Coherence of relatively quasifree algebras
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Abstract
We prove coherence of relatively quasifree algebras over noetherian rings. The Chase criterion for coherence is used.
Keywords
Coherent algebra Quasifree algebra Chase criterionMathematics Subject Classification
18G20 18G251 Introduction
An important problem in the representation theory of associative rings is to find conditions which guarantee that a noncommutative ring is coherent. Left coherence implies that the category of finitely presented left modules over a ring is abelian. This category might then be considered as being analogous to the category of coherent sheaves on an affine commutative variety. Thus, coherence is the initial point for developing noncommutative geometry in the framework of representation theory of rings in the style of the theory of coherent sheaves on algebraic varieties.
Despite its crucial importance, the problem of coherence is basically terra incognita: there are not so many classes of noncommutative rings with nice homological properties for which we can ensure coherence so far. Some recent references to results on coherence of various types of algebras the reader can find in the last paragraph of this paper.
A class of algebras for which coherence is known are the socalled quasifree algebras over fields in the sense of Cuntz and Quillen [7]. We reproduce the proof of coherence for them in the main body of the text.
In this paper, we are interested in algebras that are relatively quasifree over a commutative ring, say \({\mathbb {K}}\). By definition, this is an algebra A over \({\mathbb {K}}\) with the Abimodule of noncommutative 1forms Open image in new window being projective. Similar to the case of algebras that are quasifree over a field one can show that relatively quasifree algebras are exactly those which satisfy the lifting property for nilpotent \({\mathbb {K}}\)central extensions (Proposition 2.3).
We prove a theorem that an algebra which is relatively quasifree over a commutative noetherian ring \({\mathbb {K}}\) is left and right coherent. This result requires different and more involved techniques in comparison to the case when the ring \({\mathbb {K}}\) is a field. Our proof is based on the Chase criterion for coherence, which states that a 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 against the product of sufficiently many copies of rank one free modules over the algebra. In course of the proof, we use also a criterion of similar type for noetherian algebras.
2 Coherence of algebras and categories of finitely presented modules
All rings and algebras which we consider in this paper are unital. A left module M over a ring is said to be coherent if it is finitely generated and for every morphism \(\varphi :P\rightarrow M\) with free module P of finite rank the kernel of \(\varphi \) is finitely generated. A ring is (left) coherent if it is coherent as a left module over itself. Equivalently, a ring is coherent if every homomorphism between finitely generated projective modules over the ring has a finitely generated kernel. If ring is coherent, then finitely presented modules are the same as coherent modules and the category of finitely presented modules is abelian [4].
We will be interested in an algebra A over a commutative ring \({\mathbb {K}}\). The definition due to Cuntz and Quillen [7] of a quasifree algebra, which we adopt to the relative case (i.e. for algebras over a commutative ring \({\mathbb {K}}\) rather than over a field), is that the algebra A over a field k is quasifree if the bimodule of noncommutative differential 1forms Open image in new window (the definition is below) is a projective Open image in new window module.
The proof of the following lemma was sketched in [11, Lemma in Section 2.5].
Lemma 2.1
Hereditary rings are coherent.
Proof
This implies that every quasifree algebra is left (and, similarly, right) coherent.
It follows from the proof that the number of generators for the kernel K as a left Amodule is bounded by the number of generators for \(P_1\), which shows how special this case is. In general, even noetherian commutative rings might be of arbitrary global dimension and the number of generators of the kernel of a morphism between two finite rank free modules is not restricted by any function on ranks of the two free modules. For that reason this proof of coherence does not seem to be applicable to the case of relatively quasifree algebras that we consider below. We will need a substantially different argument.
Let us develop a bit of the theory for the case of relatively quasifree algebras. We consider an algebra A over a commutative ring \({\mathbb {K}}\). An Abimodule is said to be \({\mathbb {K}}\)central if left and right \({\mathbb {K}}\) actions coincide. Such Abimodules are identified with left Open image in new window modules.
Definition 2.2
Let \({\mathbb {K}}\) be a commutative ring. An algebra A over \({\mathbb {K}}\) is said to be quasifree over \({\mathbb {K}}\) if Open image in new window is a projective Open image in new window module.
Proposition 2.3
An algebra A is quasifree over \({\mathbb {K}}\) if and only if for any \({\widetilde{R}}\), a nilpotent extension of an algebra R by a square zero ideal in the category of algebras over \({\mathbb {K}}\), and any homomorphism \(A\rightarrow R\) there exists its lifting to a homomorphism \(A\rightarrow {\widetilde{R}}\).
Proof
Let A be a quasifree algebra. Consider a squarezero nilpotent extension \({\widetilde{R}}\rightarrow R\) and a homomorphism \(A\rightarrow R\). Denote by I the kernel of \({\widetilde{R}}\rightarrow R\). By assumptions, it is a square zero ideal in \({\widetilde{R}}\), hence it has a natural structure of \({\mathbb {K}}\)central Rbimodule. Let \({\widetilde{A}}\) be the fibred product over R of A and \({\widetilde{R}}\). This is a \({\mathbb {K}}\)algebra, which is a square zero extension of A by I, where I is endowed with a \({\mathbb {K}}\)central Abimodule structure which is the pullback of Rbimodule structure. Such extensions are classified by \(\mathrm{Ext}^2_{A\otimes _{{\mathbb {K}}} A^\mathrm{opp}}(A, I)\). This group is trivial for quasifree algebras by (2). Hence, we have a splitting homomorphism \(A\rightarrow {\widetilde{A}}\). When combined with the map \({\widetilde{A}} \rightarrow {\widetilde{R}}\) it gives the required lifting.
Conversely, if all square zero extensions allow liftings, then, by taking \(R=A\) and I an arbitrary \({\mathbb {K}}\)central Abimodule, we see that Open image in new window , i.e. Open image in new window bimodule. \(\square \)
3 Coherence for relatively quasifree algebras
Recall the following criterion of coherence due to Chase [5].
Lemma 3.1

