Twisted commutative algebras

2.1

The notion of a twisted commutative algebra goes back a long time, at least to Barratt [Reference BarrattBar78] and Joyal [Reference JoyalJoy86]. Twisted commutative algebras have received a lot of recent attention because of their role in recent work on stable representation theory by Sam, Snowden and others – see, for example, [Reference Sam and SnowdenSS15] for a possible starting point. In this section we will review the definition and set up our conventions. In particular, we caution the reader about our non-standard choice that all our twisted commutative algebras will be non-unital by default.

2.2

Let $\mathsf{Fin}_{+}$ be the groupoid of non-empty finite sets and bijections. We make $\mathsf{Fin}_{+}$ a non-unital symmetric monoidal category using the disjoint union of finite sets. A twisted commutative dg algebra (tcdga) is a lax symmetric monoidal functor $A:\mathsf{Fin}_{+}\rightarrow \mathsf{Ch}_{R}$. Explicitly, for each non-empty finite set $S$ we are given a cochain complex $A(S)$, we are given functorial chain maps $A(S)\otimes A(T)\rightarrow A(S\sqcup T)$ for any pair of non-empty finite sets, and the following two diagrams commute:

and

A morphism of tcdgas is a symmetric monoidal natural transformation. We say that a morphism $A\rightarrow A^{\prime }$ is a quasi-isomorphism if $A(S)\rightarrow A^{\prime }(S)$ is a quasi-isomorphism for all $S$.

2.3

If $A$ is a tcdga, then so is its cohomology $H^{\bullet }(A)$, where we set $H^{\bullet }(A)(S)=H^{\bullet }(A(S))$.

2.4

A skeleton of the category $\mathsf{Fin}_{+}$ is given by the disjoint union of all symmetric groups $\mathbb{S}_{n}$, $n\geqslant 1$. It follows that a tcdga can also be described as a sequence of $\mathbb{S}_{n}$-representations $A(n)$ in the category of dg $R$-modules, together with $\mathbb{S}_{n}\times \mathbb{S}_{m}$-equivariant multiplication maps

$$\begin{eqnarray}A(n)\otimes A(m)\rightarrow A(n+m)\end{eqnarray}$$

which are associative, and for which the following diagram commutes:

The left vertical arrow is the map switching the two tensor factors (and taking into account the Koszul sign rule), and the right vertical arrow is given by acting with the ‘box permutation’ in $\mathbb{S}_{n+m}$ that moves the first $n$ elements past the last $m$ elements. We refer to $A(n)$ as the arity-$n$component of the tcdga $A$. We will switch freely between both definitions of a tcdga.

2.5

Let $\unicode[STIX]{x1D6FA}$ be a not necessarily unital commutative dg algebra over $R$. We can associate to $\unicode[STIX]{x1D6FA}$ a ‘constant’ tcdga $\text{}\underline{\unicode[STIX]{x1D6FA}}$ by the rule that $\text{}\underline{\unicode[STIX]{x1D6FA}}(S)=\unicode[STIX]{x1D6FA}$ for every non-empty finite set $S$, any bijection $S\rightarrow T$ is mapped to the identity map $\unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6FA}$, and the maps $\text{}\underline{\unicode[STIX]{x1D6FA}}(S)\otimes \,\text{}\underline{\unicode[STIX]{x1D6FA}}(T)\rightarrow \text{}\underline{\unicode[STIX]{x1D6FA}}(S\sqcup T)$ are given by the multiplication in $\unicode[STIX]{x1D6FA}$. In this way we can think of the category of tcdgas as an enlargement of the category of cdgas. Whenever we say that we consider a cdga $\unicode[STIX]{x1D6FA}$ as a tcdga, this is the construction we have in mind.

2.6

Later, we will also briefly consider arbitrary lax monoidal functors $\mathsf{Fin}_{+}\rightarrow \mathsf{Ch}_{R}$, not just lax symmetric monoidal functors. A lax monoidal functor $A:\mathsf{Fin}_{+}\rightarrow \mathsf{Ch}_{R}$ is called a twisted associative dg algebra.

2.7

A definition equivalent to § 2.2 is that a tcdga is a reduced left module over the operad $\mathsf{Com}$ of non-unital commutative algebras; similarly, a twisted associative dg algebra is a reduced left module over the operad $\mathsf{Ass}$ of non-unital associative algebras. More generally, if $P$ is a dg operad, then left modules over $P$ are sometimes called twisted $P$-algebras. The terminology goes back to Barratt [Reference BarrattBar78].

Shuffle algebras

2.8

In § 6 we will study the cohomology of configuration spaces of points on $i$-acyclic spaces, with coefficients in an arbitrary ground ring $R$. There will be technical complications arising from the fact that the trivial representation of $\mathbb{S}_{n}$ is not a projective $R[\mathbb{S}_{n}]$-module. For this reason we will need to work also with a notion of a twisted commutative algebra in which one has ‘forgotten’ about the actions of the symmetric group – more precisely, we will need to work with commutative shuffle dg algebras. Commutative shuffle algebras are significantly less studied than twisted commutative algebras: shuffle algebras (not necessarily commutative) were introduced by Ronco [Reference RoncoRon11, § 2], and to my knowledge, the only explicit mention of their evident commutative analogue in the literature is in [Reference Bremner and DotsenkoBD16, § 4.6.3]. (A word of caution: there exist several completely unrelated notions of ‘shuffle algebra’ in the literature.) The reader who is only interested in working rationally may skip this subsection.

2.9

Consider the functor category $\mathsf{Ch}_{R}^{\mathsf{Fin}_{+}}$, equipped with the monoidal structure given by Day convolution:

$$\begin{eqnarray}(A\otimes B)(S)=\bigoplus _{S=T\sqcup T^{\prime }}A(T)\otimes B(T^{\prime }).\end{eqnarray}$$

It is a general fact about Day convolution that commutative monoids in this monoidal category can be canonically identified with lax symmetric monoidal functors $\mathsf{Fin}_{+}\rightarrow \mathsf{Ch}_{R}$, which gives yet another reformulation of the definition of a tcdga. Similarly, not necessarily commutative monoids in this category may be identified with twisted associative dg algebras.

2.10

Let $\mathsf{Ord}_{+}$ denote the category of finite non-empty totally ordered sets and order-preserving bijections. The functor category $\mathsf{Ch}_{R}^{\mathsf{Ord}_{+}}$ can be given a symmetric monoidal structure analogous to the one on $\mathsf{Ch}_{R}^{\mathsf{Fin}_{+}}$, namely,

$$\begin{eqnarray}(A\otimes B)(S)=\bigoplus _{S=T\sqcup T^{\prime }}A(T)\otimes B(T^{\prime }).\end{eqnarray}$$

In this formula, $S=T\sqcup T^{\prime }$ should be read as saying that $S$ is the union of two non-empty disjoint subsets $T$, $T^{\prime }$, and that the total orders on $T$ and $T^{\prime }$ are the ones inherited from $S$. But we do not insist that every element in $T$ is smaller than every element of $T^{\prime }$; that is, $S$ is not in any sense the coproduct of $T$ and $T^{\prime }$ in the category $\mathsf{Ord}_{+}$.

2.13

We can make the notion of a (commutative) shuffle dg algebra more explicit, as we did for tcdgas in § 2.4. A shuffle dg algebra is a collection of dg $R$-modules $A(n)$, $n\geqslant 1$, and for every shuffle permutation $\unicode[STIX]{x1D70B}\in \text{Sh}(n,m)\subset \mathbb{S}_{n+m}$ a composition map

$$\begin{eqnarray}\circ _{\unicode[STIX]{x1D70B}}:A(n)\otimes A(m)\rightarrow A(n+m),\end{eqnarray}$$

satisfying the following associativity law: any shuffle permutation $\unicode[STIX]{x1D70B}\in \text{Sh}(n,m,l)$ can be uniquely written as $\unicode[STIX]{x1D70B}=(\mathbf{1}_{n}\times \unicode[STIX]{x1D70E})\circ \unicode[STIX]{x1D70C}$ where $\unicode[STIX]{x1D70E}\in \text{Sh}(m,l)$ and $\unicode[STIX]{x1D70C}\in \text{Sh}(n,m+l)$, and also as $\unicode[STIX]{x1D70B}=(\unicode[STIX]{x1D70F}\times \mathbf{1}_{l})\circ \unicode[STIX]{x1D706}$ with $\unicode[STIX]{x1D70F}\in \text{Sh}(n,m)$ and $\unicode[STIX]{x1D706}\in \text{Sh}(n+m,l)$, and there is an equality

$$\begin{eqnarray}x\circ _{\unicode[STIX]{x1D70C}}(y\circ _{\unicode[STIX]{x1D70E}}z)=(x\circ _{\unicode[STIX]{x1D70F}}y)\circ _{\unicode[STIX]{x1D706}}z\end{eqnarray}$$

for any $x\in A(n)$, $y\in A(m)$, $z\in A(l)$. A shuffle dg algebra is said to be commutative if it has the following additional property: if we denote the evident bijection $\text{Sh}(n,m)\rightarrow \text{Sh}(m,n)$ by $\unicode[STIX]{x1D70B}\mapsto \unicode[STIX]{x1D70B}^{\prime }$, then we should have $x\circ _{\unicode[STIX]{x1D70B}}y=y\circ _{\unicode[STIX]{x1D70B}^{\prime }}x$ for any $\unicode[STIX]{x1D70B}\in \text{Sh}(n,m)$, $x\in A(n)$ and $y\in A(m)$.

