##
**Hochschild cohomology of complex spaces and Noetherian schemes.**
*(English)*
Zbl 1070.18010

Abstracting common features and properties of Noetherian schemes and complex spaces, Bingener-Kosarew [J. Bingener, “Local moduli spaces in analytic geometry”, Aspekte der Mathematik, D2, D3 (1987; Zbl 0644.32001)] arrived at the notion of an admissible pair of categories. These are pairs \(({\mathcal C},{\mathcal M})\) where \({\mathcal C}\) is a suitable category of algebras over a field \({\mathbb K}\) and \({\mathcal M}\) is a category of modules over algebras in \({\mathcal C}\) with a tensor product. An admissible pair is meant to describe a category of “affine” spaces.

The theme of the article under review is Hochschild cohomology for admissible pairs of categories. The author starts discussing in this framework the suitable notion of free algebras and the (graded) simplicial context. For a simplicial complex of indices, seen as a category \({\mathcal N}\), denote by \(\text{gr}({\mathcal C})^{\mathcal N}\) the associated category of simplicial (non-positively) graded algebras. For a morphism \(A\to B\) in \(\text{gr}({\mathcal C})^{\mathcal N}\), a resolvent of \(B\) over \(A\) is a free DG algebra \(R\) over \(A\) in \(\text{ gr}({\mathcal C})^{\mathcal N}\) together with a morphism \(R\to B\) which is a surjective quasi-isomorphism (qis) on each \(\alpha\in{\mathcal N}\). While existence of resolvents follows from loc. cit., the author proves that any two of them are homotopically equivalent.

Then he introduces the appropriate notion of a regular sequence \(X\subset R\) of an algebra \(R\). As expected, the Koszul complex \(K(X)\) is a DG resolvent of \(R/(X)\) over \(R\). It is clearly pointed out at which place restrictions on the characteristic of \({\mathbb K}\) come into play.

In the second section, the author introduces Hochschild cohomology associated to a morphism \(k\to a\) in \(\text{gr}({\mathcal C})^{\mathcal N}\) such that there exists a resolvent \(A\) of \(a\) over \(k\). It is defined via the simplicial Hochschild complex \({\mathbb H}_*(a/k)\) which is the object represented by \(S\otimes_Ra\) in the homotopy category \(K^-({\mathcal M}^{\mathcal N}(a))\). Here \(S\) is a resolvent of the multiplication map \(\mu:A\otimes A\to A\). The Hochschild complex \({\mathbb H}(a/k)\) is then the object represented by \(\check{C}^{\bullet}({\mathbb H}_*(a/k))\) in the derived category \(D({\mathbb K}\text{-Mod})\) for a Čech-construction \(\check{C}^{\bullet}(-)\) (transforming a simplicial graded object into a total complex) applied to \({\mathbb H}_*(a/k)\). The difficulty circumvented here resides in the fact that in the analytic context the usual bar complex \(C^{\text{bar}}(a)\) is not a complex of projective \(a\)-modules.

The main result is a HKR-type theorem for free DG algebras \(A\) in \(\text{gr}'({\mathcal C})^{\mathcal N}\). The author shows under these hypotheses the existence of a homotopy equivalence \[ \Lambda\Omega_{A/k}\to {\mathbb H}(A/k), \] where \(\text{gr}'({\mathcal C})\) is the full subcategory of \(\text{gr}({\mathcal C})\) generated by graded algebras \(A\) such that \(A^i\) is a finite \(A^0\)-module. Recall that for an algebra \(a\) over \(k\) in \({\mathcal C}^{\mathcal N}\) with resolvent \(A\) in \(\text{gr}'({\mathcal C})\), the cotangent complex \({\mathbb L}(a/k)\) is defined as the class of \(\Omega_A\otimes_Aa\) in \(K^-({\mathcal M}^{\mathcal N}(a))\). An immediate consequence of the above is a homotopy equivalence \[ \Lambda{\mathbb L}(a/k)\to{\mathbb H}(a/k) \] in \(\text{gr}'({\mathcal C})^{\mathcal N}\) over \(a\), provided \({\mathbb Q}\subset k\). The main result is then carefully translated in section \(4\) into a qis \[ {\mathbb H}(X/Y)\approx\Lambda{\mathbb L}(X/Y) \] for a morphism \(X\to Y\) of paracompact, separated complex spaces or a separated morphism of finite type of Noetherian schemes in characteristic zero.

Further results include several decomposition theorems for Hochschild cohomology: one with respect to tangent cohomology \[ \text{HH}^n(X/Y,{\mathcal M})\cong\coprod_{i+j=n}\text{Ext}^i (\Lambda^j{\mathbb L}(X/Y),{\mathcal M}), \] one with respect to sheaf cohomology with values in exterior powers of the tangent sheaf \[ \text{HH}^n(X)\cong\coprod_{i+j=n}H^i(X,\Lambda^j{\mathcal T}_X), \] and the Hodge decomposition \[ \text{HH}_n(X) \cong\prod_{i-j=n}H^j(X,\Lambda^i\Omega_X) \] thereby reproving many theorems spread over the literature in a unified way.

