On the Extension Problem and the Nil Groups of Rings of Finite Global Dimension

A vanishing result is obtained in respect of nil groups of rings of finite global dimension. Also a connection is established with the extension problem. Mathematics Subject Classification: 55N15, 55U05


Introduction
Let C be an admissible subcategory of an abelian category.We are interested in the category Nil(C) whose objects are the pairs (M, ν), where M ∈ C and ν ∈ End C (M) is nilpotent.The nil groups have geometrical significance as they occur as obstructions to geometrical problems ( [4], [7]).The Nil group vanishes for any abelian category (Proposition 6.1 on page 653 of [2]), and it is an interesting problem to determine under what conditions will the Nil group vanish for a nonabelian category [11].Some known vanishing results in respect of nil groups for rings are those for the group ring Z[G] where G is a finite group of square -free order [5], regular rings ( [7], [11]), quasi-regular rings [7], the cyclic group C n of finite order n ≥ 2 and the finite group of finite type G ∼ = F Z where F is a finite subgroup of G [8].This paper gives a solution of the above stated problem in respect of the nonabelian category of rings.We obtain that Let M , M be R -modules.The question is asked as regards the Rmodules M such that M is a submodule of M and M be its quotient.Equivalently, this question can be posed as follows: which are the R -modules M such that the sequence be exact?The classification of such R -modules M constitute what is known as the extension problem [10].In this paper, we establish a result that relates the extension problem to rings of finite global dimension.Thus giving us a supply of rings of finite global dimension.

Extensions, Global Dimensions and Nil Groups Definition 2.1 An extension of an
where each P i is projective (i.e. a sum of free R -modules).
(ii) The global dimension of R denoted by gl.dimR is defined as The finitistic global dimension of R denoted by f.gl.dimR is defined as (iv) Let G be a group, R[G] its corresponding group ring and : R[t] −→ R the augmentation map.The i -th nil group of R is defined as (vi) Let τ be the category of triples R = (R; B 0 , B 1 ), where B i , i = 0, 1 are two bimodules.A morphism in τ is a triple where φ : R −→ S is a ring homomorphism and Waldhausen nil groups are functors from the category τ to abelian groups.For an object R in τ , we first define an exact category Nil(R) with objects quadruples (P, Q, ; p, q), where P and Q are finitely generated projective right R -modules and is a pair of R -module homomorphisms such that the compositions are zero after finitely many steps.Morphisms are homomorphisms on the modules that are compatible with the maps.There is a forgetful functor φ : Nil(R) −→ P R ×P S , where P R is finitely generated projective right R -modules.Then the Waldhausen Nil -groups ( [7], [9] and [11])

Remark 2.3 There is a natural isomorphism between the NK -groups and the Waldhausen Nil -groups
Thus the vanishing results of Waldhausen Nil -groups can be applied to the NK -groups.( [3]) The group of equivalence classes of extensions of M by M under the Baer sum is denoted by Ex(M , M ) and it is isomorphic to Ext 1 (M , M ) [10].

Nil Groups of Rings of Finite Global Dimension
It is known that the relationship between the extension bifunctor and the Baer sum is illustrated in the solution to the extension problem ( [10]).Then by results from ( [12]), we relate rings of finite global dimension to the extension problem.Thus giving a condition for the supply of rings of finite global dimension.
where each P i is projective.Let p be a prime ideal of R and S a multiplicatively closed subset of R not containing 0 given by S = R − p.The ring of fractions R p = S −1 R can give an indication on whether R is regular ( [1]).Since R is of finite global dimension, it follows that R p also has finite global dimension and is regular.Thus R is regular.It is known that every R -module has a projective resolution and since R is of finite global dimension, it means that every R -module is of homological dimension ≤ n(n ∈ N).Now the category of finitely generated projective modules is a full subcategory of the category of R -modules.Therefore K i (Nil(R)) −→ K i (Nil(Mod R )) is an isomorphism.Using Corollary 6.3 on p. 654 of [2] and the fact that the groups NK i are isomorphic to the Waldhausen's groups Nil W i−1 for regular rings (see Proof of Proposition 3.8 in [3]), it follows that NK i (R) = 0 ∀ i. Theorem 3.2 Let A be a ring such that the extensions of any simple module S by S over A splits and consists of exactly only one element.Then A is of finite global dimension.

Theorem 3 . 1
Let R be a ring of finite global dimension.Then NK i (R) = 0 ∀ i. Proof: Let R be a ring of finite global dimension.Then hd R E ≤ n < ∞ for any R -module E and exact sequence 0 −→ P n ∂n the semidirect product.The nil groups of Z[V ] which takes into account the automorphism α, are called twisted nil groups and are denoted by NK α i (Z[G]) i.e.