A is left coherent.

For any family of right flat modules \(F_i\), \(i\in I\), the product Open image in new window is right flat.

For the family \(F_i\cong A\) of free modules with \(\mathrm{card}\, I= \mathrm{card}\, A\), the product Open image in new window is right flat.
We will also use a criterion in the same style for noetherianess (cf. [1]).
Lemma 3.2

A is left noetherian.

For any left Amodule M and any family of flat right modules \(F_i\), \(i\in I\), the morphism Open image in new window is mono.

For any left Amodule M and the family of rank 1 free right modules \(F_i\cong A\), \(i\in I\), \(\mathrm{card}\, I= \mathrm{card}\ A\), the morphism Open image in new window is mono.
Theorem 3.3
Let \({\mathbb {K}}\) be a commutative noetherian ring. Assume that A is a relatively quasifree algebra over \({\mathbb {K}}\) and it is flat as a \({\mathbb {K}}\)module. Then A is a left and right coherent algebra.
Proof
Consider a left Amodule M and a family of rank 1 free right Amodules \(F_i\cong A\). We shall assume all left (respectively, right) Amodules to be endowed with the right (respectively, left) \({\mathbb {K}}\)module structure identical to the left (respectively, right) \({\mathbb {K}}\)module structure.
Then, diagram (5) implies that Open image in new window , i.e Open image in new window is a right flat module. By the Chase criterion, Lemma 3.1, the algebra A is left coherent. Right coherence follows similarly. \(\square \)
Similarly, one can prove the following statement.
Theorem 3.4
Let an algebra A over a commutative ring \({\mathbb {K}}\) be of Hochschild dimension n. Then Tordimension of any product of free right modules is less than n.
Proof
Consider a projective Abimodule resolution for A of length \(n+1\). Let M be a left Amodule and \(F_i\) free right Amodules. By applying functors Open image in new window and Open image in new window to the resolution we get two complexes. Moreover, the natural transformation between the functors provides a morphism from the first complex to the second one. The same argument as in the proof of the previous theorem shows that the second complex is exact and the first one is termwise embedded into the second one. Hence, we get an exact triangle of complexes, where the third term is the quotient of the second complex by the first one. It implies a long exact sequence on cohomology of these three complexes, that shows that cohomology of the first complex in the most left term is trivial. This cohomology is identified with Open image in new window . This proves the theorem. \(\square \)
4 Examples and related results
If A is quasifree over a field k and \({\mathbb {K}}\) is a commutative kalgebra, then Open image in new window is flat and quasifree over \({\mathbb {K}}\). This gives numerous examples where Theorem 3.3 is applicable, if we take \({\mathbb {K}}\) to be a noetherian ring. Among those are free algebras, path algebras of quivers with no relation, etc.
The criterion of quasifreeness in terms of lifting the nilpotents, Proposition 2.3, implies that quasifreeness is preserved under localizations of algebras, which shows, in particular, that group algebras of free groups over noetherian rings \({\mathbb {K}}\) are coherent.
More examples come if we consider algebras which are Morita equivalent to examples, considered above, because coherence is preserved under Morita equivalence. Thus, we have