The theme of the article under review is Hochschild cohomology for admissible pairs of categories. The author starts discussing in this framework the suitable notion of free algebras and the (graded) simplicial context. For a simplicial complex of indices, seen as a category \({\mathcal N}\), denote by \(\text{gr}({\mathcal C})^{\mathcal N}\) the associated category of simplicial (non-positively) graded algebras. For a morphism \(A\to B\) in \(\text{gr}({\mathcal C})^{\mathcal N}\), a resolvent of \(B\) over \(A\) is a free DG algebra \(R\) over \(A\) in \(\text{ gr}({\mathcal C})^{\mathcal N}\) together with a morphism \(R\to B\) which is a surjective quasi-isomorphism (qis) on each \(\alpha\in{\mathcal N}\). While existence of resolvents follows from loc. cit., the author proves that any two of them are homotopically equivalent.

Then he introduces the appropriate notion of a regular sequence \(X\subset R\) of an algebra \(R\). As expected, the Koszul complex \(K(X)\) is a DG resolvent of \(R/(X)\) over \(R\). It is clearly pointed out at which place restrictions on the characteristic of \({\mathbb K}\) come into play.

In the second section, the author introduces Hochschild cohomology associated to a morphism \(k\to a\) in \(\text{gr}({\mathcal C})^{\mathcal N}\) such that there exists a resolvent \(A\) of \(a\) over \(k\). It is defined via the simplicial Hochschild complex \({\mathbb H}_*(a/k)\) which is the object represented by \(S\otimes_Ra\) in the homotopy category \(K^-({\mathcal M}^{\mathcal N}(a))\). Here \(S\) is a resolvent of the multiplication map \(\mu:A\otimes A\to A\). The Hochschild complex \({\mathbb H}(a/k)\) is then the object represented by \(\check{C}^{\bullet}({\mathbb H}_*(a/k))\) in the derived category \(D({\mathbb K}\text{-Mod})\) for a Čech-construction \(\check{C}^{\bullet}(-)\) (transforming a simplicial graded object into a total complex) applied to \({\mathbb H}_*(a/k)\). The difficulty circumvented here resides in the fact that in the analytic context the usual bar complex \(C^{\text{bar}}(a)\) is not a complex of projective \(a\)-modules.

The main result is a HKR-type theorem for free DG algebras \(A\) in \(\text{gr}'({\mathcal C})^{\mathcal N}\). The author shows under these hypotheses the existence of a homotopy equivalence \[ \Lambda\Omega_{A/k}\to {\mathbb H}(A/k), \] where \(\text{gr}'({\mathcal C})\) is the full subcategory of \(\text{gr}({\mathcal C})\) generated by graded algebras \(A\) such that \(A^i\) is a finite \(A^0\)-module. Recall that for an algebra \(a\) over \(k\) in \({\mathcal C}^{\mathcal N}\) with resolvent \(A\) in \(\text{gr}'({\mathcal C})\), the cotangent complex \({\mathbb L}(a/k)\) is defined as the class of \(\Omega_A\otimes_Aa\) in \(K^-({\mathcal M}^{\mathcal N}(a))\). An immediate consequence of the above is a homotopy equivalence \[ \Lambda{\mathbb L}(a/k)\to{\mathbb H}(a/k) \] in \(\text{gr}'({\mathcal C})^{\mathcal N}\) over \(a\), provided \({\mathbb Q}\subset k\). The main result is then carefully translated in section \(4\) into a qis \[ {\mathbb H}(X/Y)\approx\Lambda{\mathbb L}(X/Y) \] for a morphism \(X\to Y\) of paracompact, separated complex spaces or a separated morphism of finite type of Noetherian schemes in characteristic zero.

Further results include several decomposition theorems for Hochschild cohomology: one with respect to tangent cohomology \[ \text{HH}^n(X/Y,{\mathcal M})\cong\coprod_{i+j=n}\text{Ext}^i (\Lambda^j{\mathbb L}(X/Y),{\mathcal M}), \] one with respect to sheaf cohomology with values in exterior powers of the tangent sheaf \[ \text{HH}^n(X)\cong\coprod_{i+j=n}H^i(X,\Lambda^j{\mathcal T}_X), \] and the Hodge decomposition \[ \text{HH}_n(X) \cong\prod_{i-j=n}H^j(X,\Lambda^i\Omega_X) \] thereby reproving many theorems spread over the literature in a unified way.

Reviewer: Friedrich Wagemann (Nantes)

### MSC:

18G60 | Other (co)homology theories (MSC2010) |

14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |

16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |

19E20 | Relations of \(K\)-theory with cohomology theories |

18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |

32C35 | Analytic sheaves and cohomology groups |

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

16E45 | Differential graded algebras and applications (associative algebraic aspects) |

32G05 | Deformations of complex structures |