##
**Operads, algebras, modules and motives.**
*(English)*
Zbl 0840.18001

Astérisque. 233. Paris: Société Mathématique de France, 145 p. (1995).

Operads were introduced by the second author in a purely topological framework. Here they are proposed to serve the study of various kinds of algebras: associative, associative and commutative, Lie, Poisson, DGA’s, etc., occurring in algebraic topology, algebraic geometry, differential geometry and string theory. Much motivation comes from the theory of mixed Tate motives in algebraic geometry.

Fix a commutative ground ring \(k\) and write \(\Sigma_j\) for the symmetric group on \(j\) elements. Tensor products are always over \(k\). An operad (of \(k\)-modules) consists of \(k\)-modules (i.e., differential \(\mathbb{Z}\)-graded chain complexes over \(k)\) \({\mathcal C} (j)\), \(j \geq 0\), together with a unit map \(\eta : k \to {\mathcal C} (1)\), a right action of \(\Sigma_j\) on each \({\mathcal C} (j)\) for each \(j\) and maps \(\gamma : {\mathcal C} (m) \otimes {\mathcal C} (j_1) \otimes \cdots \otimes {\mathcal C} (j_m) \to {\mathcal C} (j)\), for \(m \geq 1\) and \(j_s \geq 0\), where \(\sum j_s = j\). The \(\gamma\) are supposed to satisfy suitable associativity and \(\Sigma\)-equivariance requirements and to be unital. Neglecting the \(\Sigma\)-action one gets a non-\(\Sigma\) operad. Working with unital \(k\)-algebras \(A\) one may think of \(1 \in A\) as a map \(k \to A\). It is sensible to insist that \({\mathcal C} (0) = k\), and one says that \({\mathcal C}\) is a unital operad. If \({\mathcal C}\) is unital, one has augmentations \(\varepsilon = \gamma : {\mathcal C} (j) \cong {\mathcal C} (j) \otimes {\mathcal C} (0)^j \to {\mathcal C} (0) = k\). \({\mathcal C}\) is called acyclic if the augmentations are quasi-isomorphisms, and \({\mathcal C}\) is said to be \(\Sigma\)-free (or \(\Sigma\)-projective) if \({\mathcal C} (j)\) is \(k [\Sigma_j]\)-free (or \(k [\Sigma_j]\)-projective) for each \(j\). \({\mathcal C}\) is called an \(E_\infty\) operad if it is both acyclic and \(\Sigma\)-free and if each \({\mathcal C} (j)\) is concentrated in nonnegative degrees. A non-\(\Sigma\) operad is called an \(A_\infty\) operad if it is acyclic and each \({\mathcal C} (j)\) is concentrated in nonnegative degrees.

Let \({\mathcal C}\) be an operad. A \({\mathcal C}\)-algebra is a \(k\)-module \(A\) with maps \(\theta : {\mathcal C} (j) \otimes A^j \to A\), \(j \geq 0\), that are associative, unital and equivariant in a suitable way. For a \({\mathcal C}\)-algebra \(A\) an \(A\)-module is a \(k\)-module \(M\) together with maps \(\lambda : {\mathcal C} (j) \otimes A^{j - 1} \otimes M \to M\) satisfying, as always, suitable associativity, unitality and equivariance requirements. To an operad \({\mathcal C}\) one may associate a monad \(C\), and, as a matter of fact, \({\mathcal C}\)-algebras correspond to \(C\)-algebras. Define the unital operad \({\mathcal M}\) by \({\mathcal M} (j) = k [\Sigma_j]\) as a right \(k [\Sigma_j]\)-module (concentrated in degree 0). An \({\mathcal M}\)-algebra \(A\) is now a DGA. An \(A\)-module \(M\) is then just an \(A\)-bimodule in the classical sense. Similarly, for the unital operad \({\mathcal N}\), \({\mathcal N} (j) = k\) for all \(j\), an \({\mathcal N}\)-algebra is just a commutative DGA, and an \(A\)-module is just an \(A\)-module in the classical sense. Another example is provided by the nonunital operad \({\mathcal L}\) whose algebras are just the Lie-algebras over \(k\), where one assumes that \(k\) is a field of characteristic other than 2 or 3. Here, for an \({\mathcal L}\)-algebra \(L\), an \(L\)-module is just a Lie-algebra module in the classical sense.

Actually, operads can be defined in any symmetric monoidal category, e.g. the category of topological spaces under Cartesian product. To the operad of spaces \({\mathcal E}\) one can associate the homology operad of \(k\)-modules \(H_* ({\mathcal E})\). For a space \(X\) set \(EX = \coprod {\mathcal E} (j) \times_{\Sigma_j} X^j\). Then one has the result: Let \( E\) be the monad in the category of spaces associated to \({\mathcal E}\) and let \(E_H\) be the monad in the category of \(k\)-modules associated to \(H_* ({\mathcal E})\). If \(k\) is a field of characteristic zero, then \(H_* (EX) \cong E_H H_* (X)\). Several other themes of interest are discussed, e.g. operads coming from iterated loop spaces, leading to the notion of \(n\)-Lie algebras and \(n \)-braid algebras, the little \(n\)-cubes operad \({\mathcal C}_n\) (whose homology operad is shown to be isomorphic to \({\mathcal L})\), algebras over the operad \(H_* ({\mathcal C}_{n + 1}) \) (which are shown to be just the \(n\)-braid algebras), and homology operations in characteristic \(p\).

