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Journal of Generalized Lie Theory and Applications
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Simplicial Hochschild cochains as an Amitsur complex 1


Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, U.S.A.

Corresponding Author:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70808, U.S.A.
E-mail: [email protected]

Received Date: December 12, 2007; Revised Date: April 07, 2008

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It is shown how the cochain complex of the relative Hochschild A-valued cochains of a depth two extension A|B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End BAB over the centralizer R = AB with grouplike element 1S. This specializes to finite dimensional algebras, Hopf-Galois extensions and H-separable extensions.

1 Introduction

Relative Hochschild cohomology of a subring B ⊆ A or ring homomorphism B → A is set forth in [4]. The coefficients of the general form of the cohomology theory are taken in a bimodule M over A. If M = A* is the k-dual of the k-algebra A, this gives rise to a cyclic symmetry exploited in cyclic cohomology. If M = A, this has been shown to be related to the simplicial cohomology of a finitely triangulated space via barycentric subdivision, the poset algebra of incidence relations and the separable subalgebra of simplices by Gerstenhaber and Schack in a series of papers beginning with [3]. The A-valued relative cohomology groups of (A,B) are also of interest in deformation theory. Thus we refer to the relative Hochchild cochains and cohomology groups Hn(A,B;A) as simplicial Hochschild cohomology.

In this note we will extend the following algebraic result in [6]: given a depth two ring extension A|B with centralizer R = AB and endomorphism ring S = End BAB, the simplicial Hochschild cochains under cup product are isomorphic as a graded algebra to the tensor algebra of the (R,R)-bimodule S. Since S is a left bialgebroid over R, it is in particular an R-coring with grouplike element 1S = idA. The Amitsur complex of such a coring is a differential graded algebra explained in [2, 29.2]. We note below that the algebra isomorphism in [6] extends to an isomorphism of differential graded algebras. We remark on the consequences for cohomology of
various types of Galois extensions with bialgebroid action or coaction.

2 Preliminaries on depth two extensions

All rings and algebras have a unit and are associative; homomorphisms between them preserve the unit and modules are unital. Let R be a ring, and MR, NR be two right R-modules. The notation M/N denotes that M is R-module isomorphic to a direct summand of an n-fold direct sum power of imageRecall that M and N are similar [1, p. 268] if M/N and N/M. A ring homomorphism B → A is sometimes called a ring extension Image (proper ring extension if Image

Definition 2.1. A ring homomorphism B → A is said to be a right depth two (rD2) extension if the natural (A,B)-bimodules image and A are similar.

Left D2 extension is defined similarly using the natural (B,A)-bimodule structures: a D2 extension is both rD2 and Image Note that in either case any ring extension satisfies Image Note some obvious cases of depth two: 1) A a finite dimensional algebra, B the ground field.
2) A a finite dimensional algebra, B a separable algebra, since the canonical epi Images splits. 3) image an H-separable extension. 4)image a finite Hopf-Galois extension, since the Galois isomorphism image is an (A,B)-bimodule arrow (and its twist by the antipode shows image to be Image as well).

Fix the notation S := End BAB and R = AB. Equip S with (R,R)-bimodule structure


where λ, ρ : R → S denote left and right multiplication of r, s ∈ R on A.

Lemma 2.2 ([5]). If image is rD2, then the module SR is a projective generator and



For example, if A is a finite dimensional algebra over ground field B, then S = End A, the linear endomorphism algebra. If image is H-separable, then image where Z is the center of A [5, 4.8]. If image is an H*-Hopf-Galois extension, then image the smash product where H has dual action on A restricted to R [5, 4.9].

Recall that a left R-bialgebroid H is a type of bialgebra over a possibly noncommutative base ring R. More specifically, H and R are rings with “target” and “source” ring anti-homomorphism and homomorphism R → H, commuting at all values in H, which induce an (R,R)-bimodule structure on H from the left. W.r.t. this structure, there is an R-coring structure (H,R,Δ, ε) such that 1H is a grouplike element (see the next section) and the left H-modules becomes a tensor category w.r.t. this coring structure. One of the main theorems in depth two theory is

Theorem 2.3 ([5]). Suppose A|B is a left or right D2 ring extension. Then the endomorphism ring S := End BAB is a left bialgebroid over the centralizer AB := R via the source map image ,target map image coproduct

image (2.1)

Also A under the natural action of S is a left S-module algebra with invariant subring AS image End EA, where image via image

We note in passing the measuring axiom of module algebra action from Eq. (2.1): in Sweedler notation, Σ(α)α(1)(x)α(2)(y) = α(xy).

3 Amitsur complex of a coring with grouplike

An R-coring C has coassociative coproduct image and counit ε : C → R, both mappings being (R,R)-bimodule homomorphisms. We assume that C also has a grouplike element g ∈ C, which means that image The Amitsur complex Ω(C) of (C, g) has n-cochain modules image (n times C), the zero’th given by Ω0(C) = R. The Amitsur complex is the tensor algebra image with a compatible differential d = {dn} where dn : Ωn(C) → Ωn+1(C).

