We show that faithfully flat smooth extensions of associative unital algebras are reduced flat, and therefore, fit into the Jacobi-Zariski exact sequence in Hochschild homology and cyclic (co)homology even when the algebras are noncommutative or infinite dimensional. We observe that such extensions correspond to étale maps of affine schemes, and we propose a definition for generic noncommutative fibrations using distributive laws and homological properties of the induction and restriction functors. Then we show that Galois fibrations do produce the right exact sequence in homology. We then demonstrate the versatility of our model on a geometro-combinatorial example. For a connected unramified covering of a connected graph , we construct a smooth Galois fibration and calculate the homology of the corresponding local coefficient system.
Based on homological connections between the induction and the restriction functors, in this paper we gather further evidence that the Hochschild-Jacobi-Zariski exact sequence
is the long exact sequence of associated to a fibration of ordinary (noncommutative) affine spaces when the extension is reduced flat. First, we prove in Theorem 1.8 that for faithfully flat extensions reduced flatness is equivalent to -flatness of the relative Kähler differentials . Then in Theorem LABEL:AlmostSmoothBegetsReducedFlat we obtain a faithfully flat étale descent result analogous to [WeibelGeller:EtaleDescent, Theorem (0.1)] but for all associative unital algebras not just commutative ones: We show that any faithfully flat smooth extension is reduced flat, and therefore, the geometric fiber of is homologically trivial. The result follows from the fact that now we have the Hochschild-Jacobi-Zariski exact sequence with coefficients (1.1) for faithfully flat smooth extensions, and the fact that the restriction functor already induces the correct isomorphisms in homology by Proposition 1.3.
There is an analogous Jacobi-Zariski exact sequence for extensions of commutative algebras in André-Quillen (co)homology without any further restriction on the extension [Quillen:AQCohomology, Andre:AQCohomology]. However, our (0.1) is exact for commutative and noncommutative algebras alike even when they are not finite dimensional or essentially of finite type. The results in this paper came from an observation that smooth extensions and reduced flat extensions are related in terms of homological properties of their induction and restriction functors: while the multiplication map induces a Hochschild cohomological equivalence in degrees higher than 1 for a smooth extension, for a reduced flat extension one gets a Hochschild homological equivalence for the same range for the natural -bimodule embedding . We refer the reader to Subsection LABEL:Relations for a detailed analysis of these connections.
Based on the results we obtained in Section 1, we propose that a special class of extensions of unital associative algebras that contains the class of Hopf-Galois extensions [Schneider:PrincipalHomogeneousSpaces] constitutes an appropriate model for generic smooth noncommutative fibrations. We define a noncommutative (unramified) fibration as a flat extension that admits a (bijective) distributive law [Beck:DistributiveLaws] together with an epimorphism of -bimodules that satisfies an invariance condition . Then in Theorems LABEL:MainHJZ and LABEL:MainCJZ we get the correct fibration sequence with the appropriate fiber for the map for Galois fibrations. Since we formulate our extensions in terms of distributive laws instead of cleft Hopf-Galois extensions, the extensions we consider model generic fibrations, not just principal fibrations. We refer the reader to Section LABEL:sect:Fibrations for details.
We demonstrate the versatility of our model on a geometro-combinatorial example. For a connected unramified covering of a connected graph , we construct an unramified reduced flat extension of noncommutative algebras in Subsection LABEL:HochschildOfGraphAlgebras. We then show in Theorem LABEL:UnramifiedCovering that for such extensions, we get the right analogue of the long exact sequence of a fibration in cyclic homology. Then we extend our result to local coefficient systems on graphs and their cohomology in Theorem LABEL:LocalCoefficients.
The particular result we obtain in Theorem LABEL:LocalCoefficients, combined with Burghelea’s [Burghelea:CyclicHomologyOfGroupRings], is consistent with [Milne:EtaleCohomology, Chapter III, Theorem 2.20] and [WeibelGeller:EtaleDescent, Example 2.2] where one obtains the homology of a Galois coverings of schemes from a Hochschild-Serre hyper-homology spectral sequence in which they combine the group cohomology of the structure group of the fibration and the homology of the base. This consistency indicates that our proposal is sound geometrically. Since Theorem LABEL:LocalCoefficients is a direct consequence of Theorem LABEL:MainCJZ, we also see that for cleft Hopf-Galois extensions the Hochschild homology of such an extension relative to the base is the homology of the underlying Hopf algebra. Hence our proposal is sound algebraically as well.
Plan of the article
We recall the results we need on reduced flat and smooth extensions in Section 1. Our Proposition 1.3 and Proposition 1.5 identify the reduced flat extensions and smooth extensions in terms of homological conditions on the induction and restriction functors. Then we define unramified and Galois fibrations, and discuss connections between various types of extensions and fibrations in Section LABEL:sect:Fibrations. In Subsection LABEL:subsect:Engine we prove our main technical results. First, we show that the relative Hochschild homology of a Galois fibration yields the correct homology of the fiber in Theorem LABEL:MainHJZ. Then in Theorem LABEL:MainCJZ, we show that for reduced flat Galois fibration, we have the required long exact sequences in Hochschild homology and cyclic (co)homology. We apply our main results to graph extension algebras in Section LABEL:sect:GraphCovering. In Subsection LABEL:subsect:LocalCoefficients, we define local coefficient systems on graphs, and finally in Theorem LABEL:LocalCoefficients we prove that the relative homology of a noncommutative fibration with coefficients in a local system gives us the group homology of the local coefficients.