Part II deals with partial algebras, modules, etc. \(k\) is assumed to be a Dedekind domain. For a \(\Sigma\)-projective operad \({\mathcal C}\) of simplicial \(k\)-modules and a partial \({\mathcal C}\)-algebra \(A\) there is a functor \(V\) that assigns a quasi-isomorphic \({\mathcal C}\)-algebra \(VA\) to \(A\). Similarly for a partial \(A\)-module \(M\) there exists a quasi-isomorphic \(A\)-module \(VM\). In case \(k\) is a field of characteristic zero, every operad is \(\Sigma\)-projective, so one may drop the condition in the statement. Similarly, with \(k\) a field of characteristic zero, for an acyclic operad \({\mathcal C}\) of \(k\)-modules, there is a functor \(W\) that assigns a quasi-isomorphic DGA \(WA\) to a \({\mathcal C}\)-algebra \(A\). Several results of this kind are treated. The motivating example comes from Bloch’s higher Chow groups. Let \(X\) be a smooth, quasi-projective variety over a field \(F\). Bloch defined an Adams graded simplicial abelian group \({\mathcal Z} (X)\) with free groups \({\mathcal Z}^r (X,q)\) of \(q\)-simplices in Adams grading \(r\). There is a partially defined intersection product on this graded simplicial abelian group. In Adams degree \(r\) and simplicial degree \(q\), the domain \({\mathcal Z} (X)_j\) of the \(j\)-fold product is the sum over all partitions \(\{r_1, \dots, r_j\}\) of \(r\) of the subgroups of \(\otimes^j_{i = 1} {\mathcal Z}^{r_i} (X,q)\) spanned by those \(j\)-tuples of simplices all intersections of subsets of which meet all faces (of \(X \times \Delta^q)\) properly. An essential result is the moving lemma which gives a quasi-isomorphism \({\mathcal Z} (X)_j \to {\mathcal Z} (X)^j\). Bloch’s (integral) higher Chow groups are defined by \(CH^r (X,q) = H_q ({\mathcal Z}^r (X,*); \mathbb{Z})\), and Bloch proved that \(CH^r (X,q) \otimes \mathbb{Q} \simeq (K_q (X) \otimes \mathbb{Q})^{(r)}\), where the superscript \((r)\) means the \(n^r\)-eigenspace for the Adams operations \(\psi^n\) for any \(n > 1\). One chooses an \(E_\infty\) operad \({\mathcal C}\) of simplicial abelian groups and regards the partial commutative simple rings as partial \({\mathcal C}\)-algebras. Apply the functor \(V\) to obtain genuine \({\mathcal C}\)-algebras. There is another functor \(C_\#\) which converts \({\mathcal C}\)-algebras to algebras over the associated \(E_\infty\) operad \(C_\# {\mathcal C}\) of chain complexes. Let \({\mathcal A} (X)\) be the \(E_\infty\) algebra \(C_\# V ({\mathcal Z} (X)_*)\) and let \({\mathcal A}_\mathbb{Q} (X) = C_\# W ({\mathcal Z} (X)_* \otimes \mathbb{Q})\), then one shows: \({\mathcal A} (X) \otimes \mathbb{Q}\) is an \(E_\infty\) algebra, and there is a quasi-isomorphism \({\mathcal A} (X) \otimes \mathbb{Q} \to {\mathcal A}_\mathbb{Q} (X)\), of \(E_\infty\) algebras. Writing \({\mathcal N}^{2r - p} (X) (r) = {\mathcal A}_p (X)\), a candidate for motivic cohomology becomes \(H^i_{\text{Mot}} (X, \mathbb{Q} (r)) : = H^i ({\mathcal N}_\mathbb{Q} (X)) (r)\). Let \({\mathcal N} = {\mathcal N} (\text{Spec} (F))\). The \(E_\infty\) algebras \({\mathcal N} (X)\) can be regarded as \({\mathcal N}\)-modules and thus as objects of the derived category \({\mathcal D}_{\mathcal N}\). Deligne proposed this derived category as a derived category of integral mixed Tate modules.