These are defined by d0 : R → C, d0(r) = rg − gr, and image

Some computations show that Ω(C) is a differential graded algebra [2], with defining equations,d o d = 0 as well as the graded Leibniz equation on homogeneous elements,


The name Amitsur complex comes from the case of a ring homomorphism B → A and Acoring image with coproduct image and counit image The element image is a grouplike element. We clearly obtain the classical Amitsur complex, which is acyclic if A is faithfully flat over B. In general, the Amitsur complex of a Galois Acoring (C, g) is acyclic if A is faithfully flat over the g-coinvariants B = {b ∈ A| bg = bg} [2,29.5].

The Amitsur complex of interest to this note is the following derivable from the left bialgebroid S = End BAB of a depth two ring extension A|B with centralizer AB = R. The underlying R-coring S has grouplike element 1S = idA, with (R,R)-bimodule structure, coproduct and counit defined in the previous section. In Sweedler notation, we may summarize this as follows:


4 Cup product in simplicial Hochschild cohomology

Let A|B be an extension of K-algebras. We briefly recall the B-relative Hochschild cohomology of A with coefficients in A (for coefficients in a bimodule, see the source [4]). The zero’th cochain group C0(A,B;A) = AB = R, while the n’th cochain group


(n times A in the domain). In particular, C1(A,B;A) = S = End BAB. The coboundary δn : Cn(A,B;A) → Cn+1(A,B;A) is given by


and δ0 : R → S is given by δ0(r) = λrr. The mappings satisfy δn+1 o δn = 0 for each n ≥ 0. Its cohomology is denoted by Hn(A,B;A) = ker δn/Im δn-1, and might be referred to as a simplicial Hochschild cohomology, since this cohomology is isomorphic to simplicial cohomology if A is the poset algebra of a finite simplicial complex and B is the separable subalgebra of vertices [3].

The cup product image makes use of the multiplicative stucture on A and is given by


which satisfies [3] the equation


Cup product therefore passes to a product on the cohomology. We note that image is a differential graded algebra we denote by C(A,B).

Theorem 4.1. Suppose A|B is a right or left D2 algebra extension. Then the relative Hochschild A-valued cochains C(A,B) is isomorphic as a differential graded algebra to the Amitsur complex Ω(S) of the R-coring S.

Proof. We define a mapping f by f0 = idR, f1 = idS, and for n > 1,


by image (Note that f2 is consistent with our notation in section 2.) We proved by induction on n in [6] that f is an isomorphism of graded algebras. We complete the proof by noting that f is a cochain morphism, i.e., commutes with differentials. For n = 0, we note that δ0 o f0 = f1 o d0, since d0 = δ0. For n = 1,


using Eq. (2.1). The induction step is carried out in a similar but tedious computation: this completes the proof that image

5 Applications of the theorem

We immediately note that the cohomology rings of the two differential graded algebras are isomorphic.

Corollary 5.1. Relative A-valued Hochschild cohomology is isomorphic to the cohomology of the AB-coring S = End BAB:


if A|B is a left or right depth two extension.

For example, we recover by different means the known result,

Corollary 5.2. If the ring extension A|B is H-separable and one-sided faithfully flat, then the relative Hochschild cohomology is given by


Proof. Note that the extension is necessarily proper by faithful flatness. Note that image is a Galois R-coring, since image the center of A and the isomorphism image is clearly a coring homomorphism. Whence Ω(S) is acyclic by [2, 29.5].


which is the center of the centralizer.

This will also follow from proving that an H-separable is a separable extension, a condition of trivial cohomology groups.

Corollary 5.3. Suppose A|B is a finite Hopf-H*-Galois extension. Then relative Hochschild A-valued cohomology is isomorphic to the Cartier coalgebra cohomology of H with coefficients in the bicomodule image



The author is thankful to the organizers and participants of A.G.M.F. in Gothenburg and the Norwegian algebra meeting in Oslo, 2007, for the stimulating focus on cohomology.


  1. Anderson FW, Fuller KR (1992) Rings and Categories of Modules. Springer-Verlag, 2nd Edition.
  2. Brzezi ́nski T,WisbauerR (2003) Corings and Comodules.CambridgeUniv Press.
  3. Gerstenhaber M, Schack SD (1983)Simplicialcohomology is Hochschildcohomology. J Pure ApplAlg 30: 143–156.
  4. Hochschild G (1956) Relative homological algebra.  Trans Amer Math Soc82: 246–269.
  5. Kadison L,Szlach ́anyiK (2003)Bialgebroid actions on depth two extensions and duality. Adv Math 179: 75–121.
  6. Kadison L (2006) Codepth two and related topics. ApplCategStruct14: 605–625.
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