2.14

There is an evident diagram of forgetful functors:

The functors $R\unicode[STIX]{x1D6E4}^{\otimes }$ and $R\unicode[STIX]{x1D6E4}_{c}^{\otimes }$

3.1

Before proceeding further it is perhaps useful to quickly recall some of the basics of sheaf cohomology, since this is not always part of the ‘standard toolbox’ for a working algebraic topologist. Fix a topological space $X$. We have left exact functors $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6E4}_{c}$ from sheaves of $R$-modules on $X$ to $R$-modules, given by taking global sections and global sections with compact support, respectively. Cohomology (respectively, cohomology with compact support) of a sheaf of $R$-modules is defined as the derived functors of $\unicode[STIX]{x1D6E4}$ (respectively, $\unicode[STIX]{x1D6E4}_{c}$). These derived functors can be defined using injective resolutions, but in practice they are usually computed by some other acyclic resolution. A standard choice is the Godement resolution, which has several pleasant formal properties: it is functorial, and the monoidal structure of the Godement resolution allows for the construction of functorial cup products on the chain level. The sheaves in the Godement resolution are flabby, meaning that all restriction maps are surjective. This implies that they are acyclic for both $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6E4}_{c}$. We define $R\unicode[STIX]{x1D6E4}(X,-)$ and $R\unicode[STIX]{x1D6E4}_{c}(X,-)$ to be the global sections (respectively, global sections with compact support) of the Godement resolution of a sheaf or complex of sheaves. It is somewhat more natural to consider all of the above as taking place in the derived category of sheaves of $R$-modules, that is, all operations are taken on complexes of sheaves rather than on individual sheaves, and quasi-isomorphisms are formally inverted. We will also need to consider the derived tensor product of sheaves of $R$-modules, which is instead computed by a flat resolution; a sheaf is said to be flat if all its stalks are flat $R$-modules.

3.2

Let $X$ be a topological space, and $\unicode[STIX]{x2131}$ a complex of sheaves of $R$-modules on $X$. We define two tcdgas $\unicode[STIX]{x1D6E4}^{\otimes }(X,\unicode[STIX]{x2131})$ and $\unicode[STIX]{x1D6E4}_{c}^{\otimes }(X,\unicode[STIX]{x2131})$ associated to this data: we let

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}^{\otimes }(X,\unicode[STIX]{x2131})(S)=\unicode[STIX]{x1D6E4}(X,\unicode[STIX]{x2131}^{\otimes S})\quad \text{and}\quad \unicode[STIX]{x1D6E4}_{c}^{\otimes }(X,\unicode[STIX]{x2131})(S)=\unicode[STIX]{x1D6E4}_{c}(X,\unicode[STIX]{x2131}^{\otimes S}).\end{eqnarray}$$

The multiplication maps are the obvious ones: a section of $\unicode[STIX]{x2131}^{\otimes S}$ and a section of $\unicode[STIX]{x2131}^{\otimes T}$ can be multiplied together, to produce a section of $\unicode[STIX]{x2131}^{\otimes S\sqcup T}$.

Alternatively, we may think of this construction as first taking a complex of sheaves $\unicode[STIX]{x2131}$, and constructing a sheaf of tcdgas on $X$, given by $S\mapsto \unicode[STIX]{x2131}^{\otimes S}$ (essentially the tensor algebra on $\unicode[STIX]{x2131}$). Moreover, as the functors $\unicode[STIX]{x1D6E4}(X,-)$ and $\unicode[STIX]{x1D6E4}_{c}(X,-)$ are lax symmetric monoidal,Footnote ⁴ the global sections (with or without compact support) of a sheaf of tcdgas are themselves a tcdga.

3.3

Even though $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6E4}_{c}$ are lax symmetric monoidal functors, their derived versions $R\unicode[STIX]{x1D6E4}$ and $R\unicode[STIX]{x1D6E4}_{c}$ are not – if they were, $R\unicode[STIX]{x1D6E4}(X,\mathbf{Z})$ would be a commutative cochain model for the space $X$, and it is well known that one cannot in general construct strictly commutative cochains unless one works over a field of characteristic $0$. Nevertheless we can define derived versions of the functors $\unicode[STIX]{x1D6E4}^{\otimes }$ and $\unicode[STIX]{x1D6E4}_{c}^{\otimes }$, producing a strictly commutative tcdga. This means that the ‘commutative cochain problem’ can always be solved in the larger category of twisted dg algebras.

Definition 3.5. Let $X$ be a topological space, $\unicode[STIX]{x2131}$ a bounded-below complex of sheaves of $R$-modules on $X$, $\text{gldim}(R)<\infty$. Let $\unicode[STIX]{x2112}$ be the flat and flabby resolution of $\unicode[STIX]{x2131}$ constructed in Lemma 3.4. Define

$$\begin{eqnarray}R\unicode[STIX]{x1D6E4}^{\otimes }(X,\unicode[STIX]{x2131})=\unicode[STIX]{x1D6E4}^{\otimes }(X,\unicode[STIX]{x2112})\quad \text{and}\quad R\unicode[STIX]{x1D6E4}_{c}^{\otimes }(X,\unicode[STIX]{x2131})=\unicode[STIX]{x1D6E4}_{c}^{\otimes }(X,\unicode[STIX]{x2112}).\end{eqnarray}$$

This definition is not the ‘right’ one, however, unless we impose some point-set hypotheses on $X$.

Proposition 3.6. Let $X$, $\unicode[STIX]{x2131}$ and $R$ be as in Definition 3.5. If $X$ is paracompact and Hausdorff, then

$$\begin{eqnarray}H^{\bullet }(R\unicode[STIX]{x1D6E4}^{\otimes }(X,\unicode[STIX]{x2131}))(S)\cong H^{\bullet }(X,\underbrace{\unicode[STIX]{x2131}\otimes ^{L}\cdots \otimes ^{L}\unicode[STIX]{x2131}}_{S\text{ factors}}).\end{eqnarray}$$

Moreover, if $X$ is locally compact, then

$$\begin{eqnarray}H^{\bullet }(R\unicode[STIX]{x1D6E4}_{c}^{\otimes }(X,\unicode[STIX]{x2131}))(S)\cong H_{c}^{\bullet }(X,\underbrace{\unicode[STIX]{x2131}\otimes ^{L}\cdots \otimes ^{L}\unicode[STIX]{x2131}}_{S\text{ factors}}).\end{eqnarray}$$

Commutative cochains over $\mathbf{Q}$

3.7

As remarked above, the functors $R\unicode[STIX]{x1D6E4}$ and $R\unicode[STIX]{x1D6E4}_{c}$ are not lax symmetric monoidal: if they were, one could construct strictly commutative cochains on any space. On the other hand, strictly commutative cochains do exist over $\mathbf{Q}$, as constructed by Quillen [Reference QuillenQui69] and Sullivan [Reference SullivanSul77]. Hence one might expect that when working with sheaves of $\mathbf{Q}$-vector spaces, the functors $R\unicode[STIX]{x1D6E4}$ and $R\unicode[STIX]{x1D6E4}_{c}$ can be replaced by commutative versions. This is indeed the case, using a construction due to Navarro Aznar [Reference Navarro AznarNA87, § 4].

Theorem 3.8 (Navarro Aznar).

Let $X$ be a topological space, $C_{\mathbf{Q}}^{+}(X)$ the category of bounded-below complexes of sheaves of $\mathbf{Q}$-vector spaces on $X$. There are exact lax symmetric monoidal functors

$$\begin{eqnarray}R\unicode[STIX]{x1D6E4}^{\text{TW}},R\unicode[STIX]{x1D6E4}_{c}^{\text{TW}}:C_{\boldsymbol{ Q}}^{+}(X)\rightarrow \mathsf{Ch}_{\boldsymbol{ Q}}^{+}\end{eqnarray}$$

equipped with monoidal quasi-isomorphisms of functors $R\unicode[STIX]{x1D6E4}\;\Longrightarrow \;R\unicode[STIX]{x1D6E4}^{\text{TW}}$, $R\unicode[STIX]{x1D6E4}_{c}\;\Longrightarrow \;R\unicode[STIX]{x1D6E4}_{c}^{\text{TW}}$. In particular, if $A$ is a sheaf of commutative dg $\mathbf{Q}$-algebras on $X$, then $R\unicode[STIX]{x1D6E4}^{\text{TW}}(X,A)$ and $R\unicode[STIX]{x1D6E4}_{c}^{\text{TW}}(X,A)$ are themselves commutative dg algebras.

Proof. Let $\text{Gode}^{\text{TW}}(\mathbf{Q})$ be the Godement resolution of the sheaf $\mathbf{Q}$ on $X$, defined using Navarro Aznar’s Thom–Whitney totalization, as in the proof of Theorem 3.8. Note that the lax symmetric monoidal structure of the functor $\text{Gode}^{\text{TW}}$ defines for any $n$ an $\mathbb{S}_{n}$-equivariant quasi-isomorphism

$$\begin{eqnarray}\text{Gode}^{\text{TW}}(\mathbf{Q})^{\otimes n}\stackrel{{\sim}}{\rightarrow }\text{Gode}^{\text{TW}}(\mathbf{Q}).\end{eqnarray}$$

Now we may compute the tcdga $R\unicode[STIX]{x1D6E4}^{\otimes }(X,\mathbf{Q})$ using the resolution $\text{Gode}^{\text{TW}}(\mathbf{Q})$, so that for $S\in \text{ob}\,\,\mathsf{Fin}_{+}$ we have

$$\begin{eqnarray}R\unicode[STIX]{x1D6E4}^{\otimes }(X,\mathbf{Q})(S)\simeq \unicode[STIX]{x1D6E4}(X,\text{Gode}^{\text{TW}}(\mathbf{Q})^{\otimes S})\stackrel{{\sim}}{\rightarrow }\unicode[STIX]{x1D6E4}(X,\text{Gode}^{\text{TW}}(\mathbf{Q}))\simeq R\unicode[STIX]{x1D6E4}^{\text{TW}}(X,\mathbf{Q}),\end{eqnarray}$$

where the quasi-isomorphism above is induced by the symmetric monoidal structure. ◻