Corollary 4.1
The matrix algebra over the group algebra of a free group over a noetherian ring is coherent.
This example is of interest when considering perverse sheaves and harmonic analysis on graphs (cf. [3]).
There are few results on coherence of algebras. In [6], the authors proved that if \({\mathbb {K}}\) is a noetherian algebra, and A and B are augmented coherent \({\mathbb {K}}\) algebras, then the coproduct Open image in new window over \({\mathbb {K}}\) is coherent too. Aberg in [1] was able to remove the assumption that algebras A and B are augmented by heavy use of the Chase criterion.
It makes sense to compare the Chase criterion, used in this paper, with another cohomological approach applicable to the case of local algebras. There is a simple way to control coherence of local algebras (cf. [14]). A finitely generated local algebra is coherent if, for every finitely generated ideal J, the space \(\mathrm{Tor}^A_1(k, J)\) is finite dimensional over the residue field k. In particular, it allows to prove that if a graded algebra has a doublesided ideal J, such that the quotient algebra is right noetherian, and J is free as a left Amodule, then A is right coherent [13, 14]. This criterion allows to prove coherence for some classes of algebras, see [9, 10, 15].
An interesting example of coherence, which is beyond the scope of methods of the current paper, as well as the above criterion for local algebras, is the one due to Piontkovski [13]. He showed coherence of the geometric \({\mathbb {Z}}\)algebra of the helix with the thread consisting of three elements \(({\mathscr {O}}(1), {\mathscr {O}}, {\mathscr {O}}(1))\) of an exceptional collection on \({\mathbb {P}}^3\) (see [2]). Note that any full exceptional collection on \({\mathbb {P}}^3\) has four elements. Coherence of the algebra implies existence of a tstructure in the subcategory generated by the three elements of the exceptional collection, similar to the standard geometric tstructure in the derived category of coherent sheaves on \({\mathbb {P}}^3\). This subcategory is interesting in relation to the problem of description of mathematical instantons on \({\mathbb {P}}^3\), because it contains instanton vector bundles of arbitrary topological charge. In [8], this subcategory is related to a noncommutative Grassmannian.
Note that the condition on the commutative ring \({\mathbb {K}}\) to be noetherian cannot be removed in the statement of Theorem 3.3. A counterexample due to Soublin is known, where A is a polynomial algebra in one variable over a coherent commutative ring \({\mathbb {K}}\) [16]. More precisely, if \({\mathbb {K}}\) is the direct product of a countable number of copies of Open image in new window , the ring of formal power series in two variables, then the ring Open image in new window is not coherent. Note that the ring \({\mathbb {K}}\) in this case is even uniformly coherent and has no nilpotents.
For graded rings, there is a weaker notion of graded coherence, when the category of graded modules is considered. Minamoto showed in [12] that, in contrast to the nongraded case, the algebra A[t] is graded coherent, if A is coherent and degree of t is 1.
Notes
Acknowledgments
We are grateful to Dmitry Piontkovski for useful references.
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