Coherence of relatively quasi-free algebras
Alexey Bondal, Ilya Zhdanovskiy

1Introduction

An important problem in representation theory of associative algebras is to prove coherence forvarious noncommutative algebras. Left coherence implies that the category of finitely presented
left modules is abelian. This category might then be considered as being analogous to thecategory of coherent sheaves on an affine commutative variety. Thus, coherence is the initialpoint for developing noncommutative geometry in representation theory of algebras in the stylesimilar to the theory of coherent sheaves.
Despite its crucial importance the problem of coherence is basically terra incognita: we don’tknow so far about many interesting classes of noncommutative algebras with nice homologicalproperties whether they are coherent or not.
There is a class of algebras closed to free associative algebras for which coherence is known.
These are so-called quasi-free algebras over fields in the sense of Kuntz and Quillen [CQ1]. We
reproduce the proof in the main body of the text.
In this paper, we are interested in algebras that are relatively quasi-free over a commutativealgebra, say K. By definition, such an algebra A is central over K and A-bimodule of noncommutative 1-forms Ω1A/K is projective. Similar to the case of algebras that are quasi-free over a
field one can show that relatively quasi-free algebras are exactly those which satisfy the lifting
property for nilpotent K-central extensions (proposition 1).
We prove a theorem that an algebra which is quasi-free relatively over a commutative
noetherian ring K is left and right coherent.
This result requires a different and more involved techniques in comparison to the case when
the ring K is a field. Our proof is based on the Chase criterion for coherence, which states thata ring is left coherent if and only if the product of any number of flat right modules is flat. In
fact the criterion is enough to check for the product of sufficiently many copies of the rank one
free modules over the algebra. We use also a criterion of similar style for noetherian algebras.
It would be interesting to check whether the method of this paper is applicable to proving
coherence for other important classes of algebras. In [BZ], we explore an algebra B constructed
via algebra A and an element ∆ in it. The algebra is constructed by modifying multiplicationin the vector space A via the rule:a ·B b = a∆band adjoining a unit to the resulting algebra. If element ∆ verifies some conditions, in whichcase it is called well-tempered in [BZ], we prove that if the original algebra was coherent thenso is the new one, thus providing an iterative way of constructing coherent algebras.
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We are grateful to Dmitry Piontkovsky for useful references. This work was done duringauthors visit to Kavli IPMU and was supported by World Premier International ResearchCenter Initiative (WPI Initiative), MEXT, Japan.

2Coherence of algebras and categories of finitely presented modules

A left module M of an algebra is said to be coherent if it is finitely generated and for every
morphism ϕ : P → M with free module P of finite rank the kernel of ϕ is finitely generated.
An algebra is (left) coherent if it is coherent as a left module over itself. If algebra is coherent,
then finitely presented modules are the same as coherent modules and the category of finitely
presented modules is abelian [Bou].
Let A be an algebra over a commutative ring K. The definition due to Cuntz and Quillen[CQ1] of a quasi-free algebra, which we adopt to the relative case (i.e. for algebras over acommutative ring K rather than over a field), is that algebra A over a field k is quasi-free if
the bimodule of noncommutative differential 1-forms Ω1 A (see definition below) is a projectiveA ⊗k Aopp -module.
Every quasi-free algebra is hereditary, i.e. has global dimension ≤ 1. Indeed, for any twoleft A-modules M and N we have:
ExtiA (M, N ) = ExtiA−A (A, Homk (M, N )).
Since A has a projective bimodule resolution of length 1, it follows that ExtiA (M, N ) = 0, fori ≥ 2.
Hereditary algebras are coherent. Indeed, a submodule of a projective module is projective
for such algebras. Thus, given a morphism ϕ : P1 → P2 between finitely generated projective
modules, the image I of ϕ is a submodule of a projective module, hence projective too. Thenshort exact sequence induced by ϕ:
0 → K → P1 → I → 0,
where K is the kernel of ϕ, splits. Hence, we have an epimorphism P1 → K, which proves that
K is finitely generated.
This proves that every quasi-free algebra is left (and, similarly, right) coherent. Note thatthe number of generators for the kernel K as a left A-module is bounded by the number ofgenerators of P1 , which is not true in general even for coherent commutative algebras. For thatreason this proof of coherence is not applicable to the case of relatively quasi-free algebras thatwe consider below. We will need a substantially different argument.
Let us develop the theory for the case of relatively quasi-free algebras. A K-algebra A issaid to be central over K if the image of K → A is in the center of A. An A-bimodule is said
to be K-central if left and right K actions coincide. Such A-bimodules are identified with left
A ⊗K Aopp -modules.