Before constructing the derived category of \(E_\infty\) modules, it turns out to be fruitful to treat derived categories of modules over a DGA from a topological point of view. This is done in Part III. The basic notion, to be compared with a CW-spectrum in stable homotopy theory, is that of a cell \(A\)-module, where \(A\) is an associative, unital, but not necessarily commutative DGA over \(k\). A cell \(A\)-module \(M\) is the union of an expanding sequence of sub \(A\)-modules \(M_n\) such that \(M_0 = 0\) and \(M_{n + 1}\) is the cofiber of a map \(\pi_n : F_n \to M_n\), with \(F_n\) a direct sum of ‘sphere modules’. Cofiber sequences are just exact triangles in the formalism of triangulated categories. Let \({\mathcal M}_A\) denote the category of \(A\)-modules, and write \(h {\mathcal M}_A\) for its homotopy category. The derived category \({\mathcal D}_A\) is obtained from \(h {\mathcal M}_A\) by formally inverting quasi-isomorphisms of \(A\)-modules. One has several results on cell \(A\)-modules: (i) Whidehead’s theorem which can be reformulated by saying that a quasi-isomorphism between cell \(A\)-modules is a homotopy equivalence; (ii) the cellular approximation theorem which says that for any \(A\)-module \(M\) there exists a quasi-isomorphic cell \(A\)-module \(\Gamma M\), and (iii) Brown’s representability theorem. One has a formalism for Tor and Ext as derived tensor product and Hom. For \(A\)-modules \(M\) and \(N\), one defines \(M \otimes^L_A N = M \otimes_A \Gamma N\) and \(R \operatorname{Hom}_A (M,N) = \operatorname{Hom}_A (\Gamma M,N)\). One gets the well-known formula: \({\mathcal D}_A (K \otimes^L M,N) \simeq {\mathcal D}_k (K,R \operatorname{Hom}_A (M,N))\) for \(k\)-modules \(K\). There is a corresponding formalism for relative cell \(A\)-modules. For commutative \(A\), \({\mathcal D}_A\) can be given the structure of a symmetric monoidal (= tensor) category with internal Hom’s.

Part IV is concerned with rational derived categories and mixed Tate motives. Let \(k\) be a field of characteristic zero, and let \(A\) be a DGA, here in the sense of a commutative, differential graded, and Adams graded \(k\)-algebra. Thus \(A\) is bigraded via \(k\)-submodules \(A^q (r)\), \(q \in \mathbb{Z}\) and \(r \geq 0\). One assumes \(A^q (r) = 0\) unless \(2r \geq q\), and \(A\) cohomologically connected. \({\mathcal D}_A\) will stand for the derived category of cohomologically bounded below \(A\)-modules. Let \({\mathcal H}_A\) be the full subcategory of \({\mathcal D}_A\) consisting of the cell \(A\)-modules \(M\) with \(H^q (k \otimes_A M) = 0\) for \(q \neq 0\), and let \({\mathcal F} {\mathcal H}_A\) be the full subcategory of \({\mathcal H}_A\) consisting of the modules \(M\) with finite-dimensional \(H^0 (k \otimes_A M)\). Define \(\omega (M) = H^0 (k \otimes_AM)\). The following results are proved: (i) the triangulated category \({\mathcal D}_A\) has a \(t\)-structure whose heart is \({\mathcal H}_A\). In particular, \({\mathcal H}_A\) is abelian. Also, \({\mathcal F} {\mathcal H}_A\) is a (graded) neutral Tannakian category over \(k\) with fiber functor \(\omega\); (ii) assume \(A\) is a connected DGA, then the following categories are equivalent: (a) the heart \({\mathcal H}_A\) of \({\mathcal D}_A\), (b) the category of comodules over the Hopf algebra \(\chi_A = H^0 \overline B (A)\) (the bar construction), (c) the category of generalized nilpotent representations of the co-Lie algebra \(\gamma_A\) (the \(k\)-module of indecomposable elements of \(\chi_A)\); (iii) the derived category of the bounded below complexes in \({\mathcal H}_A\) is equivalent to the derived category of modules over the DGA \(\wedge \gamma_A [-1]\) \((\wedge \gamma_A [-1]\) is the 1-minimal model of \(A)\). Now, with the notation of Part II, one may reformulate a well-known conjecture of Beilinson-Soulé: \({\mathcal N}_\mathbb{Q} (X)\) is cohomologically connected. Specializing to the case \(X = \text{Spec} (F)\), write \({\mathcal N}\) for the \(E_\infty\) algebra \({\mathcal N} (\text{Spec} (F))\) and \({\mathcal N}_\mathbb{Q}\) for the commutative DGA \({\mathcal N}_\mathbb{Q} (\text{Spec} (F))\). Write \(\chi_{\text{Mot}}\) for the Hopf algebra \(\chi_{{\mathcal N}_\mathbb{Q}} = H^0 \overline B ({\mathcal N}_\mathbb{Q})\), and define the category of (rational) mixed Tate motives over the field \(F\), \({\mathcal M} {\mathcal T} {\mathcal M} (F)\), as the category of finite-dimensional comodules over \(\chi_{\text{Mot}}\). The above results may be applied to obtain: (i) assume the Beilinson-Soulé conjecture holds for \(\text{Spec} (F)\), then \({\mathcal M} {\mathcal T} {\mathcal M}(F)\) is equivalent to \({\mathcal F} {\mathcal H}_{{\mathcal N}_\mathbb{Q}}\) (this implies the equivalence of the definitions of mixed Tate motives by Bloch and the first author on the one hand, and by Deligne on the other hand); (ii) if \({\mathcal N}_\mathbb{Q}\) is a \(K (\pi, 1)\), then \(\text{Ext}^p_{{\mathcal M} {\mathcal T} {\mathcal M} (F)} (\mathbb{Q}, \mathbb{Q} (r)) \simeq gr^r_\gamma (K_{2r - p} (F) \otimes \mathbb{Q})\).

