Algebraic K-groups Research Articles

Resolving dualities and applications to homological invariants

Abstract Dualities of resolving subcategories of module categories over rings are introduced and characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of the dualities to smaller resolving subcategories, sufficient and necessary conditions for these bimodules to be tilting are provided. This leads to the Gorenstein version of both the Miyashita’s duality and Huisgen-Zimmermann’s correspondence. An application of resolving dualities is to show that higher algebraic K-groups and semi-derived Ringel–Hall algebras of finitely generated Gorenstein-projective modules over Artin algebras are preserved under tilting.

Maximal connected 𝑘-subgroups of maximal rank in connected reductive algebraic 𝑘-groups

Let k k be any field and let G G be a connected reductive algebraic k k -group. Associated to G G is an invariant first studied in the 1960s by Satake [Ann. of Math. (2) 71 (1960), 77–110] and Tits [Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier- Villars, Paris, 1962], [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] that is called the index of G G (a Dynkin diagram along with some additional combinatorial information). Tits [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] showed that the k k -isogeny class of G G is uniquely determined by its index and the k k -isogeny class of its anisotropic kernel G a G_a . For the cases where G G is absolutely simple, all possibilities for the index of G G have been classified in by Tits [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62]. Let H H be a connected reductive k k -subgroup of maximal rank in G G . We introduce an invariant of the G ( k ) G(k) -conjugacy class of H H in G G called the embedding of indices of H ⊂ G H \subset G . This consists of the index of H H and the index of G G along with an embedding map that satisfies certain compatibility conditions. We introduce an equivalence relation called index-conjugacy on the set of k k -subgroups of G G , and observe that the G ( k ) G(k) -conjugacy class of H H in G G is determined by its index-conjugacy class and the G ( k ) G(k) -conjugacy class of H a H_a in G G . We show that the index-conjugacy class of H H in G G is uniquely determined by its embedding of indices. For the cases where G G is absolutely simple of exceptional type and H H is maximal connected in G G , we classify all possibilities for the embedding of indices of H ⊂ G H \subset G . Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when k k has cohomological dimension 1 (resp. k = R k=\mathbb {R} , k k is p \mathfrak {p} -adic).

5-Graded simple Lie algebras, structurable algebras, and Kantor pairs

Relying on the classification of simple Lie algebras over algebraically closed fields of characteristic >3, we show that any finite-dimensional central simple 5-graded Lie algebra over a field k of characteristic ≠2,3 is a simple Lie algebra of Chevalley type, i.e. a central quotient of the Lie algebra of a simple algebraic k-group. As a consequence, we prove that all central simple structurable algebras and Kantor pairs over k arise from 5-gradings on simple Lie algebras of Chevalley type.

Homotopy Mackey functors of equivariant algebraic K-theory

Given a finite group G acting on a ring R, Merling constructed an equivariant algebraic K-theory G-spectrum, and work of Malkiewich and Merling, as well as work of Barwick, provides an interpretation of this construction as a spectral Mackey functor. This construction is powerful, but highly categorical; as a result the Mackey functors comprising the homotopy are not obvious from the construction and have therefore not yet been calculated. In this work, we provide an interpretation of the homotopy Mackey functors of equivariant algebraic K-theory in terms of a purely algebraic construction. In particular, we construct Mackey functors out of the nth algebraic K-groups of group rings whose multiplication is twisted by the group action. Restrictions and transfers for these functors admit a tractable algebraic description in that they arise from restriction and extension of scalars along homomorphisms of twisted group rings. In the case where the group action is trivial, our construction recovers work of Dress and Kuku from the 1980's which constructs Mackey functors out of the algebraic K-theory of group rings. We develop many families of examples of Mackey functors, both new and old, including K-theory of endomorphism rings, the K-theory of fixed subrings of Galois extensions, and (topological) Hochschild homology of twisted group rings.

Zero-cycles with modulus and relative K-theory

Let D be an effective Cartier divisor on a regular quasiprojective scheme X of dimension d≥1 over a field. For an integer n≥0, we construct a cycle class map from the higher Chow groups with modulus {CHn+d(X|mD,n)}m≥1 to the relative K-groups {Kn(X,mD)}m≥1 in the category of pro-abelian groups. We show that this induces a proisomorphism between the additive higher Chow groups of relative 0-cycles and the reduced algebraic K-groups of truncated polynomial rings over a regular semilocal ring which is essentially of finite type over a characteristic zero field.

An elementary description of K1(R) without elementary matrices

Let R be a ring with unit. Passing to the colimit with respect to the standard inclusions GL(n,R)→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 K1(R)=GL(R)/E(R) of R, giving an elementary description that does not involve elementary matrices explicitly.

Conjugacy of Levi subgroups of reductive groups and a generalization to linear algebraic groups

We investigate Levi subgroups of a connected reductive algebraic group G, over a ground field K. We parametrize their conjugacy classes in terms of sets of simple roots and we prove that two Levi K-subgroups of G are rationally conjugate if and only if they are geometrically conjugate.These results are generalized to arbitrary connected linear algebraic K-groups. In that setting the appropriate analogue of a Levi subgroup is derived from the notion of a pseudo-parabolic subgroup.

Algebraic [formula omitted]-theory of operator rings

We produce “small” Bott rings of operators on an infinite dimensional separable complex Hilbert space. We also establish comparison results for negative algebraic and topological K-groups of Banach algebras. In our arguments we utilize two concepts that we introduce: of an algebra graded by a semilattice, and of a “secant extension” of algebras.