2Define the A ⊗K Aopp -module of relative forms by an exact sequence
0 → Ω1A/K → A ⊗K A → A → 0(1)Definition Let K be a commutative ring. An algebra A central over K is said to bequasi-free over K if Ω1A/K is a projective A ⊗K Aopp -module.
Let A be a relatively quasifree algebra over K. By applying functor Exti (−, M ) to the exact
sequence (1) for an arbitrary K-central A-bimodule M , we obtain:
Ext2A⊗K Aopp (A, M ) = Ext1A⊗K Aopp (Ω1A/K , M ) = 0.

(2)

A generalization of Cuntz-Quillen criterion for quasi-free algebras [CQ1] holds in the relativecase.
Proposition 1. Algebra A is quasi-free over K if and only if for any R̃, a nilpotent extension of the algebra R by square zero ideal in the category of algebras central over K, and anyhomomorphism A → R there exists its lifting to a homomorphism A → R̃.
Proof. Let A be a quasi-free algebra. Consider a square-zero extension R̃ → R and a homomorphism A → R. Denote by I the kernel of R̃ → R. By assumptions, it is a square zero idealin R̃, hence it has a natural structure of K-central R-bimodule. Let Ã be the fibred productover R of A and R̃. This is an algebra central over K too. It is a square zero extension of Aby I, where I is endowed with a K-central A-bimodule structure which is the pull-back of Rbimodule structure. Such extensions are classified by Ext2A⊗K Aopp (A, I). This group is trivial for
quasi-free algebras by (2). Hence, we have a splitting homomorphism A → Ã. When combinedwith the map Ã → R̃ it gives the required lifting.
Conversely, if all square zero extensions allow liftings, then, by taking R = A and I arbitraryK-central A-bimodule, we see that Ext2A⊗K Aopp (A, I) = Ext1A⊗K Aopp (Ω1A/K , I) = 0, i.e. Ω1A/K is a
projective A ⊗K Aopp -bimodule.
Recall the following criterion of coherence due to Chase [Chase].
Lemma 2. For any ring A the following are equivalent:• A is left coherent,• For any family of right flat modules Fi , i ∈ I, the product

Qi∈I Fi is right flat,

Q• For the family Fi ∼= A of free modules with card I = card A, the product i∈I Fi is rightflat.
We will use also a criterion of the same style for noetherianess (cf. [Ab]).
Lemma 3. For any ring A the following are equivalent:• A is left noetherian,3

• ForQ any left A-module
Q M and any family of flat right modules Fi , i ∈ I, the morphism( i∈I Fi ) ⊗A M → i∈I (Fi ⊗A M ) is mono,∼• For any left A-module M and theQ family of rank 1Qfree right modules Fi = A, i ∈ I,
card I = card A, the morphism ( i∈I Fi ) ⊗A M → i∈I (Fi ⊗A M ) is mono.
Theorem 4. Let K be a commutative noetherian ring. Assume that A is an algebra quasi-free
relatively over K and A is flat as a K-module. Then A is a left and right coherent algebra.
Proof. Consider left A-module M and a family of rank 1 free right A-modules Fi ∼= A. We shallconsider all left (respectively, right) A-modules to be always endowed with the right (respectively, left) K-module structure identical to theright) K-module structure.
Q left (respectively,Q
By applying functors (−) ⊗A⊗K Aopp (M ⊗K i Fi ) and i ((−) ⊗A⊗K Aopp (M ⊗K Fi )) and theirderived functors to the exact sequence (1), we obtain a commutative diagram with exact rows:
0/ TorA⊗K Aopp (A, M ⊗0/1QA⊗K Ai Tor1

