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**Lars KADISON ^{*}**

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

**Corresponding Author:**- Lars KADISON

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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

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 H^{n}(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 = A^{B} and endomorphism ring S = End_{ B}A_{B}, 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 1_{S} = id_{A}. 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.

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 M_{R}, N_{R} 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 Recall 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 (proper ring extension if

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

Left D2 extension is defined similarly using the natural (B,A)-bimodule structures: a D2 extension is both rD2 and Note that in either case any ring extension satisfies 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 splits. 3) an H-separable extension. 4) a finite Hopf-Galois extension, since the Galois isomorphism is an (A,B)-bimodule arrow (and its twist by the antipode shows to be as well).

Fix the notation S := End _{B}A_{B} and R = A^{B}. 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 is rD2, then the module S_{R} 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 is H-separable, then where Z is the center of A [5, 4.8]. If is an H*-Hopf-Galois extension, then 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 1_{H} 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 _{B}A_{B} is a left bialgebroid over the centralizer A^{B} := R via the source map ,target map coproduct

(2.1)

Also A under the natural action of S is a left S-module algebra with invariant subring A^{S} End _{E}A, where via

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).

An R-coring C has coassociative coproduct 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 The Amitsur complex Ω(C) of (C, g) has n-cochain modules (n times C), the zero’th given by Ω^{0}(C) = R. The Amitsur complex is the tensor algebra with a compatible differential d = {d^{n}} where d^{n} : Ω^{n}(C) → Ω^{n}^{+1}(C).

These are defined by d^{0} : R → C, d^{0}(r) = rg − gr, and

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 with coproduct and counit The element 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 _{B}A_{B} of a depth two ring extension A|B with centralizer A^{B} = R. The underlying R-coring S has grouplike element 1_{S} = id_{A}, with (R,R)-bimodule structure, coproduct and counit defined in the previous section. In Sweedler notation, we may summarize this as follows:

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 C^{0}(A,B;A) = A^{B} = R, while the n’th cochain group

(n times A in the domain). In particular, C^{1}(A,B;A) = S = End _{B}A_{B}. The coboundary δ^{n} : C^{n}(A,B;A) → C^{n+1}(A,B;A) is given by

and δ^{0} : R → S is given by δ^{0}(r) = λ_{r}-ρ_{r}. The mappings satisfy δ^{n+1} o δ^{n }= 0 for each n ≥ 0. Its cohomology is denoted by H^{n}(A,B;A) = ker δ^{n}/Im δ^{n-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 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 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 f_{0} = id_{R}, f_{1} = id_{S}, and for n > 1,

by (Note that f_{2} 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 δ^{0} o f_{0} = f_{1} o d^{0}, since d^{0} = δ^{0}. For n = 1,

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

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 A^{B}-coring S = End _{B}A_{B}:

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 is a Galois R-coring, since the center of A and the isomorphism is clearly a coring homomorphism. Whence Ω(S) is acyclic by [2, 29.5].

Finally,

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

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.

- Anderson FW, Fuller KR (1992) Rings and Categories of Modules. Springer-Verlag, 2nd Edition.
- Brzezi ́nski T,WisbauerR (2003) Corings and Comodules.CambridgeUniv Press.
- Gerstenhaber M, Schack SD (1983)Simplicialcohomology is Hochschildcohomology. J Pure ApplAlg 30: 143–156.
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- Kadison L (2006) Codepth two and related topics. ApplCategStruct14: 605–625.

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