In Part V it is shown that, for a commutative ring \(k\) and an \(E_\infty\) operad \({\mathcal C}\), the derived category of modules over a \({\mathcal C}\)-algebra looks very much like the derived category of modules over a commutative DGA as discussed in Part III. Let \({\mathcal C}\) be an operad, and write \(\mathbb{C}\) for \({\mathcal C} (1)\). Then \(\mathbb{C}\) is a DGA over \(k\), usually not commutative, but it is homotopy commutative if \({\mathcal C}\) is an \(E_\infty\) operad. For \(\mathbb{C}\)-modules \(M\) and \(N\) one defines the operadic tensor product \(M \boxtimes N\) to be the \(\mathbb{C}\)-module \(M \boxtimes N = {\mathcal C} (2) \otimes_{\mathbb{C} \otimes \mathbb{C}} M \otimes N\). Also, for two (left) \(\mathbb{C}\)-modules \(M\) and \(N\) one defines \(\operatorname{Hom}^\boxtimes (M,N) = \operatorname{Hom}_\mathbb{C} ({\mathcal C} (2) \otimes_\mathbb{C} M,N)\). Of fundamental importance is the existence of a special \(E_\infty\) operad \({\mathcal C}\), the linear isometries operad, for which the tensor product \(\boxtimes\) is commutative and associative, but not unital at the module level. It becomes unital in the derived category. It is shown that, up to quasi-isomorphism, it is no loss of generality to take the preferred \(E_\infty\) linear isometries operad \({\mathcal C}\). Using the cellular approximation theorem for \(\mathbb{C}\)-modules, \(\boxtimes\) and \(\operatorname{Hom}^\boxtimes\) induce derived versions \(\boxtimes^L\) and \(R \operatorname{Hom}^\boxtimes\) in \({\mathcal D}_\mathbb{C}\). The forgetful functor from \(\mathbb{C}\)-modules to \(k\)-modules induces an equivalence \({\mathcal D}_\mathbb{C} \to {\mathcal D}_k\), under which \(\boxtimes^L\) is carried to \(\otimes^L\). Similarly for \(R \operatorname{Hom}^\boxtimes\) and \(R \operatorname{Hom}\). A formula such as \({\mathcal D}_\mathbb{C} (M,N) \simeq {\mathcal D}_\mathbb{C} (M \boxtimes^L k,N) \simeq {\mathcal D}_\mathbb{C} (M,R \operatorname{Hom}^\boxtimes (k,N))\) should come as no surprise, it implies that there is a natural isomorphism \(N \to R \operatorname{Hom}^\boxtimes (k,N)\) in \({\mathcal D}_\mathbb{C}\). Three more products are introduced to deal with questions of unitality. They are used to define a tensor product \(\boxtimes_A\) and a Hom-functor \(\operatorname{Hom}^\boxtimes_A (M,N)\) for modules over \(A_\infty\) algebras. \(\boxtimes_A\) has the property that, for augmented \(A\) and a cell \(A\)-module \(N\), the unit map \(\lambda : A \boxtimes_A N\to N\) is a quasi-isomorphism. One also defines \(R \operatorname{Hom}^\boxtimes_A\). It satisfies \({\mathcal D}_A (L \boxtimes^L M,N) \simeq {\mathcal D}_\mathbb{C} (L,R \operatorname{Hom}^\boxtimes_A (M,N))\). As a matter of fact, one can carry out homological algebra in this generalized context, e.g. one may construct Hochschild homology of \(A_\infty\) algebras. One has Eilenberg-Moore spectral sequences to compute the homology of the derived tensor product \(M \boxtimes^L_A N\) and the derived Hom functor \(R \operatorname{Hom}^\boxtimes_A (M,N)\) in terms of the classical Tor and Ext groups \(\text{Tor}^*_{H^* (A)} (H^* (M)\), \(H^* (N))\) and \(\text{Ext}^*_{H^* (A)} (H^* (M)\), \(H^* (N))\). For an \(E_\infty\) algebra \(A\) and an \(A\)-module \(M\) one defines \(M^\vee = \operatorname{Hom}^\boxtimes_A (M,A)\), and one says that a cell \(A\)-module \(M\) is strongly dualizable if it has a coevaluation map \(\eta : A \to M \boxtimes_A M^\vee\) with the usual properties. For such \(M\) the natural maps \(\rho : M \to M^{\vee \vee}\) and \(\nu : M^\vee \boxtimes_A N \to \operatorname{Hom}^\boxtimes_A (M,N)\) induce isomorphisms in \({\mathcal D}_A\). It follows that, for a strongly dualizable \(M\) and any \(A\)-module \(N\), one has \(\text{Tor}^n_A (M^\vee, N) \simeq \text{Ext}^n_A (M,N)\).

After reading this highly interesting booklet of only 140 pages, the reader will be impressed by the overwhelming quantity of material hidden in it and he or she will generously acknowledge the SMF for publishing it as a double (!) volume in the Astérisque series.

