Flat Model Structures for Nonunital Algebras and Higher K-Theory

Let A be a nonunital algebra (or ring). A classical question in Homotopy Theory is to find a 'good definition' of K-theory and cyclic type homology for this type of rings and algebras. Namely, it is always possible to embed a nonunital ring A as a two-sided ideal of a unital ring R (for instance, by choosing à = Z ⋉ A to be the ring obtained by adjoining an identity to A). Thus, it is possible to define the notions of K-theory and cyclic type homologies for A in terms of this ring Ã. But this embedding of A into a unital ring R is not unique. And therefore, the different choices of R give rise to different definitions of homology theories and K-theory for A. This problem is known in the literature as the 'excision problem' in the different theories. In [35], Wodzicki proved that if R is a unitary k-pure extension of A, then it satisfies the excision property for Cyclic, Bar or Hochschild homology if and only if A is an H-unital k-algebra, in the sense that its Bar homology HB * (A, V ) = 0 for any k-module V (see e.g. [35,Theorem 3.1]). This result extended to algebras over commutative rings his remarkable result showing that if a (nonunital) ring A satisfies the excision property in rational algebraic K-theory, then the Q-algebra A ⊗ Q verifies the excision property in Cyclic homology [35] (see also [22,32]). But this H-unitality condition is rarely satisfied in practice and thus, many authors have extended these ideas and techniques to more general settings. For instance, Cuntz and Quillen have proven in [8,9,10] that arbitrary extensions of k-algebras (k ⊃ Q a field) with k-linear section satisfy excision in periodic cyclic cohomology. And Weibel in [34] constructed a homotopy invariant algebraic K-theory satisfying excision and cohomologic descent. The obstruction for the classical K-theory excision has also been studied in [5] (see also [6]).

In the present paper, we addopt the definition of h-unitary modules given by Suslin and Wodzicki (cf. [32, 7.3(ii)]). And we define the category M(A) of hunitary left A-modules as the full subcategory of all left A-modules consisting on those modules M satisfying that Tor à n (Z, M ) = 0 (or Tor à n (k, M ) = 0, if A is an algebra over a ground field k). At first sight, this definition of M(A) depends on the choosing embedding of A as a two-sided ideal of the unital ring R. And therefore we will refer to it as the h-unitary module category associated to the pair (R, A). But we will show in the last section of the paper that this h-unitary module category is independent of R.

We prove in Theorem 3.1 that any unitary left A-module M (in particular, the regular module A) has a (unique up to isomorphism) cover by an h-unitary module (equivalently, a minimal right approximation in the sense of Auslander [3]) by an h-unitary module. And then, we show that there is a very satisfactory version of relative homological algebra in the category of unbounded complexes of h-unitary modules. Namely, we prove in Section 3 that the class of all flat h-unitary Amodules imposes a cofibrantly generated Model Structure (see [27]) in the category C(M(A)) of all complexes in M(A). Let us note that, if we consider in C( Ã) the cofibrantly generated Flat Model Structure constructed in [17], then the bounded flat complexes which are generating cofibrations form a small Waldhausen category S in the sense of [33]. And the subset T ⊂ S of generating cofibrations which are flat h-unitary is a Waldhausen subcategory of it, since any cofibration in our model structure on C(M(A)) is a cofibration in C( Ã). Therefore we can use the general construction of K-theory on a Waldhausen category (see [33, Chapter IV]) to define a K-theory on A. This K-theory is going to be excisive in the sense of [35] by our results in the last section. In particular, we deduce that the group K 0 T is independent of the choice of S (see [33, Chapter II, Theorem 9.2.2]). The explicit construction of these K-groups in terms of our model structure will be developed in a forthcoming paper. We would like to stress that our construction does not restrict just to H-unital algebras, but it applies to any nonunital algebra (or ring) without the h-unitality condition.

Unfortunately, the category of h-unitary modules is far from being a monoidal category and therefore, we cannot expect to construct a monoidal model structure on C(M(A)). We solve this problem in Section 4, where we consider the wider subcategory F(A) of Ã-Mod consisting of all firm modules in the sense of [28]. The usual tensor product of A-modules is an endofunctor in F(A). And we show in Theorem 4.1 that this tensor product in F(A) induces a unitless monoidal structure in C(F(A)) which is compatible with the model structure we have induced in C(M(A)).

At this point, the question of when the additive category of h-unitary modules is abelian naturally arises. We show in Section 2 that this category is always an accessible category in the sense of [2,24]. Actual

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