arXiv subjects
Publications and source records attributed to Thomas Huettemann.
We develop a finiteness notion for unbounded chain complexes over a commutative noetherian integral domain $R$ employing the Abel summation method. The algebraic K-theory of such complexes is defined, and shown to be non-trivial. We also exhibit a natural map from the (usual) algebraic K-theory of $R$ into the new K-theory and show that its image contains a canonical infinite cyclic subgroup.
We propose a definition of an Euler characteristic for unbounded chain complexes by taking the (usual) Euler characteristics of successively longer parts of the complex, weighted inversely proportional to the length, and passing to the limit. This amounts to taking the limit of the sequence of ranks of homology modules with alternating signs in the sense of the Hölder summation method. We establish the structure of a category with cofibrations and weak equivalences on unbounded complexes for which the infinite Euler characteristic is defined, and show that its Grothendieck group is unusually large (viz., uncountable).
Let $R$ be a strongly $\mathbb{Z}^2$-graded ring, and let $C$ be a bounded chain complex of finitely generated free $R$-modules. The complex $C$ is $R_{(0,0)}$-finitely dominated, or of type FP over $R_{(0,0)}$, if it is chain homotopy equivalent to a bounded complex of finitely generated projective $R_{(0,0)}$-modules. We show that this happens if and only if $C$ becomes acyclic after taking tensor product with a certain eight rings of formal power series, the graded analogues of classical Novikov rings. This extends results of Ranicki, Quinn and the first author on Laurent polynomial rings in one and two indeterminates.
A Laurent polynomial ring $A[t,1/t]$ with coefficients in a unital ring $A$ determines a category of quasi-coherent sheaves on the projective line over $A$; its $K$-theory is known to split into a direct sum of two copies of the $K$-theory of $A$. In this paper, the result is generalised to the case of an arbitrary strongly $\mathbb{Z}$-graded ring $R$ in place of the Laurent polynomial ring. The projective line associated with $R$ is indirectly defined by specifying the corresponding category of quasi-coherent sheaves. Notions from algebraic geometry like sheaf cohomology and twisting sheaves are transferred to the new setting, and the $K$-theoretical splitting is established.
The "fundamental theorem" for algebraic $K$-theory expresses the $K$-groups of a Laurent polynomial ring $L[t,t^{-1}]$ as a direct sum of two copies of the $K$-groups of $L$ (with a degree shift in one copy), and certain "nil" groups of $L$. It is shown here that a modified version of this result generalises to strongly $\mathbb{Z}$-graded rings; rather than the algebraic $K$-groups of $L$, the splitting involves groups related to the shift actions on the category of $L$-modules coming from the graded structure. (These action are trivial in the classical case). The nil groups are identified with the reduced $K$-theory of homotopy nilpotent twisted endomorphisms, and analogues of Mayer-Vietoris and localisation sequences are established.
Let $R$ be a ring with unit. Passing to the colimit with respect to the standard inclusions $GL(n,R) \to GL(n+1,R)$ (which add a unit vector as new last row and column) yields, by definition, the stable linear group $GL(R)$; the same result is obtained, up to isomorphism, when using the "opposite" inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic $K$-group $K_1(R) = GL(R)/E(R)$ of~$R$, giving an elementary description that does not involve elementary matrices explicitly.
Let $K$ be a non-empty set of ideals of the commutative ring $R$, closed under taking smaller ideals. A subset $X$ of the group ring $R[\mathbb{Z}^s]$ is called a $K$-set if the ideal generated by the coefficients of the elements of $X$ is in $K$. For $X$ not a $K$-set we investigate the set of those homomorphisms $p \colon \mathbb{Z}^s \to \mathbb{Z}^t$ such that $p_*(X)$ is a $K$-set. We also consider corresponding notions of rank of matrices and Betti numbers of chain complexes; this includes an analysis of the case of McCoy rank. Our setup also recovers results on jump loci obtained by Kohno and Pajitnov as a special case.
