An analogue of Serre’s conjecture for a ring of distributions

Abstract The set 𝒜 := 𝔺δ0+ 𝒟+′, obtained by attaching the identity δ0 to the set 𝒟+′ of all distributions on 𝕉 with support contained in (0, ∞), forms an algebra with the operations of addition, convolution, multiplication by complex scalars. It is shown that 𝒜 is a Hermite ring, that is, every finitely generated stably free 𝒜-module is free, or equivalently, every tall left-invertible matrix with entries from 𝒜 can be completed to a square matrix with entries from 𝒜, which is invertible.


Introduction
The aim of this note is to show that the the ring A is a Hermite ring. The relevant de nitions are recalled below. For preliminaries on the distribution theory of L. Schwartz, we refer the reader to [1] and [2]. For the commutative algebraic terminology used below, we refer to [3] and [4].
Let D + denote the set of all distributions T ∈ D (R) having their distributional support contained in the half line ( , ∞). Then D + is an algebra with pointwise addition, pointwise multiplication by scalars, and with convolution taken as multiplication in the algebra. However, D + lacks the identity element with respect to multiplication. We can adjoin the identity element to the algebra D + , hence obtaining the larger algebra A := Cδ + D + , whose elements are of the form αδ + T, where α ∈ C and T ∈ D + . A is also an algebra with the same operations. We will denote the convolution operation henceforth by *.
Serre's question from 1955 was if, for the ring R = F[x , · · · , x d ] (F a eld), every nitely generated projective R-module is free. This was eventually settled in 1976, independently, by Quillen and by Suslin, and the considerations over this question gave rise to the subject of algebraic K-theory. In light of the Hilbert-Serre theorem, Serre's question for R = F[x , · · · , x d ] can be reduced to the question of whether every nitely generated stably free R-module is free. A commutative unital ring R having the property that every nitely generated stably free R-module is free is called a Hermite ring. In terms of matrices over the ring R, one has the following characterisation of Hermite rings, see for example [3,  Let R be a commutative unital ring with multiplicative identity denoted by . For m, n ∈ N = { , , , · · · }, we denote by R m×n the matrices with m rows and n columns having entries from R. The identity element in R k×k having s on the diagonal and zeroes elsewhere will be denoted by I k . A tall matrix f ∈ R K×k is said to be left-invertible if there exists a g ∈ R k×K such that gf = I k . The ring R is Hermite if and only if for all k and K ∈ N such that k < K, and for all f ∈ R K×k such that there exists a g ∈ R k×K so that gf = I k , there exists an fc ∈ R K×(K−k) and there exists a G ∈ R K×K such that G f fc = I K .
In other words, the ring R is Hermite if every left invertible matrix over R can be extended to an invertible one.
The following example is well-known, see e.g. Proof. Let A be the ring Cδ + D + . Let f ∈ A K×k be left invertible, with gf = I k δ for some g ∈ A k×K . Write where f+ ∈ (D + ) K×k , f ∈ C K×k , and g+ ∈ (D + ) k×K , g ∈ C k×K . From gf = I k δ , we obtain that g f δ + (g f+ + g+f + g+f+) = δ I k .
As the entries of f+, g+ belong to D + , it follows that there exists an ϵ > such that each of the entries of g f+ + g+f + g+f+ has its support in (ϵ, ∞). So if we act both sides (entry-wise) on a test function φ ∈ D(R) such that supp(φ) ⊂ (−∞, ϵ), then we obtain Choosing φ( ) ≠ , this now shows that g f = I k . But as C is Hermite, we can now nd a fc ∈ C K×(K−k) and a G ∈ C K×K such that As f+ ∈ (D + ) K×k , it follows that T ∈ (D + ) K×K . Suppose that ϵ > is such that each entry of T has its support contained in (ϵ , ∞). We claim that I K δ − T is invertible in (A) K×K . De ne the "geometric series" We will now show that S is well-de ned. We recall the theorem on supports for convolution of distributions [1,Theorem 8,p.120], namely that supp(T * T ) ⊂ supp(T ) + supp(T ) for any two distributions T , T ∈ D (R) whose supports satisfy the convolution condition. It follows that in our case, each entry T n has its support contained in n · supp(T) = n[ϵ , ∞) = [nϵ , ∞). So it follows that the series for S converges. Indeed, given any test function φ ∈ D(R), the series (with the action T n , φ understood to be entry-wise) ∞ n= T n , φ contains only nitely many nonzero terms. Now if Sn denotes the nth partial sum of the series I K δ + T + T + T + · · · , we have The second equality above follows from the continuity of convolution in D + ; see [1,Theorem 7 p.120]. Now, setting we have This completes the proof.

Remarks . A conjecture
Another natural convolution algebra is the algebra E (R) of all compactly supported distributions, again the usual pointwise addition and convolution taken as multiplication. We have the following: ), the answer to this question in the a rmative is equivalent to the following result (the matricial version given below is attributed to [10], and is a consequence of [8]).