On the topological computation of $$K_4$$ of the Gaussian and Eisenstein integers

In this paper we use topological tools to investigate the structure of the algebraic K-groups \(K_4(R)\) for \(R=Z[i]\) and \(R=Z[\rho ]\) where \(i := \sqrt{-1}\) and \(\rho := (1+\sqrt{-3})/2\). We exploit the close connection between homology groups of \(\mathrm {GL}_n(R)\) for \(n\le 5\) and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which \(\mathrm {GL}_n(R)\) acts. Our main result is that \(K_{4} ({\mathbb {Z}}[i])\) and \(K_{4} ({\mathbb {Z}}[\rho ])\) have no p-torsion for \(p\ge 5\).

Algebraic K-theory of Gorenstein projective modules

We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the ‘resolution theorem’ in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different algebraic K-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic K-groups along a recollement of the bounded Gorenstein derived categories of CM-nite Gorenstein algebras.

Special unipotent groups are split

We show that over any field k, a smooth unipotent algebraic k-group is special if and only if it is k-split.

Classification of quasi-homogeneous holomorphic curves and operators in the Cowen–Douglas class

In this paper we study quasi-homogeneous operators, which include the homogeneous operators, in the Cowen–Douglas class. We give two separate theorems describing canonical models (with respect to equivalence under unitary and invertible operators, respectively) for these operators using techniques from complex geometry. This considerably extends the similarity and unitary classification of homogeneous operators in the Cowen–Douglas class obtained recently by the last author and A. Korányi. In a significant generalization of the properties of the homogeneous operators, we show that quasi-homogeneous operators are irreducible and determine which of them are strongly irreducible. Applications include the equality of the topological and algebraic K-group of a quasi-homogeneous operator and an affirmative answer to a well-known question of Halmos.

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields II

ABSTRACTLet k be a separably closed field. Let G be a reductive algebraic k-group. We study Serre’s notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We show that a regular reductive k-subgroup of G is G-completely reducible over k. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.

The Bousfield localizations and colocalizations of the discrete model structure

We compute the Bousfield localizations and Bousfield colocalizations of discrete model categories, including the homotopy categories and the algebraic K-groups of these localizations and colocalizations. We prove necessary and sufficient conditions for a subcategory of a category to appear as the subcategory of fibrant objects for some such model structure. We also prove necessary and sufficient conditions for a monad to be the fibrant replacement monad of some such model structure.

On the vanishing of negative homotopy K-theory

We show that the homotopy invariant algebraic K-theory of Weibel vanishes below the negative of the Krull dimension of a noetherian scheme. This gives evidence for a conjecture of Weibel about vanishing of negative algebraic K-groups.

Stratified fiber bundles, Quinn homology and brane stability of hyperbolic orbifolds

We revisit the problem of stability of string vacua involving hyperbolic orbifolds using methods from homotopy theory and K-homology. We propose a definition of Type II string theory on such backgrounds that further carry stratified systems of fiber bundles, which generalize the more conventional orbifold and symmetric string backgrounds, together with a classification of wrapped branes by a suitable generalized homology theory. For spaces stratified fibered over hyperbolic orbifolds we use the algebraic K-theory of their fundamental groups and Quinn homology to derive criteria for brane stability in terms of an Atiyah–Hirzebruch type spectral sequence with its lift to K-homology. Stable D-branes in this setting carry stratified charges which induce new additive structures on the corresponding K-homology groups. We extend these considerations to backgrounds which support H-flux, where we use K-groups of twisted group algebras of the fundamental groups to analyze stability of locally symmetric spaces with K-amenable isometry groups, and derive stability conditions for branes wrapping the fibers of an Eilenberg–MacLane spectrum functor.

K-theory for Leavitt path algebras: Computation and classification

We show that the long exact sequence for K-groups of Leavitt path algebras deduced by Ara, Brustenga, and Cortiñas extends to Leavitt path algebras of countable graphs with infinite emitters in the obvious way. Using this long exact sequence, we compute explicit formulas for the higher algebraic K-groups of Leavitt path algebras over certain fields, including all finite fields and all algebraically closed fields. We also examine classification of Leavitt path algebras using K-theory. It is known that the K0-group and K1-group do not suffice to classify purely infinite simple unital Leavitt path algebras of infinite graphs up to Morita equivalence when the underlying field is the rational numbers. We prove for these Leavitt path algebras, if the underlying field is a number field (which includes the case when the field is the rational numbers), then the pair consisting of the K0-group and the K6-group does suffice to classify these Leavitt path algebras up to Morita equivalence.

Homogeneous spaces of Hilbert type

Let k be a global field. Let G be a connected linear algebraic k-group, assumed reductive when k is a function field. It follows from a result of a paper by Bary-Soroker, Fehm and Petersen that when H is a smooth connected k-subgroup of G, the quotient space G/H is of Hilbert type. We prove a similar result for certain non-connected k-subgroups H of G. In particular, we prove that if G is a simply connected k-group over a number field k, and H is an abelian k-subgroup of G, not necessarily connected, then G/H is of Hilbert type.

Algebraic proofs of some fundamental theorems in algebraic $K$-theory

We present new proofs of the additivity, resolution and cofinality theorems for the algebraic K-theory of exact categories. These proofs are entirely algebraic, based on Grayson's presentation of higher algebraic K-groups via binary complexes.

Generalized 1-motivic sheaves

Let k be a field of characteristic 0. We extend the definition of the category of 1-motivic sheaves, introduced by Barbieri-Viale and Kahn, allowing quotients of connected algebraic k-groups by formal k-groups. We show that its bounded derived category is equivalent to the bounded derived category of the category of generalized 1-motives with torsion.