}om_A^*(P^*,\epsilon)\) is a quasi-isomorphism. The natural dg algebra structure on \(\mathcal{H}om_A^*(P^*,P^*)\) defines the algebra structure on \(\operatorname{\sf Ext}^*_A(A/I,A/I)\):

We now consider \(P^*\) the resolution of \(A/I\) obtained in Theorem 12. It is a PD dg algebra. We define the set of PD dg derivations on \(P^*\): \[\begin{array}{rl} \mathcal{D}er_A^{*,\operatorname{pd}}(P^*,P^*)=\displaystyle\bigoplus_{n \in \mathbb{Z}} & \{ \varphi \in \mathcal{H}om_A^n(P^*,P^*) \;| \;\varphi(ab)=\varphi(a)b+(-1)^{|a|n}a\varphi(b) \\ & \quad \text{and } \varphi(a^{(k)})=\varphi(a)a^{(k-1)} \text{ if } |a| \text{ even} \} \end{array}\] where we write \(a^{(k)}=\gamma_k(a)\) to lighten notation. It is a routine to verify that \(\mathcal{D}er_A^{*,\operatorname{pd}}(P^*,P^*)\) is a subcomplex of \(\mathcal{H}om_A^*(P^*,P^*)\).

An immediate consequence of the previous proposition is the following corollary:

Consider two PD dg \(A\)-algebras \(P^*\) and \(Q^*\) equipped with a PD dg algebra morphism \(\theta:P^* \to Q^*\). Then \(\mathcal{H}om_A^{*}(P^*,Q^*)\) is a cochain complex with differential \(d(f)=d_Q\circ f-(-1)^{|f|}f\circ d_P\). In fact \(\mathcal{H}om_A^{*}(P^*,Q^*)\) is a right dg \(\mathcal{H}om_A^{*}(P^*,P^*)\)-module and left dg \(\mathcal{H}om_A^{*}(Q^*,Q^*)\)-module with the actions defined by \[\begin{align} \label{eq:action95dg} f\cdot \varphi=f \circ \varphi \quad \text{ and } \quad \psi\cdot f=\psi\circ f \end{align}\tag{16}\] for any \(\varphi \in \mathcal{H}om_A^{|\varphi|}(P^*,P^*)\), \(\psi \in \mathcal{H}om_A^{|\psi|}(Q^*,Q^*)\) and \(f \in \mathcal{H}om_A^{|f|}(P^*,Q^*)\).

Let \(\theta_*:\mathcal{H}om_A^{*}(P^*,P^*)\to \mathcal{H}om_A^{*}(P^*,Q^*)\) and \(\theta^*:\mathcal{H}om_A^{*}(Q^*,Q^*)\to \mathcal{H}om_A^{*}(P^*,Q^*)\) be the maps defined by \[\theta_*(\varphi)=\theta\circ \varphi \quad \text{and } \quad \theta^*(\psi)=\psi\circ \theta\] for all \(\varphi \in \mathcal{H}om_A^{*}(P^*,P^*)\) and \(\psi\in \mathcal{H}om_A^{*}(Q^*,Q^*)\). Then both \(\theta^*\) and \(\theta_*\) are chain maps.

We can then define \[\begin{array}{rl} \mathcal{D}er_{A, \theta}^{*,\operatorname{pd}}(P^*,Q^*)=\displaystyle\bigoplus_{n \in \mathbb{Z}} & \{ f \in \mathcal{H}om_A^n(P^*,Q^*) \;| \;f(ab)=f(a)\theta(b)+(-1)^{|a|n}\theta(a)f(b) \\ & \quad \text{and } f(a^{(k)})=f(a)\theta(a^{(k-1)}) \text{ if } |a| \text{ even} \}. \end{array}\] It can be similarly verified that \(\mathcal{D}er_{A,\theta}^{*,\operatorname{pd}}(P^*,Q^*)\) is a subcomplex of \(\mathcal{H}om_A^*(P^*,Q^*)\) and that \[\begin{align} \label{eq:derivation-transfer} \theta_*(\mathcal{D}er_A^{*,\operatorname{pd}}(P^*,P^*))\subseteq \mathcal{D}er_{A,\theta}^{*,\operatorname{pd}}(P^*,Q^*) \;\text{ and } \; \theta^*(\mathcal{D}er_A^{*,\operatorname{pd}}(Q^*,Q^*))\subseteq \mathcal{D}er_{A,\theta}^{*,\operatorname{pd}}(P^*,Q^*) \end{align}\tag{17}\]