Proof. Let $j:(X\setminus A)\rightarrow X$ and $i:A\rightarrow X$ be the inclusions. We get a short exact sequence

$$\begin{eqnarray}0\rightarrow j_{!}\mathbf{Q}\rightarrow \mathbf{Q}\rightarrow i_{\ast }\mathbf{Q}\rightarrow 0\end{eqnarray}$$

of sheaves on $X$. Applying the functor $R\unicode[STIX]{x1D6E4}^{\text{TW}}(X,-)$ gives the result, since $R\unicode[STIX]{x1D6E4}_{c}^{\text{TW}}(X\setminus A,\mathbf{Q})=R\unicode[STIX]{x1D6E4}^{\text{TW}}(X,j_{!}\mathbf{Q})$ (here we use $X$ compact) and $R\unicode[STIX]{x1D6E4}^{\text{TW}}(A,\mathbf{Q})=R\unicode[STIX]{x1D6E4}^{\text{TW}}(X,i_{\ast }\mathbf{Q})$.◻

3.11

We claim that the functor $R\unicode[STIX]{x1D6E4}^{\text{TW}}(-,\mathbf{Q})$ is a cochain theory on the category of CW complexes in the sense of Mandell [Reference MandellMan02]. Indeed, the functor $R\unicode[STIX]{x1D6E4}$ is homotopy invariant [Reference IversenIve86, IV.1], hence a weak homotopy invariant of CW complexes by Whitehead’s theorem. The wedge axiom is clear. It is well known that in the presence of the wedge axiom/direct limit axiom, it is enough to verify exactness/excision for finite CW complexes. But if $(X,A)$ is a pair of finite CW complexes, then exactness and excision are immediate from the preceding lemma.

The relationship with $\mathbb{E}_{\infty }$-algebras

3.14

There is a close relationship between twisted commutative dg algebras and $\mathbb{E}_{\infty }$-algebras. We briefly state the main results here; these are principally due to Sagave and Schlichtkrull [Reference Sagave and SchlichtkrullSS12] (who only treated $\mathbb{E}_{\infty }$-algebras in the category of spaces; see Richter and Shipley [Reference Richter and ShipleyRS17] for the analogous results for cochain complexes) and Pavlov and Scholbach [Reference Pavlov and ScholbachPS18]. The results of this section will not be used in the sequel and are included only to put our constructions into a broader context. We will first state the results in the non-unital setting, since the statements are cleaner in this case, and the algebras of interest to us are non-unital.

3.16

Every dg $\mathsf{I}_{+}$-algebra has an underlying tcdga, given by restriction along the functor $\mathsf{Fin}_{+}\rightarrow \mathsf{I}_{+}$. The forgetful functor from dg $\mathsf{I}_{+}$-algebras to tcdgas has a left adjoint given by the left Kan extension.

3.17

The following theorem can be deduced as a very special case of general results of Pavlov and Scholbach [Reference Pavlov and ScholbachPS18], although they do not state it in the precise form we need it. Indeed, their goal is to construct a model structure on operadic algebras in abstract symmetric spectra with respect to the stable model structure on spectra, whereas Theorem 3.18 concerns the unstable model structure, from which the stable model structure is obtained by localization. But all the properties that they use of the unstable model structure are first verified for the stable model structure. Moreover, they consider unital objects throughout, which in this context means that they consider the categories $\mathsf{Ch}_{R}^{\mathsf{Fin}}$ and $\mathsf{Ch}_{R}^{\mathsf{I}}$, where the empty set is included in the domain category. This forces them to work with a ‘positive’ model structure, in which a map $A\rightarrow B$ is a weak equivalence (fibration) if $A(S)\rightarrow B(S)$ is a weak equivalence (fibration) for non-empty $S$. The categories $\mathsf{Ch}_{R}^{\mathsf{Fin}_{+}}$ and $\mathsf{Ch}_{R}^{\mathsf{I}_{+}}$ are Quillen equivalent as symmetric monoidal non-unital model categories to $\mathsf{Ch}_{R}^{\mathsf{Fin}}$ and $\mathsf{Ch}_{R}^{\mathsf{I}}$, respectively, with their positive model structures, and this implies with only a little bit of work also a Quillen equivalence between categories of $P$-algebras in the respective categories.

Sketch of proof of Theorem 3.18.

Let $C$ denote either of the two categories $\mathsf{Ch}_{R}^{\mathsf{Fin}_{+}}$ or $\mathsf{Ch}_{R}^{\mathsf{I}_{+}}$. The strategy is to lift the projective model structure on $C$ along the forgetful-free adjunction

$$\begin{eqnarray}U:\text{Alg}(P)\leftrightarrows C:F_{P}.\end{eqnarray}$$

The non-trivial property that needs to be checked in order for the model structure to lift is that if $M\rightarrow N$ is a trivial cofibration in $C$, $A$ is a $P$-algebra, and $F_{P}(M)\rightarrow A$ is any $P$-algebra morphism then the pushout

$$\begin{eqnarray}A\rightarrow A\amalg _{F_{P}(M)}F_{P}(N)\end{eqnarray}$$

is a trivial cofibration. We may in fact take $M\rightarrow N$ to be a generating trivial cofibration, so $0=M=F_{P}(M)$ and we are computing a coproduct. Now for any $P$-algebra $A$ in $C$ there is an operad $O_{A}$ in $C$ such that $O_{A}$-algebras are equivalent to $P$-algebras with a morphism from $A$ [Reference Getzler and JonesGJ94, Lemma 1.18]. The arity-$0$ component of $O_{A}$ is isomorphic to $A$, since it is the initial object in the category of $O_{A}$-algebras. The free $O_{A}$-algebra on an object $N$ of $C$ is the pushout $A\amalg F_{P}(N)$. Now

$$\begin{eqnarray}F_{O_{A}}(N)=A\oplus \bigoplus _{n\geqslant 1}O_{A}(n)\otimes _{\mathbb{S}_{n}}N^{\otimes n}\end{eqnarray}$$

and we claim that $O_{A}(n)\otimes _{\mathbb{S}_{n}}N^{\otimes n}$ is acyclic for all $n\geqslant 1$. Indeed,

$$\begin{eqnarray}N^{\otimes n}(k)=\!\!\bigoplus _{\{1,\ldots ,k\}=S_{1}\sqcup \cdots \sqcup S_{n}}\!\!N(S_{1})\otimes \cdots \otimes N(S_{n})=R[\mathbb{S}_{n}]\otimes \!\!\bigoplus _{\substack{ \{1,\ldots ,k\}=S_{1}\sqcup \cdots \sqcup S_{n} \\ \min (S_{1})<\cdots <\min (S_{n})}}\!\!N(S_{1})\otimes \cdots \otimes N(S_{n})\end{eqnarray}$$

and each $N(S_{i})$ is an acyclic complex of free $R$-modules. Crucially, we used the fact that $N(0)=0$ in the last step to ensure that none of the subsets $S_{i}$ were empty.◻

3.20

Theorem 3.18 is perhaps surprising at first, since $R$ is a completely arbitrary ring. Hinich [Reference HinichHin97] and Harper [Reference HarperHar10] proved analogous results for $P$-algebras in $\mathsf{Ch}_{R}$ and in $\mathsf{Ch}_{R}^{\mathsf{Fin}}$ (i.e. left $P$-modules) only when $R$ contains $\mathbf{Q}$, and without this assumption there is in general no such model structure [Reference Goerss and SchemmerhornGS07, Example 3.7]. Theorem 3.18 shows that the arity-$0$ component is in fact the only obstruction to transferring the model structure: once we impose the condition that $M(0)=0$, everything works without a hitch over an arbitrary base ring. Thus twisted commutative dg algebras are ‘better behaved’ than ordinary commutative algebras in positive characteristic, as evidenced in particular by the fact that the commutative cochain problem can always be solved in the category of tcdgas, as we explained in § 3.3. There are in fact several earlier results in the literature of the following flavor: for a dg operad $P$, left $P$-modules $M$ with $M(0)=0$ exhibit ‘homotopically correct’ behavior in arbitrary characteristic, even in situations where $P$-algebras do not. We mention three examples.

1. (i) Stover [Reference StoverSto93] proved that reduced left $\mathsf{Lie}$ modules satisfy the Milnor–Moore theorem over an arbitrary base ring; the classical Milnor–Moore theorem for Lie algebras requires working over a $\mathbf{Q}$-algebra.

2. (ii) Richter [Reference RichterRic15] proved that if $A$ is a reduced left $\mathsf{Com}$-module (i.e. a non-unital tcdga), then the Harrison homology, André–Quillen homology and Gamma homology of $A$ all coincide, and give a summand of the Hochschild homology. For ordinary commutative algebras this is in general only true over $\mathbf{Q}$.

3. (iii) Fresse [Reference FresseFre00] introduced the notion of a divided power structure on a $P$-algebra. The definition admits an evident generalization to a divided power structure on a left $P$-module $M$. If $M(0)=0$, then it follows from [Reference StoverSto93, 9.10] that $M$ is always canonically equipped with a divided power structure.

3.21

We are now ready to state the relationship between $\mathsf{I}_{+}$-algebras and $\mathbb{E}_{\infty }$-algebras, which can be summed up in the following two theorems, which are simple modifications of the statements of [Reference Richter and SagaveRS18, Theorem 2.13] and [Reference Richter and SagaveRS18, Theorem 4.10].