Fix a commutative ground ring \(k\) and write \(\Sigma_j\) for the symmetric group on \(j\) elements. Tensor products are always over \(k\). An operad (of \(k\)-modules) consists of \(k\)-modules (i.e., differential \(\mathbb{Z}\)-graded chain complexes over \(k)\) \({\mathcal C} (j)\), \(j \geq 0\), together with a unit map \(\eta : k \to {\mathcal C} (1)\), a right action of \(\Sigma_j\) on each \({\mathcal C} (j)\) for each \(j\) and maps \(\gamma : {\mathcal C} (m) \otimes {\mathcal C} (j_1) \otimes \cdots \otimes {\mathcal C} (j_m) \to {\mathcal C} (j)\), for \(m \geq 1\) and \(j_s \geq 0\), where \(\sum j_s = j\). The \(\gamma\) are supposed to satisfy suitable associativity and \(\Sigma\)-equivariance requirements and to be unital. Neglecting the \(\Sigma\)-action one gets a non-\(\Sigma\) operad. Working with unital \(k\)-algebras \(A\) one may think of \(1 \in A\) as a map \(k \to A\). It is sensible to insist that \({\mathcal C} (0) = k\), and one says that \({\mathcal C}\) is a unital operad. If \({\mathcal C}\) is unital, one has augmentations \(\varepsilon = \gamma : {\mathcal C} (j) \cong {\mathcal C} (j) \otimes {\mathcal C} (0)^j \to {\mathcal C} (0) = k\). \({\mathcal C}\) is called acyclic if the augmentations are quasi-isomorphisms, and \({\mathcal C}\) is said to be \(\Sigma\)-free (or \(\Sigma\)-projective) if \({\mathcal C} (j)\) is \(k [\Sigma_j]\)-free (or \(k [\Sigma_j]\)-projective) for each \(j\). \({\mathcal C}\) is called an \(E_\infty\) operad if it is both acyclic and \(\Sigma\)-free and if each \({\mathcal C} (j)\) is concentrated in nonnegative degrees. A non-\(\Sigma\) operad is called an \(A_\infty\) operad if it is acyclic and each \({\mathcal C} (j)\) is concentrated in nonnegative degrees.

Let \({\mathcal C}\) be an operad. A \({\mathcal C}\)-algebra is a \(k\)-module \(A\) with maps \(\theta : {\mathcal C} (j) \otimes A^j \to A\), \(j \geq 0\), that are associative, unital and equivariant in a suitable way. For a \({\mathcal C}\)-algebra \(A\) an \(A\)-module is a \(k\)-module \(M\) together with maps \(\lambda : {\mathcal C} (j) \otimes A^{j - 1} \otimes M \to M\) satisfying, as always, suitable associativity, unitality and equivariance requirements. To an operad \({\mathcal C}\) one may associate a monad \(C\), and, as a matter of fact, \({\mathcal C}\)-algebras correspond to \(C\)-algebras. Define the unital operad \({\mathcal M}\) by \({\mathcal M} (j) = k [\Sigma_j]\) as a right \(k [\Sigma_j]\)-module (concentrated in degree 0). An \({\mathcal M}\)-algebra \(A\) is now a DGA. An \(A\)-module \(M\) is then just an \(A\)-bimodule in the classical sense. Similarly, for the unital operad \({\mathcal N}\), \({\mathcal N} (j) = k\) for all \(j\), an \({\mathcal N}\)-algebra is just a commutative DGA, and an \(A\)-module is just an \(A\)-module in the classical sense. Another example is provided by the nonunital operad \({\mathcal L}\) whose algebras are just the Lie-algebras over \(k\), where one assumes that \(k\) is a field of characteristic other than 2 or 3. Here, for an \({\mathcal L}\)-algebra \(L\), an \(L\)-module is just a Lie-algebra module in the classical sense.

Actually, operads can be defined in any symmetric monoidal category, e.g. the category of topological spaces under Cartesian product. To the operad of spaces \({\mathcal E}\) one can associate the homology operad of \(k\)-modules \(H_* ({\mathcal E})\). For a space \(X\) set \(EX = \coprod {\mathcal E} (j) \times_{\Sigma_j} X^j\). Then one has the result: Let \( E\) be the monad in the category of spaces associated to \({\mathcal E}\) and let \(E_H\) be the monad in the category of \(k\)-modules associated to \(H_* ({\mathcal E})\). If \(k\) is a field of characteristic zero, then \(H_* (EX) \cong E_H H_* (X)\). Several other themes of interest are discussed, e.g. operads coming from iterated loop spaces, leading to the notion of \(n\)-Lie algebras and \(n \)-braid algebras, the little \(n\)-cubes operad \({\mathcal C}_n\) (whose homology operad is shown to be isomorphic to \({\mathcal L})\), algebras over the operad \(H_* ({\mathcal C}_{n + 1}) \) (which are shown to be just the \(n\)-braid algebras), and homology operations in characteristic \(p\).