The homomorphism \(\sigma \circ (-):\mathcal{H}om_A^{*}(Q^*,Q^*) \to \mathcal{H}om_A^{*}(Q^*,Q^*)\) induces the identity on \(H^*(\mathcal{H}om_A^{*}(Q^*,Q^*))\) since \(\sigma - \operatorname{id}_{Q^*}=d_{\mathcal{H}om}(h)\) for some \(h \in \mathcal{H}om_A^{-1}(Q^*,Q^*)\).

The composition of chain maps: \[\mathcal{D}er_A^{*,\operatorname{pd}}(P^*,P^*) \hookrightarrow \mathcal{H}om_A^{*}(P^*,P^*) \to \mathcal{H}om^{*}(P^*,A/I)\] induces an \(A\)-linear map of graded Lie algebras \[\begin{align} \label{eq:morphism} \Psi_P: H^*(\mathcal{D}er_A^{*,\operatorname{pd}}(P^*,P^*)) \to \operatorname{Ext}_A^*(A/I,A/I) \end{align}\tag{18}\] where the Lie bracket is the graded commutator.

Set \(\varphi \in Z^0(\mathcal{D}er_A^{*,\operatorname{pd}}(P^*,P^*))\), then \(\varphi_0:P^0 \to P^0\) is an \(A\)-module morphism defined by \(\varphi_0(1)\) as \(P^0=A\). But \(\varphi_0(1)=0\) due to the Leibniz rule, and so \(\varphi_0=0\). As \(P^*\) is exact and \(\varphi\) is a chain map, one can construct by induction on \(n\) an \(A\)-module homomorphism \(h_n:P^{n} \to P^{n-1}\) such that \(\varphi_n=d_{P^*} \circ h_n+h_{n-1} \circ d_{P^*}=d_{\mathcal{H}om}(h)_n\). Hence the cohomology class of \(\varphi\) is zero in \(H^0(\mathcal{H}om_A^{*,\operatorname{pd}}(P^*,P^*))\). Combining that with the result of Lemma 7 with \(X^*=P^*\), we get that

\[\begin{align} \label{eq:neg95hom} \Psi_P(H^{\leq 0}(\mathcal{D}er_A^{*,\operatorname{pd}}(P^*,P^*)))=0. \end{align}\tag{19}\]

We now introduce the following definition:

We will thus omit the subscript and write \(\Psi\) in the rest of this paper.

A complex \(X^*\) of \(A\)-modules is called \(I\)-minimal if \(d(X^*) \subseteq I X^*\). In this case, we have \(H^*(\mathcal{H}om_A(X^*,A/I))=\mathcal{H}om_A(X^*,A/I)\) and \(H^*(X^* \otimes A/I)=X^* \otimes A/I\).

We now assume that \(P^*\). We also assume that \(F_n\) is a free \(A\)-module of finite rank for each \(n \geq 0\). It follows that as a PD dg algebra, \(P^*\) might have infinitely many generators, but for each degree, there will be only finitely many of them.

Following the proof of Theorem 12, let \(\mathcal{B}(-(n+1))\) be an \(A\)-basis of \(F_{n+1}[n+1]\) and set \(\mathcal{B}(0)=0\). We also set an order on each \(\mathcal{B}(-n)\). It follows that \(\mathcal{B}=\bigcup_{n \geq 0} \mathcal{B}(-n)\) is a set of generators of the PD dg algebra \(P^*\). Finally we set an order on \(\mathcal{B}\) such that \(b'<b\) for any \(b \in \mathcal{B}(-n)\) and \(b' \in \mathcal{B}(-(n+1))\).