3.24

Concretely, Theorem 3.23 says that we can think of the category of non-unital $\mathbb{E}_{\infty }$-algebras as a full subcategory of dg $\mathsf{I}_{+}$-algebras. In particular, there is then also a natural forgetful functor from non-unital $\mathbb{E}_{\infty }$-algebras to twisted commutative dg algebras, by Theorem 3.18. If $\unicode[STIX]{x1D6FA}$ is a non-unital $\mathbb{E}_{\infty }$-algebra, then by Theorem 3.23 we may associate to $\unicode[STIX]{x1D6FA}$ a fibrant dg $\mathsf{I}_{+}$-algebra $\text{}\underline{\unicode[STIX]{x1D6FA}}$, well defined up to quasi-isomorphism. Since all the maps $\text{}\underline{\unicode[STIX]{x1D6FA}}(S)\rightarrow \text{}\underline{\unicode[STIX]{x1D6FA}}(T)$ are quasi-isomorphisms, each component $\text{}\underline{\unicode[STIX]{x1D6FA}}(S)$ is in fact quasi-isomorphic to the homotopy colimit $\unicode[STIX]{x1D6FA}$, and the multiplication maps $\text{}\underline{\unicode[STIX]{x1D6FA}}(S)\otimes \text{}\underline{\unicode[STIX]{x1D6FA}}(T)\rightarrow \text{}\underline{\unicode[STIX]{x1D6FA}}(S\sqcup T)$ are given up to homotopy by the multiplication in $\unicode[STIX]{x1D6FA}$. Thus we may think of this as a homotopy coherent version of the construction of § 2.5. Composing with the forgetful functor from dg $\mathsf{I}_{+}$-algebras to twisted commutative algebras, we may in particular think of a non-unital $\mathbb{E}_{\infty }$-algebra as defining a ‘homotopically constant’ tcdga.

3.25

Let us explain how Theorems 3.22 and 3.23, which concern non-unital $\mathbb{E}_{\infty }$-algebras, follow from analogous theorems for unital $\mathbb{E}_{\infty }$-algebras. We briefly recall these results here; for a more complete account with detailed references see [Reference Richter and SagaveRS18, § 4]. Let $\mathsf{I}$ denote the unital symmetric monoidal category of all finite sets and injections. The category of functors $\mathsf{I}\rightarrow \mathsf{Ch}_{R}$ can be given a positive model structure in which $F\rightarrow G$ is a fibration (weak equivalence) if $F(S)\rightarrow G(S)$ is so for all non-empty sets $S$; no assumption is made on $F(\varnothing )\rightarrow G(\varnothing )$. We say that $F\rightarrow G$ is an $\mathsf{I}$-equivalence if $\text{hocolim}_{\mathsf{I}}\,F\rightarrow \text{hocolim}_{\mathsf{I}}\,G$ is a weak equivalence. The positive model structure admits a Bousfield localization with weak equivalences the $\mathsf{I}$-equivalences, which we call the positive $\mathsf{I}$-model structure. A functor $F$ is fibrant for this model structure if $F(S)\rightarrow F(T)$ is a weak equivalence for all $\varnothing \neq S{\hookrightarrow}T$. By an $\mathsf{I}$-algebra we mean a (unital) lax symmetric monoidal functor $\mathsf{I}\rightarrow \mathsf{Ch}_{R}$, that is, a commutative monoid in $\mathsf{Ch}_{R}^{\mathsf{I}}$. There is an induced model structure on $\mathsf{I}$-algebras, with $A\rightarrow B$ a morphism of $\mathsf{I}$-algebras a fibration (weak equivalence) when $A\rightarrow B$ is a fibration (weak equivalence) for the positive $\mathsf{I}$-model structure. Then the category of $\mathsf{I}$-algebras is Quillen equivalent to the category of unital $\mathbb{E}_{\infty }$-algebras, and if $A$ is an $\mathsf{I}$-algebra then $\text{hocolim}_{\mathsf{I}}\,A$ is an $\mathbb{E}_{\infty }$-algebra.

Now the category of non-unital $\mathbb{E}_{\infty }$-algebras in $\mathsf{Ch}_{R}$ is Quillen equivalent to the category of augmented $\mathbb{E}_{\infty }$-algebras $A\rightarrow R$, where the latter category has its natural model structure as a slice category. By the opposite of [Reference HoveyHov99, Theorem 1.3.17] and the results recalled in § 3.25, non-unital $\mathbb{E}_{\infty }$-algebras are then Quillen equivalent to the category of $\mathsf{I}$-algebras with an augmentation $A\rightarrow \text{}\underline{R}$, where $\text{}\underline{R}$ denotes the constant $\mathsf{I}$-algebra at $R$. If $A$ is such an augmented $\mathsf{I}$-algebra then its augmentation ideal $\overline{A}$, defined by $\overline{A}(S)=\text{Ker}(A(S)\rightarrow R)$, is a well-defined $\mathsf{I}_{+}$-algebra. The functor from augmented $\mathsf{I}$-algebras to $\mathsf{I}_{+}$-algebras has a left adjoint given by adjoining a unit freely in each arity. This is in fact a Quillen adjunction between augmented $\mathsf{I}$-algebras (with the positive $\mathsf{I}$-model structure) and $\mathsf{I}_{+}$-algebras (with the model structure of Theorem 3.18). The category of $\mathsf{I}_{+}$-algebras admits a Bousfield localization in which the weak equivalences are the maps inducing a weak equivalence of homotopy colimits, and the Quillen adjunction between augmented $\mathsf{I}$-algebras and $\mathsf{I}_{+}$-algebras induces a Quillen equivalence between augmented $\mathsf{I}$-algebras and this Bousfield localization, proving Theorem 3.23. The homotopy colimit of an $\mathsf{I}_{+}$-algebra is the augmentation ideal in the homotopy colimit of the $\mathsf{I}$-algebra obtained by freely adjoining a unit in each arity, proving Theorem 3.22.

Remark 3.28. A direct proof is also possible. Let us just sketch the argument. We take $X$ to be paracompact, Hausdorff and locally contractible. First one shows, using local contractibility, that there is a weak equivalence of sheaves of $\mathbb{E}_{\infty }$-algebras on $X$ between the constant sheaf $R$ and the sheafification of the complex of presheaves $C^{\bullet }(-,R)$ of singular cochains. Then their derived global sections are also weakly equivalent as $\mathbb{E}_{\infty }$-algebras. But the global sections of the sheafification of $C^{\bullet }(-,R)$ are the quotient

$$\begin{eqnarray}C^{\bullet }(X,R)/C_{0}^{\bullet }(X,R)\end{eqnarray}$$

by the subcomplex of singular cochains which are zero on some open cover of $X$, and the quotient map $C^{\bullet }(X,R)\rightarrow C^{\bullet }(X,R)/C_{0}^{\bullet }(X,R)$ is a weak equivalence of $\mathbb{E}_{\infty }$-algebras. Also, the sheafification of $C^{\bullet }(-,R)$ is flabby and its derived global sections are just its global sections. On the other hand, we claim that $\text{hocolim}_{\mathsf{I}}\,R\unicode[STIX]{x1D6E4}^{\otimes }(X,R)\cong R\unicode[STIX]{x1D6E4}(X,R)$ as $\mathbb{E}_{\infty }$-algebras; indeed, this follows from $R\unicode[STIX]{x1D6E4}^{\otimes }(X,R)$ being a fibrant $\mathsf{I}$-algebra.

Order complex

4.1

Throughout this section we let $(P,\preccurlyeq )$ denote a finite poset. We define the order complex of $P$, $\Vert P\Vert$, to be the simplicial complex whose $p$-simplices are chains $x_{0}\prec x_{1}\prec \cdots \prec x_{p}$ of comparable elements of $P$. Equivalently, $\Vert P\Vert$ is the geometric realization of the nerve of $P$, when we think of $P$ as a category.

4.2

If $P$ has either a largest or smallest element, its order complex is contractible. In this case it is common to remove the top and/or bottom element before taking the order complex; however, the following related construction will be more convenient for us. We always use $\hat{0}$ and $\hat{1}$ to denote a smallest and a largest element of $P$, respectively. Define

$$\begin{eqnarray}\displaystyle P & = & \displaystyle \Vert P\Vert /\Vert P\setminus \{\hat{1}\}\Vert ,\nonumber\\ \displaystyle P & = & \displaystyle \Vert P\Vert /\Vert P\setminus \{\hat{0}\}\Vert ,\nonumber\\ \displaystyle \unicode[STIX]{x27E6}P\unicode[STIX]{x27E7} & = & \displaystyle \Vert P\Vert /(\Vert P\setminus \{\hat{0}\}\Vert \cup \Vert P\setminus \{\hat{1}\}\Vert ).\nonumber\end{eqnarray}$$

For example, $\unicode[STIX]{x27E6}P\unicode[STIX]{x27E7}$ is the based CW complex obtained from $\Vert P\Vert$ by collapsing to a point the subcomplex consisting of all simplices not containing both $\hat{0}$ and $\hat{1}$. We always follow the standard convention of interpreting the quotient space $X/A$ as the pushout $X\leftarrow A\rightarrow \text{pt}$, so that when $A$ is empty we obtain $X$ with a disjoint basepoint.

Lemma 4.3. If $P$ has a largest element $\hat{1}$ or a smallest element $\hat{0}$ respectively, then

$$\begin{eqnarray}\displaystyle P & \cong & \displaystyle \unicode[STIX]{x1D6F4}\Vert P\setminus \{\hat{1}\}\Vert ,\nonumber\\ \displaystyle P & \cong & \displaystyle \unicode[STIX]{x1D6F4}\Vert P\setminus \{\hat{0}\}\Vert ,\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D6F4}$ denotes the unreduced suspension. If $P$ has both a top and bottom element, then

$$\begin{eqnarray}\unicode[STIX]{x27E6}P\unicode[STIX]{x27E7}\cong \left\{\begin{array}{@{}ll@{}}S^{0}\quad & \text{if }P=\{\ast \},\\ \unicode[STIX]{x1D6F4}^{2}\Vert P\setminus \{\hat{0},\hat{1}\}\Vert \quad & \text{if }|P|\geqslant 2.\end{array}\right.\end{eqnarray}$$

Lemma 4.5. Let $P$ and $Q$ be posets with largest and smallest elements. Then

$$\begin{eqnarray}\unicode[STIX]{x27E6}P\times Q\unicode[STIX]{x27E7}\cong \unicode[STIX]{x27E6}P\unicode[STIX]{x27E7}\wedge \unicode[STIX]{x27E6}Q\unicode[STIX]{x27E7},\end{eqnarray}$$

$\wedge$ denoting the smash product.