Part II deals with partial algebras, modules, etc. \(k\) is assumed to be a Dedekind domain. For a \(\Sigma\)-projective operad \({\mathcal C}\) of simplicial \(k\)-modules and a partial \({\mathcal C}\)-algebra \(A\) there is a functor \(V\) that assigns a quasi-isomorphic \({\mathcal C}\)-algebra \(VA\) to \(A\). Similarly for a partial \(A\)-module \(M\) there exists a quasi-isomorphic \(A\)-module \(VM\). In case \(k\) is a field of characteristic zero, every operad is \(\Sigma\)-projective, so one may drop the condition in the statement. Similarly, with \(k\) a field of characteristic zero, for an acyclic operad \({\mathcal C}\) of \(k\)-modules, there is a functor \(W\) that assigns a quasi-isomorphic DGA \(WA\) to a \({\mathcal C}\)-algebra \(A\). Several results of this kind are treated. The motivating example comes from Bloch’s higher Chow groups. Let \(X\) be a smooth, quasi-projective variety over a field \(F\). Bloch defined an Adams graded simplicial abelian group \({\mathcal Z} (X)\) with free groups \({\mathcal Z}^r (X,q)\) of \(q\)-simplices in Adams grading \(r\). There is a partially defined intersection product on this graded simplicial abelian group. In Adams degree \(r\) and simplicial degree \(q\), the domain \({\mathcal Z} (X)_j\) of the \(j\)-fold product is the sum over all partitions \(\{r_1, \dots, r_j\}\) of \(r\) of the subgroups of \(\otimes^j_{i = 1} {\mathcal Z}^{r_i} (X,q)\) spanned by those \(j\)-tuples of simplices all intersections of subsets of which meet all faces (of \(X \times \Delta^q)\) properly. An essential result is the moving lemma which gives a quasi-isomorphism \({\mathcal Z} (X)_j \to {\mathcal Z} (X)^j\). Bloch’s (integral) higher Chow groups are defined by \(CH^r (X,q) = H_q ({\mathcal Z}^r (X,*); \mathbb{Z})\), and Bloch proved that \(CH^r (X,q) \otimes \mathbb{Q} \simeq (K_q (X) \otimes \mathbb{Q})^{(r)}\), where the superscript \((r)\) means the \(n^r\)-eigenspace for the Adams operations \(\psi^n\) for any \(n > 1\). One chooses an \(E_\infty\) operad \({\mathcal C}\) of simplicial abelian groups and regards the partial commutative simple rings as partial \({\mathcal C}\)-algebras. Apply the functor \(V\) to obtain genuine \({\mathcal C}\)-algebras. There is another functor \(C_\#\) which converts \({\mathcal C}\)-algebras to algebras over the associated \(E_\infty\) operad \(C_\# {\mathcal C}\) of chain complexes. Let \({\mathcal A} (X)\) be the \(E_\infty\) algebra \(C_\# V ({\mathcal Z} (X)_*)\) and let \({\mathcal A}_\mathbb{Q} (X) = C_\# W ({\mathcal Z} (X)_* \otimes \mathbb{Q})\), then one shows: \({\mathcal A} (X) \otimes \mathbb{Q}\) is an \(E_\infty\) algebra, and there is a quasi-isomorphism \({\mathcal A} (X) \otimes \mathbb{Q} \to {\mathcal A}_\mathbb{Q} (X)\), of \(E_\infty\) algebras. Writing \({\mathcal N}^{2r - p} (X) (r) = {\mathcal A}_p (X)\), a candidate for motivic cohomology becomes \(H^i_{\text{Mot}} (X, \mathbb{Q} (r)) : = H^i ({\mathcal N}_\mathbb{Q} (X)) (r)\). Let \({\mathcal N} = {\mathcal N} (\text{Spec} (F))\). The \(E_\infty\) algebras \({\mathcal N} (X)\) can be regarded as \({\mathcal N}\)-modules and thus as objects of the derived category \({\mathcal D}_{\mathcal N}\). Deligne proposed this derived category as a derived category of integral mixed Tate modules.

Before constructing the derived category of \(E_\infty\) modules, it turns out to be fruitful to treat derived categories of modules over a DGA from a topological point of view. This is done in Part III. The basic notion, to be compared with a CW-spectrum in stable homotopy theory, is that of a cell \(A\)-module, where \(A\) is an associative, unital, but not necessarily commutative DGA over \(k\). A cell \(A\)-module \(M\) is the union of an expanding sequence of sub \(A\)-modules \(M_n\) such that \(M_0 = 0\) and \(M_{n + 1}\) is the cofiber of a map \(\pi_n : F_n \to M_n\), with \(F_n\) a direct sum of ‘sphere modules’. Cofiber sequences are just exact triangles in the formalism of triangulated categories. Let \({\mathcal M}_A\) denote the category of \(A\)-modules, and write \(h {\mathcal M}_A\) for its homotopy category. The derived category \({\mathcal D}_A\) is obtained from \(h {\mathcal M}_A\) by formally inverting quasi-isomorphisms of \(A\)-modules. One has several results on cell \(A\)-modules: (i) Whidehead’s theorem which can be reformulated by saying that a quasi-isomorphism between cell \(A\)-modules is a homotopy equivalence; (ii) the cellular approximation theorem which says that for any \(A\)-module \(M\) there exists a quasi-isomorphic cell \(A\)-module \(\Gamma M\), and (iii) Brown’s representability theorem. One has a formalism for Tor and Ext as derived tensor product and Hom. For \(A\)-modules \(M\) and \(N\), one defines \(M \otimes^L_A N = M \otimes_A \Gamma N\) and \(R \operatorname{Hom}_A (M,N) = \operatorname{Hom}_A (\Gamma M,N)\). One gets the well-known formula: \({\mathcal D}_A (K \otimes^L M,N) \simeq {\mathcal D}_k (K,R \operatorname{Hom}_A (M,N))\) for \(k\)-modules \(K\). There is a corresponding formalism for relative cell \(A\)-modules. For commutative \(A\), \({\mathcal D}_A\) can be given the structure of a symmetric monoidal (= tensor) category with internal Hom’s.