A basis of \(P^*\) as an \(A\)-module is given through \[\begin{align} \label{eq:basis95P} \mathcal{X}=\{f : \mathcal{B} \to \mathbb{N} \;| \;f \text{ has finite support and }f(b)=0, 1 \text{ if } |b| \text{ odd for }b \in \mathcal{B} \}. \end{align}\tag{20}\] For \(f \in \mathcal{X}\), we write \(\operatorname{deg}f=\sum_{b \in \mathcal{B}}f(b)|b|\) for the degree of \(f\). We also write \(\operatorname{poldeg}b^{(f)}=\sum_{b \in \mathcal{B}} f(b)\) for its polynomial degree. There is a decomposition \(\mathcal{X}=\bigcup_{n \in \mathbb{N}}\mathcal{X}(-n)\) where \(\mathcal{X}(-n)=\{f \in \mathcal{X} \;| \;\operatorname{deg}f=-n\}\). The basis element of \(P^*\) corresponding to \(f \in \mathcal{X}\) will be written \(b^{(f)}=\prod_{b \in \mathcal{B}} b^{(f(b))}\) where the \(b\)’s are in descending order and \(b^{(f(b))}=\gamma_{f(b)}(b)\). Set \(b^{(f)} \in \mathcal{X}\) and define the \(A\)-linear map \((b^{(f)})^\vee:P^{\operatorname{deg} f} \to A/I\) that sends all elements of \(\mathcal{X}(\operatorname{deg}f)\) to zero except for \(b^{(f)}\) which is sent to \(1\) (\(A/I\) is an algebra so it has a unit).

Similarly, we write \((b^\vee)^{f}=\prod_{b \in \mathcal{B}} (b^\vee)^{f(b)}\) for the ordered product in \(\operatorname{Ext}_A^*(A/I,A/I)\) (notice that \((b^\vee)^{f}\) uses the product without divided powers).

We cannot lift morphisms corresponding to elements of the form \((b^{(f)})^\vee\) if there exist \(b_1 \neq b_2\) in \(\mathcal{B}\) with \(f(b_1) \neq 0 \neq f(b_2)\), as they would not satisfy the PD relations. For example, consider \(b^{(k)}\) for an even variable \(b\) and \(k \geq 2\). We want to lift \((b^{(k)})^\vee\). As in Theorem 25, we define \(\widetilde{\varphi}_n=0\) for \(n >|b^{(k)}|\) and \(\widetilde{\varphi}_{|b^{(k)}|}:P^{|b^{(k)}|} \to P^0\) where \(\widetilde{\varphi}_{|b^{(k)}|}(b^{(k)})=1\). But we want \(\widetilde{\varphi}_{|b^{(k)}|}\) to satisfy the PD relations, so \(1=\widetilde{\varphi}_{|b^{(k)}|}(b^{(k)})=\widetilde{\varphi}_{|b|}(b)b^{(k-1)}=0\) as \(\widetilde{\varphi}_{|b|}=0\), which is a contradiction. Likewise, if \(Z=b^{(f_1)}b^{(f_2)}\) as in the previous proof. Then \(\widetilde{\varphi}_{|b^{(f_1)}b^{(f_2)}|}(b^{(f_1)}b^{(f_2)})=1\), \(\widetilde{\varphi}_{|b^{(f_1)}|}(b^{(f_1)})=0\), and \(\widetilde{\varphi}_{|b^{(f_2)}|}(b^{(f_2)})=0\). Hence \(1=\widetilde{\varphi}_{|b^{(f_1)}b^{(f_2)}|}(b^{(f_1)}b^{(f_2)})=\widetilde{\varphi}_{|b^{(f_1)}|}(b^{(f_1)})b^{(f_2)}+(-1)^{|b^{(f_1)}b^{(f_2)}||b^{(f_1)}|}b^{(f_1)}\widetilde{\varphi}_{|b^{(f_2)}|}(b^{(f_2)})=0\), which is also a contradiction.

Based on Theorem 26 and Eq.@eq:eq:neg95hom , we have the following situation: \[\begin{array}{cccc} \Psi: & H^*(\mathcal{D}er_A^{*,\operatorname{pd}}(P^*,P^*)) & \to& H^*(\mathcal{H}om_A^{*}(P^*,P^*)) \\ &\{\text{degree }\leq 0\} & \mapsto & 0 \\ &\text{degree }n \geq 1 & \mapsto & A/I\text{-linear span of }\{b^\vee \;| \;b \in \mathcal{B}(-n)\} \subset H^n(\mathcal{H}om_A^{*}(P^*,P^*)). \end{array}\] where \(H^{<0}(\mathcal{H}om_A^{*}(P^*,P^*))=0\) by Lemma 7.