4.6

We say that $P$ is a lattice if every subset of $P$ has a unique least upper bound (join) and greatest lower bound (meet). In particular, $P$ must have a largest and smallest element, which are the empty join and meet, respectively. In a finite poset, the existence of joins implies the existence of meets, and vice versa: the meet of a subset can be defined as the join of the set of all its lower bounds.

4.7

Suppose that $P$ has a smallest element $\hat{0}$. The minimal elements of $P\setminus \{\hat{0}\}$ are called atoms.

Sheaf cohomology on posets

4.9

We say that a subset $U\subseteq P$ is upwards closed if $x\in U$ and $x\preccurlyeq y$ implies $y\in U$. We let $U(P)$ denote the collection of upwards closed subsets of $P$. The set $U(P)$ is itself a finite poset, partially ordered by inclusion. The upwards closed subsets form a topology on $P$, the Alexandrov topology.

4.10

Let $\unicode[STIX]{x1D6F7}:P\rightarrow \mathsf{Ch}_{R}$ be a functor from $P$ to cochain complexes. Let $U\subseteq P$ be upwards closed. We define a double complex

$$\begin{eqnarray}\mathfrak{B}(U,\unicode[STIX]{x1D6F7})=\bigoplus _{\varnothing \neq C\subseteq U\text{ a chain}}\unicode[STIX]{x1D6F7}(\max C),\end{eqnarray}$$

where the direct sum is taken over all strictly increasing chains of elements in $U$. The vertical grading and vertical differentials are given by the internal grading and differential of $\unicode[STIX]{x1D6F7}(x)$. The horizontal grading is given by $|C|-1$, and the horizontal differential is an alternating sum over all ways of adding an extra element to the chain; note that if $C\subset C^{\prime }$, then $\max C\preccurlyeq \max C^{\prime }$, which produces a map $\unicode[STIX]{x1D6F7}(\max C)\rightarrow \unicode[STIX]{x1D6F7}(\max C^{\prime })$. Equivalently, $\mathfrak{B}(U,\unicode[STIX]{x1D6F7})$ is the bar resolution computing the homotopy limit $\operatorname{holim}_{U}\unicode[STIX]{x1D6F7}$.

Remark 4.11. For our purposes we will never need the interpretation of $\mathfrak{B}(U,\unicode[STIX]{x1D6F7})$ as a homotopy limit. Nevertheless it may be helpful psychologically to know that it is indeed a homotopy limit, and the reader may at this point object that the bar resolution is usually only used to compute homotopy limits in simplicial model categories. But the model category of unbounded cochain complexes is not simplicial. Let us for completeness argue that this is a homotopy limit. If $\unicode[STIX]{x1D6F7}:P\rightarrow \mathsf{Ch}_{R}$ is a functor, then define

$$\begin{eqnarray}R\unicode[STIX]{x1D6F7}(x)=\bigoplus _{\substack{ \varnothing \neq C\subseteq P\text{ a chain} \\ \min (C)\succcurlyeq x}}\unicode[STIX]{x1D6F7}(\max (C))\end{eqnarray}$$

with a differential as above given by adding elements to the chain. If $x\preccurlyeq y$ then we obtain $R\unicode[STIX]{x1D6F7}(x)\rightarrow R\unicode[STIX]{x1D6F7}(y)$ by projecting onto the summands for which $\min (C)\succcurlyeq y$. We get a natural transformation $\unicode[STIX]{x1D6F7}\rightarrow R\unicode[STIX]{x1D6F7}$, mapping $\unicode[STIX]{x1D6F7}(x)$ diagonally into the summands given by singleton chains, which is a quasi-isomorphism pointwise (argue as in Proposition 4.18). Since fibrations in $\mathsf{Ch}_{R}$ are surjections, it follows that $R\unicode[STIX]{x1D6F7}$ is Reedy fibrant (i.e. all maps $R\unicode[STIX]{x1D6F7}(x)\rightarrow R\unicode[STIX]{x1D6F7}(y)$ are surjections), so $\unicode[STIX]{x1D6F7}\rightarrow R\unicode[STIX]{x1D6F7}$ is a fibrant replacement. The limit of $R\unicode[STIX]{x1D6F7}$ is precisely the bar resolution, and $P$ (being a poset) has cofibrant constants so that the homotopy limit is computed by any Reedy fibrant replacement.

4.12

Assume now that $P$ has a smallest element $\hat{0}$. We define similarly a double complex

$$\begin{eqnarray}\widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7})=\bigoplus _{C\subseteq U\text{ a chain}}\unicode[STIX]{x1D6F7}(\max C),\end{eqnarray}$$

where we now allow also the empty chain; $\max (\varnothing )=\hat{0}$ is the minimal element of $P$. Again the vertical differential and grading come from $\unicode[STIX]{x1D6F7}$, the horizontal differential is an alternating sum over all ways of adding an element to a chain, but the horizontal grading is given by the cardinality $|C|$. The two cochain complexes $\mathfrak{B}(U,\unicode[STIX]{x1D6F7})$ and $\widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7})$ differ only by an ‘augmentation’ by $\unicode[STIX]{x1D6F7}(\hat{0})$ and by a degree shift; there is a short exact sequence

$$\begin{eqnarray}0\rightarrow \unicode[STIX]{x1D6F7}(\hat{0})\rightarrow \widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7})\rightarrow \mathfrak{B}(U,\unicode[STIX]{x1D6F7})[1]\rightarrow 0.\end{eqnarray}$$

4.13

If $\unicode[STIX]{x1D6F7}$ is the constant functor taking each element to the $R$-module $R$, then the complexes $\mathfrak{B}(U,\unicode[STIX]{x1D6F7})$ and $\widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7})$ compute the cohomology (respectively, the reduced cohomology) of the order complex $\Vert U\Vert$. Indeed, there is an evident isomorphism with the complexes of (reduced) cellular chains of $\Vert U\Vert$.

Two spectral sequences

4.16

If $x\in P$, then we denote by $P^{\preccurlyeq x}$ the the lower interval $\{y\in P:y\preccurlyeq x\}$.

4.17

Let $P,U$ and $\unicode[STIX]{x1D6F7}$ be as in §§ 4.10 and 4.12. Suppose, moreover, that we are given a strictly increasing function $\unicode[STIX]{x1D70C}:P\rightarrow \mathbf{Z}$. We may define decreasing filtrations of the complexes $\mathfrak{B}(U,\unicode[STIX]{x1D6F7})$ and $\widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7})$ as

$$\begin{eqnarray}F^{p}\mathfrak{B}(U,\unicode[STIX]{x1D6F7})=\bigoplus _{\substack{ \varnothing \neq C\subseteq U\text{ a chain} \\ \unicode[STIX]{x1D70C}(\max (C))\geqslant p}}\unicode[STIX]{x1D6F7}(\max C)\end{eqnarray}$$

and

$$\begin{eqnarray}F^{p}\widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7})=\bigoplus _{\substack{ C\subseteq U\text{ a chain} \\ \unicode[STIX]{x1D70C}(\max (C))\geqslant p}}\unicode[STIX]{x1D6F7}(\max C).\end{eqnarray}$$

Proposition 4.18. The above filtration on $\mathfrak{B}(U,\unicode[STIX]{x1D6F7})$ has the property that

$$\begin{eqnarray}H^{\bullet }(\text{Gr}^{p}\mathfrak{B}(U,\unicode[STIX]{x1D6F7}))=\bigoplus _{\substack{ x\in U \\ \unicode[STIX]{x1D70C}(x)=p}}\widetilde{H}^{\bullet }(U^{\preccurlyeq x},\unicode[STIX]{x1D6F7}(x)).\end{eqnarray}$$

For $U\subsetneq P$, let $U_{0}=U\cup \{\hat{0}\}$. Then

$$\begin{eqnarray}H^{\bullet }(\text{Gr}^{p}\widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7}))=\bigoplus _{\substack{ x\in U_{0} \\ \unicode[STIX]{x1D70C}(x)=p}}\widetilde{H}^{\bullet }(\unicode[STIX]{x27E6}U_{0}^{\preccurlyeq x}\unicode[STIX]{x27E7},\unicode[STIX]{x1D6F7}(x)).\end{eqnarray}$$

Proof. Clearly we have

$$\begin{eqnarray}\text{Gr}^{p}\mathfrak{B}(U,\unicode[STIX]{x1D6F7})=\bigoplus _{\substack{ \varnothing \neq C\subseteq U\text{ a chain} \\ \unicode[STIX]{x1D70C}(\max C)=p}}\unicode[STIX]{x1D6F7}(\max C).\end{eqnarray}$$

If two chains in the complex have different maximal elements, then the two chains are incomparable. (This is where we use the fact that $\unicode[STIX]{x1D70C}$ was assumed strictly increasing.) Thus this complex becomes a direct sum of complexes indexed by the possible maximal elements, that is, the elements $x\in U$ with $\unicode[STIX]{x1D70C}(x)=p$. The summand corresponding to $x$ is given by

$$\begin{eqnarray}\bigoplus _{\substack{ C\subseteq U\text{ a chain} \\ \max C=x}}\unicode[STIX]{x1D6F7}(x).\end{eqnarray}$$

But the chains in $U$ whose largest element equal $x$ can be identified with the reduced cellular cochains of $U^{\preccurlyeq x}$, so this summand is equal to $\widetilde{C}^{\bullet }(U^{\preccurlyeq x})\otimes \unicode[STIX]{x1D6F7}(x)$. The cohomology of this complex is $\widetilde{H}^{\bullet }(U^{\preccurlyeq x},\unicode[STIX]{x1D6F7}(x))$.