Let R be a strongly Z-graded ring with degree-0 subring S, and let C be a chain complex of modules over the subring P of elements of non-negative degree. We show that there are non-commutative localisations of P which detect whether the complex C is S-finitely dominated or S-contractible, respectively, and that these localisations are universal among P-rings making S-finitely dominated and S-contractible complexes contractible. This generalises known results for polynomial rings to a much wider class of rings. We show by example that in general C need not be P-homotopy finite even if C is S-finitely dominated; this differs from the case of polynomial rings.
It has been shown by McCoy that a right ideal of a polynomial ring with several indeterminates has a non-trivial homogeneous right annihilator of degree 0 provided its right annihilator is non-trivial to begin with. In this note, it is documented that any $\mathbb{N}$-graded ring $R$ has a slightly weaker property: the right annihilator of a right ideal contains a homogeneous non-zero element, if it is non-trivial to begin with. If $R$ is a subring of a $\mathbb{Z}^k$ -graded ring $S$ satisfying a certain non-annihilation property (which is the case if $S$ is strongly graded, for example), then it is possible to find annihilators of degree 0.
It is well-known that the Jacobson radical of a unital ring is its largest superfluous right ideal. It is recorded here that the result carries over to non-unital rings provided the notion of "superfluous" is taken relative to all regular ideals.
Let L be a strongly Z-graded ring, and let C be a bounded chain complex of finitely generated L-modules. We give a homological characterisation of when C is homotopy equivalent, over L_0, to a bounded complex of finitely generated projective L_0-modules, generalising known results for twisted Laurent polynomial rings.
We investigate modules over "systematic" rings. Such rings are "almost graded" and have appeared under various names in the literature; they are special cases of the G-systems of Grzeszczuk. We analyse their K-theory in the presence of conditions on the support, and explain how this generalises and unifies calculations of graded and filtered K-theory scattered in the literature. Our treatment makes systematic use of the formalism of idempotent completion and a theory of triangular objects in additive categories, leading to elementary and transparent proofs throughout.
We present a homological characterisation of those chain complexes of modules over a Laurent polynomial ring in several indeterminates which are finitely dominated over the ground ring (that is, are a retract up to homotopy of a bounded complex of finitely generated free modules). The main tools, which we develop in the paper, are a non-standard totalisation construction for multi-complexes based on truncated products, and a high-dimensional mapping torus construction employing a theory of cubical diagrams that commute up to specified coherent homotopies.
Finitely dominated chain complexes over a Laurent polynomial ring in one indeterminate are characterised by vanishing of their Novikov homology. We present an algebro-geometric approach to this result, based on extension of chain complexes to sheaves on the projective line. We also discuss the K-theoretical obstruction to extension.
Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,1/x,y,1/y]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterises R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are obtained by tensoring C over L with R[[x,y]][1/xy] and R[x,1/x][[y]][1/y], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.
We consider a non-standard totalisation functor to produce a cochain complex from a given double complex: instead of sums or products, totalisation is defined via truncated products of modules. We give an elementary proof of the fact that a double complex with exact rows (resp, columns) yields an acyclic cochain complex under totalisation using right (resp, left) truncated products. As an application we consider the algebraic mapping torus T(h) of a self map h of a cochain complex C. We show that if C consists of finitely presented modules then T(h) has trivial negative Novikov cohomology; if in addition h is a quasi-isomorphism, then T(h) has trivial positive Novikov cohomology as well. As a consequence we obtain a new proof that a finitely dominated cochain complex over a Laurent polynomial ring has trivial Novikov cohomology.
Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,1/x]. Then C is R-finitely dominated, ie, homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules, if and only if the two chain complexes C((x)) and C((1/x)) are acyclic, as has been proved by Ranicki. Here C((x)) is the tensor product over L of C with the Novikov ring R((x)) = R[[x]][1/x] (also known as the ring of formal Laurent series in x); similarly, C((1/x)) is the tensor product over L of C with the Novikov ring R((1/x)) = R[[1/x]][x]. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.
Suppose X is a projective toric scheme defined over a commutative ring R equipped with an ample line bundle L. We prove that its K-theory has k+1 direct summands K(R) where k is minimal among non-negative integers such that the twisted line bundle L(-k-1) is not acyclic. In fact, using a combinatorial description of quasi-coherent sheaves throughout we prove the result for a ring R which is either commutative, or else left noetherian.