We have seen in Theorem 26 that \(\Psi\) is not surjective as products of \(b^\vee\) cannot be reached. Moreover, \(\Psi\) might not be injective. Indeed, using Theorem 26, if we write \(\widetilde{\varphi^{b}}\) for the lift of \(b^\vee\) and set \(a \in A\), then we see that \(\Psi(a \widetilde{\varphi^{b}})=\overline{a}b^\vee\). Hence if \(\overline{a}=\overline{a'}\) with \(a \neq a'\), then \(\Psi(a \widetilde{\varphi^{b}})=\Psi(a' \widetilde{\varphi^{b}})\) but \(a \widetilde{\varphi^{b}} \neq a' \widetilde{\varphi^{b}}\).

It was stated in Theorem 22 that \(H^*(\mathcal{D}er_A^{*,\operatorname{pd}}(P^*,P^*))\) is a restricted graded Lie algebra with \(q(D)=D \circ D\). However, we see from Lemma 4 that \(\Psi(q(D))=\Psi(D) \circ \Psi(D)=-\Psi(D) \cdot \Psi(D)\) as \(|D|\) odd. On \(\operatorname{Im}(\Psi)\) we define two quadratic maps \(q_{yon}:\Psi(D) \mapsto \Psi(D) \cdot \Psi(D)\) and \(q_{comp}:\Psi(D) \mapsto \Psi(D) \circ \Psi(D)\). We also write \([\cdot,\cdot]_{yon}\) and \([\cdot,\cdot]_{comp}\) for the super commutator with respect to the Yoneda and composition products, respectively. Then we can verify that \(([\cdot,\cdot]_{yon}, q_{yon})\) and \(([\cdot,\cdot]_{comp}, q_{comp})\) provide \(\pi^*(A,I)\) with two different restricted graded Lie algebra structures. In what follows, we will only consider the structure coming from the Yoneda product and write only \(([\cdot,\cdot], q)\) for the Lie bracket and the quadratic map.

We give below the last theorem of this section, providing a connection between the homotopy Lie algebra of \(A\) and the PD dg derivations of the resolution \(P^*\).

5 Homotopy Lie algebras and complete intersections↩︎

5.1 The Yoneda algebra of a complete intersection algebra↩︎

Let \(\mathsf{k}\) be a commutative Noetherian ring and set \(A=\mathsf{k}[x_1,\dots, x_n]/(c_1,\dots, c_k)\) a complete intersection \(\mathsf{k}\)-algebra, i.e., \(c_1, \dots, c_k \in \mathsf{k}[x_1, \dots, x_n]\) is a regular sequence. This means that \(c_{i+1}\) is not a zero divisor in \(\mathsf{k}[x_1,\dots, x_n]/( c_1,\dots, c_i)\) for all \(i\). Set \(\mathfrak{m}=(x_1, \dots , x_n) \subset k[x_1, \dots, x_n]\). The polynomial ring \(\mathsf{k}[x_1, \dots, x_n]\) is graded with \(|x_i|=1\), and so is \(\mathfrak{m}\). We will take about polynomial degree (not to be confused with the polynomial degree of Section 4, but the difference will be clear by the context). We assume that for all \(1 \leq p \leq k\), we have \(c_p \in \mathfrak{m}^2\) and write \(c_p=\sum_{i=1}^n c_{p, i}x_i\) with \(c_{p, i} \in \mathfrak{m}\). Let \(\mathfrak{m}_x=(\overline{x_1}, \dots, \overline{x_{n}})\) be the ideal of \(A\) which is the image of \(\mathfrak{m}\). We are interested in the \(A\)-module \(\mathsf{k}=A/\mathfrak{m}_x\). Using a reasoning similar to that of [19], a corollary of Theorem 12 in the complete intersection case states that the free resolution \(P^*\) of the \(A\)-module \(\mathsf{k}\) is given by \(P_2\), i.e., the complex \(P_2\) is already exact. Hence \(P^*\) is generated as a PD dg algebra by the set \[\{T_i, S_p \;| \;1 \leq i \leq n, \, 1 \leq p \leq k, \, \operatorname{deg}T_i=-1, \, \operatorname{deg}S_p=-2\}\] and the differential \(d\) is given by:

• \(d(T_i) = \overline{x_i}\) for all \(1 \leq i \leq n\),

• \(d(S_p) = \displaystyle \sum_{i=1}^n \overline{c_{p, i}} \, T_i\) for all \(1 \leq p \leq k\).