The argument for $\widetilde{\mathfrak{B}}$ is similar.◻

Proposition 4.19. Under the above assumptions, there are spectral sequences

$$\begin{eqnarray}E_{1}^{pq}=\bigoplus _{\substack{ x\in U \\ \unicode[STIX]{x1D70C}(x)=p}}\widetilde{H}^{p+q}(U^{\preccurlyeq x};\unicode[STIX]{x1D6F7}(x))\;\Longrightarrow \;H^{p+q}(\mathfrak{B}(U,\unicode[STIX]{x1D6F7}))\end{eqnarray}$$

and

$$\begin{eqnarray}E_{1}^{pq}=\bigoplus _{\substack{ x\in U_{0} \\ \unicode[STIX]{x1D70C}(x)=p}}\widetilde{H}^{p+q}(\unicode[STIX]{x27E6}U_{0}^{\preccurlyeq x}\unicode[STIX]{x27E7};\unicode[STIX]{x1D6F7}(x))\;\Longrightarrow \;H^{p+q}(\widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7})).\end{eqnarray}$$

4.20

These spectral sequences, in the context of sheaf cohomology on posets, were first considered by Bacławski [Reference BacławskiBac75, § 4].

Proposition 4.21. Let $P,U$ and $\unicode[STIX]{x1D6F7}$ be as in §§ 4.10 and 4.12, and assume that $P$ is a lattice. Let $\unicode[STIX]{x1D70C}:P\rightarrow \mathbf{Z}$ be strictly increasing. Let $U\subsetneq P$ be upwards closed and let $J_{U}$ be subposet consisting of all joins of minimal elements of $U$. Let $U_{0}=U\cup \{\hat{0}\}$ and $J_{U_{0}}=J_{U}\cup \{\hat{0}\}$. The spectral sequences of Proposition 4.19 may be simplified to

$$\begin{eqnarray}E_{1}^{pq}=\bigoplus _{\substack{ x\in J_{U} \\ \unicode[STIX]{x1D70C}(x)=p}}\widetilde{H}^{p+q}(J_{U}^{\preccurlyeq x};\unicode[STIX]{x1D6F7}(x))\;\Longrightarrow \;H^{p+q}(\mathfrak{B}(U,\unicode[STIX]{x1D6F7}))\end{eqnarray}$$

and

$$\begin{eqnarray}E_{1}^{pq}=\bigoplus _{\substack{ x\in J_{U_{0}} \\ \unicode[STIX]{x1D70C}(x)=p}}\widetilde{H}^{p+q}(\unicode[STIX]{x27E6}J_{U_{0}}^{\preccurlyeq x}\unicode[STIX]{x27E7};\unicode[STIX]{x1D6F7}(x))\;\Longrightarrow \;H^{p+q}(\widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7})).\end{eqnarray}$$

Semidirect product

4.22

Let $G$ be a group acting on $P$. Define the semidirect product $G\ltimes P$ as the following category: the objects of $G\ltimes P$ are the elements of $P$, and a morphism $x\rightarrow y$ is an element $g\in G$ such that $g\cdot x\preccurlyeq y$. Composition of morphisms is given by group multiplication. This is an instance of the Grothendieck construction, when we think of $P$ as a category and the action of $G$ on $P$ as defining a functor $G\rightarrow \mathsf{Cat}$. Note that there is a functor $G\ltimes P\rightarrow G$, and that $P$ sits inside $G\ltimes P$ as the subcategory consisting of all morphisms given by the identity element in $G$. Hence we get a ‘short exact sequence’ of sorts,

$$\begin{eqnarray}P\rightarrow G\ltimes P\rightarrow G.\end{eqnarray}$$

4.23

Suppose that $\unicode[STIX]{x1D6F7}$ is a functor $G\ltimes P\rightarrow \mathsf{Ch}_{R}$. We may in particular restrict $\unicode[STIX]{x1D6F7}$ to $P$, which allows us to make sense of $\mathfrak{B}(U,\unicode[STIX]{x1D6F7})$ and $\widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7})$ for $U\subseteq P$ upwards closed. However, the action of $G$ furnishes an additional functoriality: if $g\cdot U=V$, for $U,V\subseteq P$ upwards closed, then multiplication by $g$ induces a map $\mathfrak{B}(U,\unicode[STIX]{x1D6F7})\rightarrow \mathfrak{B}(V,\unicode[STIX]{x1D6F7})$. Indeed, if $g\cdot x=y$ then $g$ gives a morphism $x\rightarrow y$ in $G\ltimes P$, so $\unicode[STIX]{x1D6F7}(g)$ is a morphism $\unicode[STIX]{x1D6F7}(x)\rightarrow \unicode[STIX]{x1D6F7}(y)$; we use this to map each summand of $\mathfrak{B}(U,\unicode[STIX]{x1D6F7})$ to a corresponding summand of $\mathfrak{B}(V,\unicode[STIX]{x1D6F7})$. We record this as a proposition.

Sheaves on the partition lattice

5.1

Let $S$ be a finite set. Let $\unicode[STIX]{x1D6F1}_{S}$ denote the poset consisting of all equivalence relations on $S$, ordered by refinement: we say that ${\sim}\,\,\preccurlyeq \,\,{\sim}^{\prime }$ if $x\sim y$ implies $x{\sim}^{\prime }y$. The poset $\unicode[STIX]{x1D6F1}_{S}$ is a lattice: the join of ${\sim}$ and ${\sim}^{\prime }$ is the relation ${\sim}^{\prime \prime }$ defined by $x{\sim}^{\prime \prime }y\;\Longleftrightarrow \;x\sim y$ or $x{\sim}^{\prime }y$; the meet of ${\sim}$ and ${\sim}^{\prime }$ is given by $x{\sim}^{\prime \prime }y\;\Longleftrightarrow \;x\sim y$ and $x{\sim}^{\prime }y$. We call $\unicode[STIX]{x1D6F1}_{S}$ the partition lattice. When $S=\{1,\ldots ,n\}$ we denote it simply by $\unicode[STIX]{x1D6F1}_{n}$.

5.2

We usually prefer to think of the elements of $\unicode[STIX]{x1D6F1}_{S}$ as partitions of $S$ into non-empty disjoint subsets. The smallest element is the partition of $S$ into one-element blocks, and the largest element is the partition of $S$ into a single block. In an abuse of notation we denote a typical partition $S=\coprod _{i\in I}T_{i}$ by the symbol $T$.

5.3

Let $A$ be a tcdga (§ 2.2). Then we obtain a functor $\unicode[STIX]{x1D6F7}_{A}:\unicode[STIX]{x1D6F1}_{S}\rightarrow \mathsf{Ch}_{R}$, for any finite set $S$. Specifically, if $T$ denotes the partition $S=\coprod _{i\in I}T_{i}$, then

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{A}(T)=\bigotimes _{i\in I}A(T_{i}).\end{eqnarray}$$

If $T\preccurlyeq T^{\prime }$ then $T^{\prime }$ is obtained from $T$ by merging some blocks of the partition. The map $\unicode[STIX]{x1D6F7}_{A}(T)\rightarrow \unicode[STIX]{x1D6F7}_{A}(T^{\prime })$ is given by applying the maps $A(T_{i})\otimes A(T_{j})\rightarrow A(T_{i}\sqcup T_{j})$ furnished by the tcdga structure.

5.4

In fact to define a functor $\unicode[STIX]{x1D6F1}_{S}\rightarrow \mathsf{Ch}_{R}$ we do not need the full data of a tcdga. Let us instead suppose that $S$ is totally ordered. Then every subset of $S$ inherits a natural total order as well, so that for each partition $\coprod _{i\in I}T_{i}$ of $S$ we get an induced total ordering on each of the blocks $T_{i}$. If $A$ is a commutative dg shuffle algebra (§2.11) then we may similarly define

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{A}(T)=\bigotimes _{i\in I}A(T_{i}),\end{eqnarray}$$

where the maps $\unicode[STIX]{x1D6F7}_{A}(T)\rightarrow \unicode[STIX]{x1D6F7}_{A}(T^{\prime })$ are now given by merging blocks using the shuffle algebra structure on $A$.

5.5

Note that if $A$ and $A^{\prime }$ are quasi-isomorphic as tcdgas (or as commutative dg shuffle algebras), then $\unicode[STIX]{x1D6F7}_{A}$ and $\unicode[STIX]{x1D6F7}_{A^{\prime }}$ are quasi-isomorphic as functors.

5.6

The difference between the constructions of §§ 5.3 and 5.4 is that when $A$ is a tcdga we will not only get a functor $\unicode[STIX]{x1D6F1}_{n}\rightarrow \mathsf{Ch}_{R}$ for all $n$, but in fact a functor

$$\begin{eqnarray}\mathbb{S}_{n}\ltimes \unicode[STIX]{x1D6F1}_{n}\rightarrow \mathsf{Ch}_{R}\end{eqnarray}$$

from the semidirect product (§ 4.22) of $\unicode[STIX]{x1D6F1}_{n}$ with $\mathbb{S}_{n}$, where $\mathbb{S}_{n}$ acts in the obvious way on the partition lattice. This will have the consequence that the various spectral sequences we write down for the cohomology of configuration spaces are $\mathbb{S}_{n}$-equivariant.