KQ/ Ω1

A/K ⊗A⊗K Ai Fi )

opp

/

(A, M ⊗K Fi )

Q

(M ⊗K

opp

Qi Fi )

(3)

1i (ΩA/K ⊗A⊗K Aopp (M ⊗K Fi ))For any left A-module M and right A-module N , we have an isomorphism of objects in thederived categories:
A ⊗LA⊗K Aopp (M ⊗LK N ) = M ⊗LA N,which implies a spectral sequence:
KATorA⊗iopp

A(A, TorKj (M, N )) =⇒ Tori+j (M, N )For a flat K-module N , it implies that
Tor1A⊗K Aopp

(A, M ⊗K N ) = TorA
1 (M, N ).

(4)

Since Fi are rank 1 free as A-modules and A is a flat K-module, Fi are also flat K-modules.
AHenceQ Tor1 (M, Fi ) = 0. Since K is noetherian, it is coherent, hence by Chase criterion, lemma
2, i Fi is also flat. In view of (4), diagram (3) reads:
0/ TorA (M,10Q/ Ω1A/K ⊗A⊗K Ai Fi )/Q(M ⊗KoppQi Fi )(5)1i (ΩA/K ⊗A⊗K Aopp (M ⊗K Fi ))Q
Q
Let us show that Ω1A/K ⊗A⊗K Aopp (M ⊗K i Fi ) → i (Ω1A/K ⊗A⊗Aopp (M ⊗K Fi )) is an embedding.
Q
Q
By criterion of noetherianess, lemma 3, for K, we have that M ⊗K i Fi → i (M ⊗K Fi ) is anembedding.

4Let D = ⊕j (A ⊗K Aopp ) be a free A ⊗K Aopp -module, thenYYD ⊗A⊗K Aopp (M ⊗KFi ) = ⊕j (M ⊗KFi )iiandYY(D ⊗A⊗K Aopp (M ⊗K Fi )) =(⊕j (M ⊗K Fi )).
i

i

The morphism
D ⊗A⊗K Aopp (M ⊗KYFi ) →iY(D ⊗A⊗K Aopp (M ⊗K Fi ))iis the composite of two morphism:
YYY⊕j (M ⊗KFi ) → ⊕j(M ⊗K Fi ) →(⊕j (M ⊗K Fi )).
ii

i

Both morphism are readily embeddings, hence so is the composite.
Since Ω1A/K is a projective bimodule, we have an imbeddingΩ1A/K → ⊕j A ⊗K Aopp
as a direct summand. This implies a diagram:
QΩ1A/K ⊗A⊗K Aopp (M ⊗K i Fi )Q1i (ΩA/K ⊗A⊗K Aopp (M ⊗K Fi ))// ⊕j (M ⊗KQQi Fi )(6)

i (⊕j (M ⊗K Fi ))The upper horizontal arrow is an embeddingQbecause it obtained by tensoring up an embeddingof a direct summand with a module M ⊗K i Fi . We have shown above that the right verticalarrow is an embedding too.
Q Therefore, theQ left vertical arrow is an embedding. Then, diagram(5) implies that TorA
(M,F)
=0,i.e1i ii Fi is a right flat module. By Chase criterion, lemma2, algebra A is left coherent.
Right coherence follows similarly.

References[Ab] H. Aberg, Coherence of Amalgamations, Journal of Algebra 78 (1982), 372–385.
[Bou] N. Bourbaki, Algebre Homologique, Masson, Paris, New-York, Barcelone, Milan, 1980.
[BZ] A. Bondal, I. Zhdanovskiy, Representation theory for systems of projectors anddiscrete Laplace operators, IPMU13-0001, IPMU, Kashiwa, Japan, 2013, 48 .,http://db.ipmu.jp/ipmu/sysimg/ipmu/1012.pdf
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[Chase] S. Chase, Direct Products of Modules, Trans. Amer. Math. Soc. 97 (1960), 457–473.
[CQ1] J. Cuntz, and D. Quillen, Algebra extensions and nonsingularity, J. of Amer. Math. Soc.
8 (1995), no. 2, 251–289.

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