To lighten notations, we will remove the \(\overline{\color{white}x}\) so it is assumed that we are in the quotient ring. Moreover, we will write \(S_i^{(k)}=\gamma_k(S_i)\) for the divided power structure. We see that \(d(P^*) \subseteq \mathfrak{m}_xP^{*}\), and so the resolution is \(\mathfrak{m}_x\)-minimal, implying that \(\operatorname{Ext}_A^*(\mathsf{k},\mathsf{k})=\mathcal{H}om_{A}^*(P^*, \mathsf{k})\).

For all \(1 \leq i \leq n\) and \(1 \leq p \leq k\), we define \(\alpha_i=T_i^\vee\) and \(\beta_p=S_p^\vee\) (see before Theorem 25 for the notation).

The above example illustrates a particular property of \(\pi^*(A,\mathsf{k})\), in that it is not impacted by terms of degree higher than \(2\) in the relations of \(A\). We indeed see that for the simple singularities of types \(D_{n+2}\), \(E_6\), \(E_7\) and \(E_8\), we obtain the same \(\pi^*(A,\mathsf{k})\), even if the rings of functions are different. However, those ring differences appear in terms of polynomial degree \(3\) and above in the relations. This particularity of \(\pi^*(A,\mathsf{k})\) is due to the fact that we are looking at the trivial \(A\)-module \(\mathsf{k}\). Indeed, if we were to add to the relation \(c\) of \(A_X\) a term \(x^{a_x}y^{a_y}z^{a_z}\) with \(a_x+a_y+a_z \geq 3\) and \(a_x \geq 1\), then we could choose the differential of the corresponding element \(S\) of degree \(-2\) in the resolution so that it contains \(x^{a_x-1}y^{a_y}z^{a_z}T_x\). It follows that if we lift the morphism \(\alpha_x\), then \((\widetilde{\alpha_x})_{-2}(S)\) contains \(x^{a_x-2}y^{a_y}z^{a_z}T_x\) if \(a_x \geq 2\), or \(y^{a_y-1}z^{a_z}T_y\), or \(y^{a_y}z^{a_z-1}T_z\) otherwise, depending on the values of \(a_y\) and \(a_z\). In all cases, we see that \((\widetilde{\alpha_x})_{-2}(S)\) contains a term of the form \(p_iT_i\) where \(p_i\) is a monomial of degree \(\geq 1\). Hence \(\alpha_i \circ (\widetilde{\alpha_x})_{-2}(S)\) will contain \(p_i\alpha_i(T_i)=p_i1=0\) because \(\mathsf{k}\) is the trivial \(A\)-module. Therefore if we were to modify the exponents in \(x^{a_x}y^{a_y}z^{a_z}\), as long as their sum remains above \(3\), the Yoneda product will remain the same. This can be seen more generally in Theorem 29, because in the presentation of \(\pi^*(A,\mathsf{k})\), the relations are constructed from the terms of polynomial degree \(2\) in the relations, and higher degree terms do not interfere.

Let \(A=\mathsf{k}[x_1,\dots, x_n]/(c_1,\dots, c_k)\) be a \(\mathsf{k}\)-algebra such that for all \(1 \leq p \leq k\), we have \(c_p \in \mathfrak{m}^2\). Let \(\lfloor c_p \rfloor \in \mathfrak{m}^2\) be the homogeneous polynomial degree \(2\) part of \(c_p\). We define \(\lfloor A \rfloor= \mathsf{k}[x_1,\dots, x_n]/(\lfloor c_1 \rfloor,\dots, \lfloor c_k \rfloor)\), which is a quadratic algebra. Even if \(A\) is a complete intersection, the algebra \(\lfloor A \rfloor\) might not be a complete intersection. Due to the above discussion, we obtain a corollary to the previous theorem.