The functors $C\unicode[STIX]{x2131}$ and $CD$, and the connection with configuration spaces

Definition 5.7. Let $U\subset \unicode[STIX]{x1D6F1}_{n}$ be upwards closed, and let $A$ be a tcdga. We define

$$\begin{eqnarray}\displaystyle CD(U,A) & = & \displaystyle \mathfrak{B}(U,\unicode[STIX]{x1D6F7}_{A}),\nonumber\\ \displaystyle C\unicode[STIX]{x2131}(U,A) & = & \displaystyle \widetilde{\mathfrak{B}}(U,\unicode[STIX]{x1D6F7}_{A}),\nonumber\end{eqnarray}$$

where $\mathfrak{B}$ and $\widetilde{\mathfrak{B}}$ are defined in §§ 4.10 and 4.12, and $\unicode[STIX]{x1D6F7}_{A}$ is defined in § 5.3.

Remark 5.8. We observe that $CD$ and $C\unicode[STIX]{x2131}$ are functors

$$\begin{eqnarray}(\mathbb{S}_{n}\ltimes U(\unicode[STIX]{x1D6F1}_{n}))\times \mathsf{TCDGA}_{R}\rightarrow \mathsf{Ch}_{R}.\end{eqnarray}$$

Note also that quasi-isomorphic tcdgas give rise to quasi-isomorphic functors. Finally, we remark that if $A$ is only a commutative shuffle dg algebra, then we can define $CD$ and $C\unicode[STIX]{x2131}$ similarly; the resulting functors are just not defined on the semidirect product with $\mathbb{S}_{n}$.

5.9

Let $X$ be a locally compact Hausdorff space space and let $\unicode[STIX]{x2131}$ be a complex of sheaves on $X^{n}$. For $T\in \unicode[STIX]{x1D6F1}_{n}$, let $j(T):X(T)\rightarrow X^{n}$ be the corresponding locally closed subset of $X^{n}$, and let $i(T):\overline{X(T)}\rightarrow X^{n}$ be the inclusion of its closure. Note that each closed subset $\overline{X(T)}$ is a cartesian product of $X$ with itself: if $T$ has blocks $T_{1},\ldots ,T_{k}$, then $\overline{X(T)}\cong X^{k}$. Let $U\subseteq \unicode[STIX]{x1D6F1}_{n}$ be upwards closed, $j:F(X,U)\rightarrow X^{n}$ the open inclusion, and $i:D(X,U)\rightarrow X^{n}$ its closed complement. The following result is a special case of [Reference PetersenPet17, Proposition 3.1]; see also the results of [Reference GetzlerGet99b] when $F(X,U)=F(X,n)$.

Proposition 5.10. There are quasi-isomorphisms

$$\begin{eqnarray}j_{!}j^{\ast }\unicode[STIX]{x2131}\simeq \bigoplus _{C\subseteq U~\text{a chain}}~i(\max C)_{\ast }i(\max C)^{\ast }\unicode[STIX]{x2131}\end{eqnarray}$$

and

$$\begin{eqnarray}i_{\ast }i^{\ast }\unicode[STIX]{x2131}\simeq \bigoplus _{\varnothing \neq C\subseteq U~\text{a chain}}i(\max C)_{\ast }i(\max C)^{\ast }\unicode[STIX]{x2131},\end{eqnarray}$$

where the right-hand sides are considered as double complexes of sheaves: the bigradings and the differentials on the right-hand side are given by exactly the same formulas as for the complexes $C\unicode[STIX]{x2131}$ and $CD$; see §§ 4.10 and 4.12.

Proof. Consider $X^{n}$ as a stratified space with a single open stratum given by $F(X,U)$; the remaining strata are the locally closed subsets $X(T)\subset X^{n}$ where $T\in U$. Then the poset of strata is given by $U\cup \{\hat{0}\}$. In [Reference PetersenPet17, § 3], a general construction is explained for resolving sheaves of the form $j_{!}j^{\ast }\unicode[STIX]{x2131}$ in this situation, and the double complex $\bigoplus _{C\subseteq U\text{ a chain}}i(\max C)_{\ast }i(\max C)^{\ast }\unicode[STIX]{x2131}$ is exactly equal to the complex $L^{\bullet }(\unicode[STIX]{x2131})$ defined in [Reference PetersenPet17, § 3].

Now there is also a distinguished triangle

$$\begin{eqnarray}j_{!}j^{\ast }\unicode[STIX]{x2131}\rightarrow \unicode[STIX]{x2131}\rightarrow i_{\ast }i^{\ast }\unicode[STIX]{x2131},\end{eqnarray}$$

and the double complex $\bigoplus _{C\subseteq U\text{ a chain}}(i^{\max C})_{\ast }(i^{\max C})^{\ast }\unicode[STIX]{x2131}$ differs only by a shift in grading and a coaugmentation by the additional summand $\unicode[STIX]{x2131}$, corresponding to the empty chain. The result follows.◻

5.11

Since the proof is short, let us outline the proof of the quasi-isomorphism $j_{!}j^{\ast }\unicode[STIX]{x2131}\rightarrow L^{\bullet }(\unicode[STIX]{x2131})$ from [Reference PetersenPet17]. Consider the summand of $\bigoplus _{C\subseteq U\text{ a chain}}i(\max C)_{\ast }i(\max C)^{\ast }\unicode[STIX]{x2131}$ corresponding to the empty chain; this gives a copy of $\unicode[STIX]{x2131}$. The natural map $j_{!}j^{\ast }\unicode[STIX]{x2131}\rightarrow \unicode[STIX]{x2131}$ provides a map of complexes

$$\begin{eqnarray}j_{!}j^{\ast }\unicode[STIX]{x2131}\rightarrow \bigoplus _{C\subseteq U\text{ a chain}}i(\max C)_{\ast }i(\max C)^{\ast }\unicode[STIX]{x2131}.\end{eqnarray}$$

We may consider this as an augmented complex and it is enough to prove that it is acyclic. This can be checked on stalks. If $x\in F(X,U)$ then the stalk of both sides at $x$ is just $\unicode[STIX]{x2131}_{x}$; note that all summands except the empty chain on the right-hand side have support outside $F(X,U)$. If $x$ lies in some non-open stratum $X(T)$, then the stalk of the left-hand side is zero and the stalk of the right-hand side is given by the tensor product

$$\begin{eqnarray}\unicode[STIX]{x2131}_{x}\otimes \widetilde{C}^{\bullet }(U_{0}^{\preccurlyeq T}),\end{eqnarray}$$

where $U_{0}=U\cup \{\hat{0}\}$. But this order complex is contractible since $U_{0}^{\preccurlyeq T}$ has a largest element.

Theorem 5.12. Let $X$ be a paracompact Hausdorff space, $\unicode[STIX]{x2131}$ a complex of sheaves of $R$-modules on $X$, $R$ a ring of finite global dimension. Let $U\subset \unicode[STIX]{x1D6F1}_{n}$ be upwards closed. There are quasi-isomorphisms of cochain complexes

$$\begin{eqnarray}\displaystyle C\unicode[STIX]{x2131}(U,R\unicode[STIX]{x1D6E4}^{\otimes }(X,\unicode[STIX]{x2131})) & \simeq & \displaystyle C^{\bullet }(X^{n},D(X,U);\unicode[STIX]{x2131}^{\boxtimes n}),\nonumber\\ \displaystyle CD(U,R\unicode[STIX]{x1D6E4}^{\otimes }(X,\unicode[STIX]{x2131})) & \simeq & \displaystyle C^{\bullet }(D(X,U),\unicode[STIX]{x2131}^{\boxtimes n}),\nonumber\end{eqnarray}$$

which are natural in $X$, $\unicode[STIX]{x2131}$ and $U$. If $X$ is in additional locally compact, then

$$\begin{eqnarray}\displaystyle C\unicode[STIX]{x2131}(U,R\unicode[STIX]{x1D6E4}_{c}^{\otimes }(X,\unicode[STIX]{x2131})) & \simeq & \displaystyle C_{c}^{\bullet }(F(X,U),\unicode[STIX]{x2131}^{\boxtimes n}),\nonumber\\ \displaystyle CD(U,R\unicode[STIX]{x1D6E4}_{c}^{\otimes }(X,\unicode[STIX]{x2131})) & \simeq & \displaystyle C_{c}^{\bullet }(D(X,U),\unicode[STIX]{x2131}^{\boxtimes n}).\nonumber\end{eqnarray}$$

Proof. Let $\unicode[STIX]{x2112}$ be the flat and flabby resolution of $\unicode[STIX]{x2131}$ constructed in Lemma 3.4. Apply Proposition 5.10 to get quasi-isomorphisms

$$\begin{eqnarray}j_{!}j^{\ast }\unicode[STIX]{x2131}^{\boxtimes n}\simeq \bigoplus _{C\subseteq U\text{ a chain}}i(\max C)_{\ast }i(\max C)^{\ast }\unicode[STIX]{x2112}^{\boxtimes n}\end{eqnarray}$$

and

$$\begin{eqnarray}i_{\ast }i^{\ast }\unicode[STIX]{x2131}^{\boxtimes n}\simeq \bigoplus _{\varnothing \neq C\subseteq U\text{ a chain}}i(\max C)_{\ast }i(\max C)^{\ast }\unicode[STIX]{x2112}^{\boxtimes n}.\end{eqnarray}$$

Note now that the right-hand sides are complexes of soft sheaves (respectively, $c$-soft if $X$ is locally compact), so these resolutions can be used to compute the cohomology of $j_{!}j^{\ast }\unicode[STIX]{x2131}$ and $i_{\ast }i^{\ast }$.