Part IV is concerned with rational derived categories and mixed Tate motives. Let \(k\) be a field of characteristic zero, and let \(A\) be a DGA, here in the sense of a commutative, differential graded, and Adams graded \(k\)-algebra. Thus \(A\) is bigraded via \(k\)-submodules \(A^q (r)\), \(q \in \mathbb{Z}\) and \(r \geq 0\). One assumes \(A^q (r) = 0\) unless \(2r \geq q\), and \(A\) cohomologically connected. \({\mathcal D}_A\) will stand for the derived category of cohomologically bounded below \(A\)-modules. Let \({\mathcal H}_A\) be the full subcategory of \({\mathcal D}_A\) consisting of the cell \(A\)-modules \(M\) with \(H^q (k \otimes_A M) = 0\) for \(q \neq 0\), and let \({\mathcal F} {\mathcal H}_A\) be the full subcategory of \({\mathcal H}_A\) consisting of the modules \(M\) with finite-dimensional \(H^0 (k \otimes_A M)\). Define \(\omega (M) = H^0 (k \otimes_AM)\). The following results are proved: (i) the triangulated category \({\mathcal D}_A\) has a \(t\)-structure whose heart is \({\mathcal H}_A\). In particular, \({\mathcal H}_A\) is abelian. Also, \({\mathcal F} {\mathcal H}_A\) is a (graded) neutral Tannakian category over \(k\) with fiber functor \(\omega\); (ii) assume \(A\) is a connected DGA, then the following categories are equivalent: (a) the heart \({\mathcal H}_A\) of \({\mathcal D}_A\), (b) the category of comodules over the Hopf algebra \(\chi_A = H^0 \overline B (A)\) (the bar construction), (c) the category of generalized nilpotent representations of the co-Lie algebra \(\gamma_A\) (the \(k\)-module of indecomposable elements of \(\chi_A)\); (iii) the derived category of the bounded below complexes in \({\mathcal H}_A\) is equivalent to the derived category of modules over the DGA \(\wedge \gamma_A [-1]\) \((\wedge \gamma_A [-1]\) is the 1-minimal model of \(A)\). Now, with the notation of Part II, one may reformulate a well-known conjecture of Beilinson-Soulé: \({\mathcal N}_\mathbb{Q} (X)\) is cohomologically connected. Specializing to the case \(X = \text{Spec} (F)\), write \({\mathcal N}\) for the \(E_\infty\) algebra \({\mathcal N} (\text{Spec} (F))\) and \({\mathcal N}_\mathbb{Q}\) for the commutative DGA \({\mathcal N}_\mathbb{Q} (\text{Spec} (F))\). Write \(\chi_{\text{Mot}}\) for the Hopf algebra \(\chi_{{\mathcal N}_\mathbb{Q}} = H^0 \overline B ({\mathcal N}_\mathbb{Q})\), and define the category of (rational) mixed Tate motives over the field \(F\), \({\mathcal M} {\mathcal T} {\mathcal M} (F)\), as the category of finite-dimensional comodules over \(\chi_{\text{Mot}}\). The above results may be applied to obtain: (i) assume the Beilinson-Soulé conjecture holds for \(\text{Spec} (F)\), then \({\mathcal M} {\mathcal T} {\mathcal M}(F)\) is equivalent to \({\mathcal F} {\mathcal H}_{{\mathcal N}_\mathbb{Q}}\) (this implies the equivalence of the definitions of mixed Tate motives by Bloch and the first author on the one hand, and by Deligne on the other hand); (ii) if \({\mathcal N}_\mathbb{Q}\) is a \(K (\pi, 1)\), then \(\text{Ext}^p_{{\mathcal M} {\mathcal T} {\mathcal M} (F)} (\mathbb{Q}, \mathbb{Q} (r)) \simeq gr^r_\gamma (K_{2r - p} (F) \otimes \mathbb{Q})\).