We conclude this section by drawing a parallel between Theorem 29 and the following result of G. Sjödin:

When the algebra \(A\) in Theorem 29 is a complete intersection with the relations in \(\mathfrak{m}^2\), we know from Theorem 28 that the Yoneda algebra is finitely generated. So we want to focus on the numerical condition in Sjödin’s result. The number \(\binom{n+1}{2}\) is the dimension of the space of homogeneous polynomials of degree \(2\) in \(n\) variables. On the other hand, \(\operatorname{dim}_{\mathsf{k}} \mathfrak{m}_x^2/\mathfrak{m}_x^3\) is the dimension of the space of homogeneous polynomials of degree \(2\) in \(A\). Hence Sjödin’s criteria means that the degree \(2\) homogeneous subspace of \(A\) is as big as possible. A consequence of Theorem 29 is that the Yoneda algebra is strictly graded commutative if and only if \(c_p \in \mathfrak{m}^3\) for all \(p\). However, if this condition is verified, then all monomials of degree \(2\) in \(A\) are linearly independent, and so Sjödin’s criteria is verified. If a relation \(c_p\) contains terms of degree \(2\), it can be written as \(c_p=\lfloor c_p \rfloor+(c_p)_{\geq3}\) with \((c_p)_{\geq3} \in \mathfrak{m}^3\). But as \(\overline{c_p}=0\) in \(A\), we see that \(\overline{\lfloor c_p \rfloor}= -\overline{(c_p)_{\geq3}}\) and so \(\overline{\lfloor c_p \rfloor} \in \mathfrak{m}_x^3\), creating a linear dependency between elements in \(\mathfrak{m}_x^2/\mathfrak{m}_x^3\). Thus \(\mathfrak{m}_x^2/\mathfrak{m}_x^3\) cannot have the highest possible dimension. It follows that we have an equivalence: \[\operatorname{dim}_{\mathsf{k}} \mathfrak{m}_x^2/\mathfrak{m}_x^3=\binom{n+1}{2} \Longleftrightarrow c_p \in \mathfrak{m}^3 \;\forall \;p.\] Moreover, we know from Theorem 29 that for a complete intersection ring, having \(c_p \in \mathfrak{m}^3\) for all \(p\) is equivalent to \(\pi^*(R, \mathsf{k})\) having trivial Lie bracket. We thus obtain the following corollary to Theorem 31:

The advantage of Sjödin’s theorem is that \(R\) being a complete intersection is not a hypothesis but a part of the result. However, his result does not give any information about the structure of \(\pi^*(R, \mathsf{k})\) when the above equality is not verified. On the other hand, Theorem 29 gives a presentation of the restricted graded Lie algebra even if the equality is not verified.

5.2 Reconstruction of complete intersections↩︎

In this section, we show how to reconstruct an algebra from its homotopy Lie algebra. We will first need results related to Gröbner bases.

Let \(\operatorname{Pol}=\mathsf{k}[x_1, \dots, x_n]\) be a polynomial ring with \(\mathsf{k}\) a field. For \(\mathbf{a}=(a_1,\dots,a_n) \in \mathbb{N}^n\), we write \(x^\mathbf{a}=x_1^{a_1} \cdots x_n^{a_n} \in \operatorname{Pol}\). We will refer to notations of [51].

The following result seems to be well-known to the commutative algebra community, but we failed to find a reference. We provide a proof in Appendix 6 for the sake of completeness.

Let \(\mathfrak{g}=\mathfrak{g}_1 \oplus \mathfrak{g}_2\) be a finite dimensional restricted graded Lie algebra over a field \(\mathsf{k}\) such that \(\dim_{\mathsf{k}} \mathfrak{g}_1 \geq \dim_{\mathsf{k}} \mathfrak{g}_2\). We write \(\{\alpha_1, \dots, \alpha_n \}\) and \(\{\beta_1, \dots, \beta_k\}\) for bases of \(\mathfrak{g}_1\) and \(\mathfrak{g}_2\) respectively. For each \(1 \leq p \leq k\), we construct the matrix \(N^p=(n^{p}_{i,j})\in M_n(\mathsf{k})\) given by the structure constants of \(\mathfrak{g}\), i.e., \([\alpha_i,\alpha_j]=\sum_{p=1}^k (n^p_{i,j}+n^p_{j,i})\beta_p\) for \(i \neq j\) and \(q(\alpha_i)=\sum_{p=1}^k n^p_{i,i}\beta\). We can now prove the following result:

5.3 Finite generation of the Yoneda algebra↩︎

Let \(\mathsf{k}\) be a commutative ring and let \(A=\mathsf{k}[x_1,\dots, x_n]/(c_1,\dots, c_k)\) be a finitely generated commutative \(\mathsf{k}\)-algebra such that the resolution \(P^*\) of Theorem 12 of the trivial \(A\)-module \(\mathsf{k}=A/\mathfrak{m}_x\) is obtained after finitely many steps and each \(F_n\) is of finite rank over \(A\) (see the proof of Theorem 12 for the notation). It follows that \(\mathcal{B}\) is a finite set and we write \(N\) for its cardinal. We also assume that \(P^*\) is an \(\mathfrak{m}_x\)-minimal resolution of \(\mathsf{k}\), i.e., \(d(P^*) \subset \mathfrak{m}_x P^*\), and so \(\operatorname{Ext}_{A}^*(\mathsf{k}, \mathsf{k})=\mathcal{H}om^*_A(P^*, \mathsf{k})\). We know from Proposition 27 that for any \(n >0\), the space \(\operatorname{Ext}_A^n(\mathsf{k}, \mathsf{k})\) is spanned by elements of the form \((b^\vee)^f\) with \(f \in \mathcal{B}(-n)\). In particular, we have seen that \(\operatorname{Ext}_{A}^*(\mathsf{k}, \mathsf{k})\) is generated over \(\mathsf{k}\) (with the Yoneda product) by the subspace \(V=\bigoplus_{b \in \mathcal{B}} \mathsf{k} b^\vee\), and any element of \(\operatorname{Ext}_{A}^*(\mathsf{k}, \mathsf{k})\) is an ordered polynomial in the \(b^\vee\). Assuming \(\mathsf{k}\) is a field, it follows that \[\bigoplus_{s=0}^k\dim_{\mathsf{k}} V^s \leq \displaystyle \bigoplus_{s=0}^k \binom{N-1+s}{N-1} \leq \displaystyle (k+1)(N+k)^{N-1},\] where \(V^s\) is the subspace of \(\operatorname{Ext}_{A}^*(\mathsf{k}, \mathsf{k})\) spanned by the \((b^\vee)^f\) with \(\operatorname{poldeg}f=s\). It follows that \(\operatorname{GKdim}_{\mathsf{k}}\operatorname{Ext}_{A}^*(\mathsf{k}, \mathsf{k}) \leq N\) (cf. [52]) and the Yoneda algebra has finite Gelfand-Kirillov dimension. Based on [17], it follows that the local ring \(A_{\mathfrak{m}_x}\) is a complete intersection. Furthermore, as stated in Section 5.1, if \(A\) is a complete intersection, then there exists a resolution \(P^*\) obtained after adjoining variables of degree \(-2\). This resolution is \(\mathfrak{m}_x\)-minimal if the relations of \(A\) are at least of polynomial degree \(2\). By combining these results, we obtain:

We now assume that \(A\) is local and not a complete intersection. Hence its \(\mathfrak{m}_x\)-minimal PD dg resolution (if it exists) will need an infinite set \(\mathcal{B}\). It follows that its Yoneda algebra will be generated by infinitely many variables. If we assume that there are relations so that \(\operatorname{Ext}_A^*(\mathsf{k},\mathsf{k})\) only needs a finite subset \(\mathcal{B}'\) of \(\mathcal{B}\), then the previous reasoning still applies and so any element will be a linear combination of elements of the form \((b'^\vee)^f\) with \(b' \in \mathcal{B}'\). But then we can use the same reasoning as the proof of Theorem 35 to see that \(A\) is a complete intersection, which is a contradiction. Hence \(\operatorname{Ext}_A^*(\mathsf{k},\mathsf{k})\) is generated by an infinite set \(\mathcal{B}^\vee=\{b^\vee \;| \;b \in \mathcal{B}\}\), and for each \(b^\vee \in \mathcal{B}^\vee\), we have \(b^\vee \notin (b'^\vee \;| \;b' \in \mathcal{B}\backslash \{b\})\). It follows that \(\pi^*(A, \mathsf{k})\) will have infinitely many homogeneous components. In particular, we get the following corollary:

6 Computations on PD dg algebras↩︎

Consider the settings and notation of Section 5.1.

We give below the proof of a lemma in Section 5.2.

7 Some homological exercises↩︎

Let \(\mathcal{C}\) be an abelian category. We denote by \(\mathcal{C}h(\mathcal{C})\) the category of cochain complexes in \(\mathcal{C}\) with morphisms being chain maps, and \(\mathcal{H}(\mathcal{C})\) the homotopy category. The proof of the following lemma is a straightforward exercise by doing a downward induction on the cohomological degree.

8 Square of PD derivations of odd degrees↩︎

References↩︎