Adams completion and symmetric algebra

Deleanu, Frei and Hilton have developed the notion of generalized Adams completion in a categorical context. In this paper, the symmetric algebra of a given algebra is shown to be the Adams completion of the algebra by considering a suitable set of morphisms in a suitable category. 
 
 Key words: Category of fraction, calculus of left fraction, symmetric algebra, tensor algebra, Adams completion.


INTRODUCTION
The notion of generalized completion (Adams completion) arose from a categorical completion process suggested by Adams (1973Adams ( , 1975)).Originally this was considered for admissible categories and generalized homology (or cohomology) theories.Subsequently, this notion has been considered in a more general framework by Deleanu et al. (1974) where an arbitrary category and an arbitrary set of morphisms of the category are considered; moreover they have also suggested the dual notion, namely the completion (Adams completion) of an object in a category.
The notion of Let be an arbitrary category and a set of morphisms of .Let denote the category of fractions of with respect and be the canonical function.Let denote the category of sets and functions.Then for a given object of , defines a covariant function.If this function is representable by an object of , that is, .then is called the (generalized) Adams completion of with respect to the set of morphisms or simply the -cocompletion of We shall often refer to as the completion of (Deleanu et al., 1974).Given a set of morphisms of , the saturation of is defined as the set of all morphisms in such that is an isomorphism in . is said to be saturated if (Deleanu et al., 1974).
Theorem 1 Behera and Nanda (1987) Let be a complete smallcategory ( is a fixed Grothendeick universe) and a set of morphisms of that admits a calculus of left *Corresponding author.E-mail: mitaray8@gmail.com.

Symmetric algebra
Let be a commutative ring.Let be a -module and be the tensor algebra of over . is a graded -algebra with the graded piece of degree being the additive subgroup , which we denote by .The map defined by is a morphism of -modules, which gives an isomorphism of -modules of with its image (Murfet, 2006).Let denote the -th symmetric algebra (Grinberg, 2013).The map is a subjective -module homomorphism.We prove the following for our need.

The category
Let be a fixed Grothendieck universe (Schubert, 1972).Let denote the category of all free modules and free module homomorphisms where is a commutative ring with unit 1.We assume that the underlying sets of the elements of are elements of .Let denote the set of all free -module homomorphisms such that is subjective.

Proposition
Let be a subset of ; where the index set is an element of , then is an element of .

Proof 3
The proof is trivial.
We will show that the set of free -module homorphisms of the category of free -modules and free -modules homomorphisms admits a calculus of left fraction.Hence condition (i) of Theorem 2 (Deleanu et al., 1974) holds.In order to prove condition (ii) of Theorem 2 (Deleanu et al., 1974) Subject Classification: 15A72, 55P60.Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License fractions.Suppose that the following compatibility condition with co-product is satisfied.If each , of left fractions.Then an object of is thecompletion of the object with respect to if and only if there exists a morphism in which is couniversal with respect to morphisms of : words, the following diagram is commutative (Deleanu et al., 1974): Theorem 3 Let be a set of morphisms in a category admitting a calculus of left fractions.Let be the canonical morphism as defined in Theorem 2 Letbe a commutative ring with unit 1.Let and be free -modules and let be a free -module subjective homomorphism.Then has the following property: given a free - Propositionadmits a calculus of left fractions.