Now the key observation is that the complex of global sections of the complex of sheaves

$$\begin{eqnarray}\bigoplus _{C\subseteq U\text{ a chain}}i(\max C)_{\ast }i(\max C)^{\ast }\unicode[STIX]{x2112}^{\boxtimes n}\end{eqnarray}$$

is isomorphic to $C\unicode[STIX]{x2131}(U,R\unicode[STIX]{x1D6E4}^{\otimes }(X,\unicode[STIX]{x2131}))$. Indeed, the summands in the double complex given by $C\unicode[STIX]{x2131}(U,R\unicode[STIX]{x1D6E4}^{\otimes }(X,\unicode[STIX]{x2131}))$ correspond in exactly the same way to chains in the upwards closed subset $U$, and each summand is given precisely by global sections of tensor powers of the resolution $\unicode[STIX]{x2112}$. Moreover, general sheaf theory tells us that there is a quasi-isomorphism $R\unicode[STIX]{x1D6E4}(X^{n},j_{!}j^{\ast }\unicode[STIX]{x2131}^{\boxtimes n})\simeq C^{\bullet }(X^{n},D(X,U);\unicode[STIX]{x2131}^{\boxtimes n})$; this proves the first quasi-isomorphism of the theorem. Similarly the compactly supported global sections of the same complex of sheaves are isomorphic to$C\unicode[STIX]{x2131}(U,R\unicode[STIX]{x1D6E4}_{c}^{\otimes }(X,\unicode[STIX]{x2131}))$, and general sheaf theory also tells us that there is a quasi-isomorphism $R\unicode[STIX]{x1D6E4}_{c}(X^{n},j_{!}j^{\ast }\unicode[STIX]{x2131}^{\boxtimes n})\simeq C_{c}^{\bullet }(F(X,U),\unicode[STIX]{x2131}^{\boxtimes n})$, etc. This finishes the proof.◻

5.13

Let us record some special cases of Theorem 5.12.

Corollary 5.14. Let $X$ be a paracompact and locally compact Hausdorff space. Let $A$ be a cdga model for the compactly supported cochains $C_{c}^{\bullet }(X,\mathbf{Q})$. If $A$ is considered as a constant tcdga, then $C\unicode[STIX]{x2131}(U,A)\simeq H_{c}^{\bullet }(F(X,U),\mathbf{Q})$. The functor

$$\begin{eqnarray}\mathbb{S}_{n}\ltimes U(\unicode[STIX]{x1D6F1}_{n})\rightarrow \mathsf{Ch}_{R}\end{eqnarray}$$

which assigns to $U\subset \unicode[STIX]{x1D6F1}_{n}$ upwards closed the compactly supported cohomology $H_{c}^{\bullet }(F(X,U),\mathbf{Q})$ depends up to quasi-isomorphism only on a cdga model of algebra $C_{c}^{\bullet }(X,\mathbf{Q})$.

Spectral sequences

Theorem 5.18. Let $X$, $\unicode[STIX]{x2131}$ and $U$ be as in Theorem 5.12. Each of the four cases of Theorem 5.12 produces a spectral sequence, which can be written as

$$\begin{eqnarray}\displaystyle E_{1}^{pq} & = & \displaystyle \bigoplus _{\substack{ T\in J_{U_{0}} \\ |T|=n-p}}\bigoplus _{i+j=p+q}\widetilde{H}^{i}(\unicode[STIX]{x27E6}J_{U_{0}}^{\preccurlyeq T}\unicode[STIX]{x27E7};H^{j}(\overline{X(T)},i(T)^{\ast }\unicode[STIX]{x2131}^{\boxtimes n}))\;\Longrightarrow \;H^{p+q}(X^{n},D(X,U);\unicode[STIX]{x2131}^{\boxtimes n}),\nonumber\\ \displaystyle E_{1}^{pq} & = & \displaystyle \bigoplus _{\substack{ T\in J_{U} \\ |T|=n-p}}\bigoplus _{i+j=p+q}\widetilde{H}^{i}(J_{U}^{\preccurlyeq T};H^{j}(\overline{X(T)},i(T)^{\ast }\unicode[STIX]{x2131}^{\boxtimes n}))\;\Longrightarrow \;H^{p+q}(D(X,U),\unicode[STIX]{x2131}^{\boxtimes n}),\nonumber\\ \displaystyle E_{1}^{pq} & = & \displaystyle \bigoplus _{\substack{ T\in J_{U_{0}} \\ |T|=n-p}}\bigoplus _{i+j=p+q}\widetilde{H}^{i}(\unicode[STIX]{x27E6}J_{U_{0}}^{\preccurlyeq T}\unicode[STIX]{x27E7};H_{c}^{j}(\overline{X(T)},i(T)^{\ast }\unicode[STIX]{x2131}^{\boxtimes n}))\;\Longrightarrow \;H_{c}^{p+q}(F(X,U),\unicode[STIX]{x2131}^{\boxtimes n}),\nonumber\\ \displaystyle E_{1}^{pq} & = & \displaystyle \bigoplus _{\substack{ T\in J_{U} \\ |T|=n-p}}\bigoplus _{i+j=p+q}\widetilde{H}^{i}(J_{U}^{\preccurlyeq T};H_{c}^{j}(\overline{X(T)},i(T)^{\ast }\unicode[STIX]{x2131}^{\boxtimes n}))\;\Longrightarrow \;H_{c}^{p+q}(D(X,U),\unicode[STIX]{x2131}^{\boxtimes n}),\nonumber\end{eqnarray}$$

respectively. Here $|T|$ denotes the number of blocks in the partition $T$.

Remark 5.19. Suppose that the partition $T$ has blocks $T_{1},\ldots ,T_{k}$ of size $n_{1},\ldots ,n_{k}$, respectively. Then $\overline{X(T)}\cong X^{k}$, and the sheaf $i(T)^{\ast }\unicode[STIX]{x2131}^{\boxtimes n}$ is given by

$$\begin{eqnarray}\unicode[STIX]{x2131}^{\otimes n_{1}}\boxtimes \cdots \boxtimes \unicode[STIX]{x2131}^{\otimes n_{k}}.\end{eqnarray}$$

Hence in all cases the $E_{1}$-differential of the spectral sequence depends only on: (a) the reduced cohomology of lower intervals in the poset $J_{U}$; (b) the (compactly supported) cohomology of $X$ with coefficients in the tensor powers of the sheaf $\unicode[STIX]{x2131}$. The $E_{1}$-differential is given by multiplication in the twisted commutative algebra given by the direct sum

$$\begin{eqnarray}\bigoplus _{n\geqslant 1}H^{\bullet }(X,\unicode[STIX]{x2131}^{\otimes n}),\end{eqnarray}$$

that is, the cohomology of the tcdga $R\unicode[STIX]{x1D6E4}^{\otimes }(X,\unicode[STIX]{x2131})$.

Comparison with the Cohen–Taylor–Totaro spectral sequence

5.20

Suppose that $M$ is an oriented $d$-dimensional manifold. By Poincaré duality there is a natural isomorphism

$$\begin{eqnarray}H_{d-k}(M,\mathbf{Z})\cong H_{c}^{k}(M,\mathbf{Z})\end{eqnarray}$$

relating homology to cohomology with compact supports. Thus if $X$ is an oriented manifold, the third spectral sequence of Theorem 5.18 yields a spectral sequence computing the homology of the configuration space $F(X,n)$. We claim that the resulting spectral sequence is the dual of the Cohen–Taylor–Totaro spectral sequence [Reference Cohen and TaylorCT78, Reference TotaroTot96] computing the cohomology of $F(X,n)$. More precisely, Totaro constructed this spectral sequence simply as the Leray spectral sequence in sheaf cohomology associated to the inclusion $F(X,n){\hookrightarrow}X^{n}$, and our claim is that the third spectral sequence of Theorem 5.18 coincides with the homological Leray spectral sequence of $F(X,n){\hookrightarrow}X^{n}$, up to reindexing and a degree shift.

5.21

Before proving this claim, let us recall how the Leray spectral sequence is constructed. Let $Y$ be a space, $f:Y\rightarrow \text{pt}$ the map to a point. Then $H^{\bullet }(Y,\mathbf{Z})$ is the cohomology of the complex $Rf_{\ast }\mathbf{Z}$ in the derived category $D^{+}(\text{pt})$, where $\mathbf{Z}$ denotes the constant sheaf on $Y$ (which might be more naturally denoted by $f^{\ast }\mathbf{Z}$). Any map from $Y$ to another space $Z$ gives us a factorization

$$\begin{eqnarray}Y\stackrel{g}{\longrightarrow }Z\stackrel{h}{\longrightarrow }\text{pt},\end{eqnarray}$$

so that we can write $Rf_{\ast }\mathbf{Z}\simeq Rh_{\ast }Rg_{\ast }\mathbf{Z}$. Now the object $Rg_{\ast }\mathbf{Z}$ of $D^{+}(Z)$ has a canonical filtration induced by the $t$-structure on $D^{+}(Z)$, which is uniquely determined in the filtered derived category by the fact that its $q$th graded summand is isomorphic to $R^{q}g_{\ast }\mathbf{Z}[-q]$. After applying $Rh_{\ast }$ we also obtain a filtration of $Rf_{\ast }\mathbf{Z}$, and the spectral sequence of this filtered complex is the Leray spectral sequence $E_{2}^{pq}=H^{p}(Z,R^{q}g_{\ast }\mathbf{Z})\;\Longrightarrow \;H^{p+q}(Y,\mathbf{Z})$. If the spaces involved are of finite type then the homology of $Y$ is given by the Verdier dual $\mathbb{D}Rf_{\ast }\mathbf{Z}\simeq Rf_{!}\mathbb{D}\mathbf{Z}$, and the filtration of $\mathbb{D}Rf_{\ast }\mathbf{Z}\simeq \mathbb{D}Rh_{\ast }Rg_{\ast }\mathbf{Z}$ induced by the filtration of $Rg_{\ast }\mathbf{Z}$ is called the homological Leray spectral sequence.

5.22

We will also need to recall the relationship between the cohomology of the partition lattice and the cohomology of configuration spaces of points in Euclidean space. Consider the configuration space $F(\mathbf{R}^{d},n)$. Its cohomology is non-zero only in degrees divisible by $d-1$, and there is an isomorphism