In Part V it is shown that, for a commutative ring \(k\) and an \(E_\infty\) operad \({\mathcal C}\), the derived category of modules over a \({\mathcal C}\)-algebra looks very much like the derived category of modules over a commutative DGA as discussed in Part III. Let \({\mathcal C}\) be an operad, and write \(\mathbb{C}\) for \({\mathcal C} (1)\). Then \(\mathbb{C}\) is a DGA over \(k\), usually not commutative, but it is homotopy commutative if \({\mathcal C}\) is an \(E_\infty\) operad. For \(\mathbb{C}\)-modules \(M\) and \(N\) one defines the operadic tensor product \(M \boxtimes N\) to be the \(\mathbb{C}\)-module \(M \boxtimes N = {\mathcal C} (2) \otimes_{\mathbb{C} \otimes \mathbb{C}} M \otimes N\). Also, for two (left) \(\mathbb{C}\)-modules \(M\) and \(N\) one defines \(\operatorname{Hom}^\boxtimes (M,N) = \operatorname{Hom}_\mathbb{C} ({\mathcal C} (2) \otimes_\mathbb{C} M,N)\). Of fundamental importance is the existence of a special \(E_\infty\) operad \({\mathcal C}\), the linear isometries operad, for which the tensor product \(\boxtimes\) is commutative and associative, but not unital at the module level. It becomes unital in the derived category. It is shown that, up to quasi-isomorphism, it is no loss of generality to take the preferred \(E_\infty\) linear isometries operad \({\mathcal C}\). Using the cellular approximation theorem for \(\mathbb{C}\)-modules, \(\boxtimes\) and \(\operatorname{Hom}^\boxtimes\) induce derived versions \(\boxtimes^L\) and \(R \operatorname{Hom}^\boxtimes\) in \({\mathcal D}_\mathbb{C}\). The forgetful functor from \(\mathbb{C}\)-modules to \(k\)-modules induces an equivalence \({\mathcal D}_\mathbb{C} \to {\mathcal D}_k\), under which \(\boxtimes^L\) is carried to \(\otimes^L\). Similarly for \(R \operatorname{Hom}^\boxtimes\) and \(R \operatorname{Hom}\). A formula such as \({\mathcal D}_\mathbb{C} (M,N) \simeq {\mathcal D}_\mathbb{C} (M \boxtimes^L k,N) \simeq {\mathcal D}_\mathbb{C} (M,R \operatorname{Hom}^\boxtimes (k,N))\) should come as no surprise, it implies that there is a natural isomorphism \(N \to R \operatorname{Hom}^\boxtimes (k,N)\) in \({\mathcal D}_\mathbb{C}\). Three more products are introduced to deal with questions of unitality. They are used to define a tensor product \(\boxtimes_A\) and a Hom-functor \(\operatorname{Hom}^\boxtimes_A (M,N)\) for modules over \(A_\infty\) algebras. \(\boxtimes_A\) has the property that, for augmented \(A\) and a cell \(A\)-module \(N\), the unit map \(\lambda : A \boxtimes_A N\to N\) is a quasi-isomorphism. One also defines \(R \operatorname{Hom}^\boxtimes_A\). It satisfies \({\mathcal D}_A (L \boxtimes^L M,N) \simeq {\mathcal D}_\mathbb{C} (L,R \operatorname{Hom}^\boxtimes_A (M,N))\). As a matter of fact, one can carry out homological algebra in this generalized context, e.g. one may construct Hochschild homology of \(A_\infty\) algebras. One has Eilenberg-Moore spectral sequences to compute the homology of the derived tensor product \(M \boxtimes^L_A N\) and the derived Hom functor \(R \operatorname{Hom}^\boxtimes_A (M,N)\) in terms of the classical Tor and Ext groups \(\text{Tor}^*_{H^* (A)} (H^* (M)\), \(H^* (N))\) and \(\text{Ext}^*_{H^* (A)} (H^* (M)\), \(H^* (N))\). For an \(E_\infty\) algebra \(A\) and an \(A\)-module \(M\) one defines \(M^\vee = \operatorname{Hom}^\boxtimes_A (M,A)\), and one says that a cell \(A\)-module \(M\) is strongly dualizable if it has a coevaluation map \(\eta : A \to M \boxtimes_A M^\vee\) with the usual properties. For such \(M\) the natural maps \(\rho : M \to M^{\vee \vee}\) and \(\nu : M^\vee \boxtimes_A N \to \operatorname{Hom}^\boxtimes_A (M,N)\) induce isomorphisms in \({\mathcal D}_A\). It follows that, for a strongly dualizable \(M\) and any \(A\)-module \(N\), one has \(\text{Tor}^n_A (M^\vee, N) \simeq \text{Ext}^n_A (M,N)\).

After reading this highly interesting booklet of only 140 pages, the reader will be impressed by the overwhelming quantity of material hidden in it and he or she will generously acknowledge the SMF for publishing it as a double (!) volume in the Astérisque series.

Reviewer: W.W.J.Hulsbergen (Haarlem)

### MSC:

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

55Uxx | Applied homological algebra and category theory in algebraic topology |

19-02 | Research exposition (monographs, survey articles) pertaining to \(K\)-theory |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

18G99 | Homological algebra in category theory, derived categories and functors |

19D99 | Higher algebraic \(K\)-theory |

19E99 | \(K\)-theory in geometry |

14A20 | Generalizations (algebraic spaces, stacks) |

18E30 | Derived categories, triangulated categories (MSC2010) |