Notation and conventions
Throughout this article, we are going assume is a ground field of characteristic 0. All unadorned tensor products are taken over . All algebras are assumed to be over , and all are unital and associative. However, they need not be commutative or finite dimensional. All modules are assumed to be left modules unless otherwise stated. We use to denote a small category of -modules. For an algebra , we use to denote the enveloping algebra . Thus modules over are exactly bimodules over . We use the homological convention for complexes: all complexes are positively graded and differentials reduce the degree by one. We use and to denote the derived bifunctors of the tensor product and -bifunctors, respectively. We are going to use to denote the bar complex, and to denote the Hochschild complex. Also, we use for the Hochschild homology, and for the cyclic homology functors. All graphs are assumed to be undirected and simple, but they need not be finite. In particular, we have no loops on a vertex, and no multiple edges between any two vertices.
1. Reduced flat and almost smooth extensions
For this section, we assume we have an extension of unital associative algebras such that viewed as a left and right -module is flat.
1.1. Relative Hochschild (co)homology
Given an extension , the relative two sided bar complex is defined to be the graded -module
For every , the differentials are defined as
for every homogeneous tensor , then extended linearly. Since is a -projective resolution of as a -module, for any -module we write the relative Hochschild chain and cochain complexes as
that yield the relevant relative Hochschild homology and cohomology groups and , respectively. In the case , we simply write and instead of and .
1.2. Induction and restriction
We have two related functors:
Induction , and
Restriction where we view as an -bimodule via the inclusion
for every and .
For every we have
for every .
We observe that the is a free resolution of the -bimodule . Since we assumed is a right and left flat -module, we also have that is a flat resolution of the -bimodule . Then
as we wanted to prove. ∎
By Lemma 1.1 we have a sequence of natural morphisms
for every , and . In the following subsections, we are going to show that viewed as a natural transformation of functors is an isomorphism when the extension is reduced flat, and again viewed as a natural transformation of functors is an isomorphism when the extension is (almost) smooth, both for a certain range of . Moreover, we are also going to show that when the extension is faithfully flat then the fact that is an isomorphism implies so is .
1.3. Almost smooth extensions
For an extension of -algebras , we define to be the kernel of the relative multiplication map as a morphism of -modules. We call a flat extension (almost) smooth if is a projective (resp. flat) -module [Schelter:SmoothAlgebras]. Notice that when an extension is smooth then it is also almost smooth.
A flat extension is almost smooth if and only if we have for every and for every .
The proof follows from the fact that is almost smooth if and only if we have a sequence of isomorphisms of the form
for every and . ∎
There is a version of Proposition 1.3 for smooth extensions that works with Hochschild cohomology instead of homology that says is smooth if and only if
for every and .
1.4. Reduced flat extensions
We now recall from [Kaygun:JacobiZariski] that we call an extension as reduced flat when the cokernel of the -bimodule inclusion is flat as a -bimodule. We also observe that reduced flatness of the extension is equivalent to the fact that the Hochschild homology of of with coefficients in any , and the torsion groups are isomorphic for all . Combining this result with Lemma 1.1 we get
A flat extension is reduced flat if and only if we have natural isomorphisms of the form for every and for every .
We will say that an extension satisfies Hochschild-Jacobi-Zariski (resp. cyclic-Jacobi-Zariski) condition [Loday:CyclicHomology, 18.104.22.168] if we have a long exact sequence in Hochschild homology (resp. cyclic homology) of the form
for every , and for every .
Proposition 1.6 ([Kaygun:JacobiZariski, Theorem 4.1 and Theorem 4.2]).
If a flat extension is reduced flat then the extension satisfies both Hochschild-Jacobi-Zariski and cyclic-Jacobi-Zariski conditions for every .
Recall that the relative homology groups measure the failure of the extension of being smooth since we have both the exact sequence by Proposition 1.3 and Proposition 1.6 for a almost smooth reduced flat extensions . This is rather subtle: almost smoothness does imply relative homology vanishes since absolute -flatness of implies that its relative -flatness as a -module. However, the converse need not be true in general. The fact that the relative homology vanishes for implies is only -flat relative to . The fact that is reduced flat over gives us (1.1), and then we get the isomorphisms for the required range, and then Proposition 1.3 gives us the absolute -flatness.
1.5. Faithfully flat almost smooth extensions
Assume is faithfully flat over . Then is reduced flat over if and only if is a flat -bimodule.
We start by observing that there is a natural isomorphism of -modules of the form coming from the diagram