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Let R be a strong n -coherent ring such that each finitely n -presented R -module has finite projective dimension. We consider ℱ n (R) the full subcategory of R -Mod of finitely n -presented modules. We prove that ℱ n (R) is an exact category, K i (R) = K i (ℱ n (R)) for every i ≥ 0 and we obtain an expression of Nil i (R). [ABSTRACT FROM AUTHOR]

Let ℚ be the rational filed. For a square-free integer d with d ≠ 0 , 1 , we denote by K = ℚ (d) the quadratic field. Let K be the ring of algebraic integers of K. In this paper, we completely determine the unit group of the quotient ring K / 〈 ξ n 〉 of K for an arbitrary prime ξ in ℚ (d) , where ℚ (d) has the unique factorization property, and n > 1 is a rational integer. [ABSTRACT FROM AUTHOR]

We show that each of the three K-theory multifunctors from small permutative categories to G ∗ -categories, G ∗ -simplicial sets, and connective spectra, is an equivalence of homotopy theories. For each of these K-theory multifunctors, we describe an explicit homotopy inverse functor. As a separate application of our general results about pointed diagram categories, we observe that the right-induced homotopy theory of Bohmann–Osorno E ∗ -categories is equivalent to the homotopy theory of pointed simplicial categories. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that homotopy pro-nilpotent structured ring spectra are TQ -local, where structured ring spectra are described as algebras over a spectral operad O . Here, TQ is short for topological Quillen homology, which is weakly equivalent to O -algebra stabilization. An O -algebra is called homotopy pro-nilpotent if it is equivalent to a limit of nilpotent O -algebras. Our result provides new positive evidence to a conjecture by Francis–Gaisgory on Koszul duality for general operads. As an application, we simultaneously extend the previously known 0-connected and nilpotent TQ -Whitehead theorems to a homotopy pro-nilpotent TQ -Whitehead theorem. [ABSTRACT FROM AUTHOR]

We show that the Gersten complex for the (improved) Milnor K-sheaf on a smooth scheme over an excellent discrete valuation ring is exact except at the first place and that exactness at the first place may be checked at the discrete valuation ring associated to the generic point of the special fibre. This complements results of Gillet-Levine for K -theory, Geisser for motivic cohomology and Schmidt-Strunk and the author for étale cohomology. [ABSTRACT FROM AUTHOR]

We discuss a categorical approach to the theory of discriminants in the combinatorial language introduced by Gelfand, Kapranov and Zelevinsky. Our point of view is inspired by homological mirror symmetry and provides K -theoretic evidence for a conjecture presented by Paul Aspinwall in a conference talk in Banff in March 2016 and later in a joint paper with Plesser and Wang. [ABSTRACT FROM AUTHOR]

We introduce a notion of Milnor square of stable ∞ -categories and prove a criterion under which algebraic K-theory sends such a square to a cartesian square of spectra. We apply this to prove Milnor excision and proper excision theorems in the K-theory of algebraic stacks with affine diagonal and nice stabilizers. This yields a generalization of Weibel's conjecture on the vanishing of negative K-groups for this class of stacks. [ABSTRACT FROM AUTHOR]

We study a symplectic variant of algebraic K-theory of the integers, which comes equipped with a canonical action of the absolute Galois group of Q . We compute this action explicitly. The representations we see are extensions of Tate twists Z p (2 k - 1) by a trivial representation, and we characterize them by a universal property among such extensions. The key tool in the proof is the theory of complex multiplication for abelian varieties. [ABSTRACT FROM AUTHOR]

We prove that if - is a group of polynomial growth, then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cocycle, we thus define a higher analogue of Lott's delocalized eta invariant and prove its convergence for invertible differential operators.We also use a determinant map construction of Xie and Yu to prove that if ' is of polynomial growth, then there is a well-defined pairing between delocalized cyclic cocycles and K-theory classes of C*-algebraic secondary higher invariants. When this K-theory class is that of a higher rho invariant of an invertible differential operator, we show this pairing is precisely the aforementioned higher analogue of Lott's delocalized eta invariant. As an application of this equivalence, we provide a delocalized higher Atiyah--Patodi--Singer index theorem, given that M is a compact spin manifold with boundary, equipped with a positive scalar metric g and having fundamental group which is finitely generated and of polynomial growth. [ABSTRACT FROM AUTHOR]

Let G be a linear Lie group acting properly on a G-spinc manifold M with compact quotient. We give a short proof that Poincaré duality holds between G-equivariant K-theory of M, defined using finite-dimensional G-vector bundles, and G-equivariant K-homology of M, defined through the geometric model of Baum and Douglas. [ABSTRACT FROM AUTHOR]

The group |$G = \textrm{GL}_r(k) \times (k^\times)^n$| acts on |$\textbf{A}^{r \times n}$|⁠ , the space of |$r$| -by- |$n$| matrices: |$\textrm{GL}_r(k)$| acts by row operations and |$(k^\times)^n$| scales columns. A matrix orbit closure is the Zariski closure of a point orbit for this action. We prove that the class of such an orbit closure in |$G$| -equivariant |$K$| -theory of |$\textbf{A}^{r \times n}$| is determined by the matroid of a generic point. We present two formulas for this class. The key to the proof is to show that matrix orbit closures have rational singularities. [ABSTRACT FROM AUTHOR]

We consider Bott-Samelson varieties BS c (s) for a semisimple compact Lie group C corresponding to sequences of (not necessarily simple) reflections s. Let n be the length of s , K be a maximal torus in C and W be the Weyl group of C. For any set R of not overlapping integer pairs (i , j) such that 1 ⩽ i ⩽ j ⩽ n and a function v : R → W , we consider the subspace BS c (s , v) ⊂ BS c (s) of solutions of the equations in C / K requiring that the K -orbit of the product of coordinates counted from i to j be equal to the K -orbit of v evaluated at (i , j) ∈ R. We decompose BS c (s , v) into a twisted product (in the sense of iterated fibre bundles) of smaller Bott-Samelson varieties BS c (t) and the fibres of the canonical projections from BS c (t) to the flag variety. Finally, we prove the tensor product decomposition for the K -equivariant cohomology of BS c (s , v). [ABSTRACT FROM AUTHOR]

We introduce a global equivariant refinement of algebraic K‐theory; here 'global equivariant' refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global Ω$\Omega$‐spectrum that keeps track of genuine G$G$‐equivariant infinite loop spaces, for all finite groups G$G$. The resulting global algebraic K‐theory spectrum is a rigid way of packaging the representation K‐theory, or 'Swan K‐theory' into one highly structured object. [ABSTRACT FROM AUTHOR]

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in Pic (X) ⊗ C . We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples. [ABSTRACT FROM AUTHOR]

Let G be a semisimple Lie group with finite center, K ⊂ G a maximal compact subgroup, and P ⊂ G a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use G-invariant differential forms on G / K × G / P to construct G-equivariant Poisson transforms mapping differential forms on G/P to differential forms on G/K. Such invariant forms can be constructed using finite-dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on G/P to the associated Bernstein–Gelfand–Gelfand (or BGG) complex in a well defined sense. The main part of this paper is devoted to an explicit construction of such transforms with additional favorable properties in the case that G = S U (n + 1 , 1). Thus, G/P is S 2 n + 1 with its natural CR structure and the relevant BGG complex is the Rumin complex, while G/K is complex hyperbolic space of complex dimension n + 1. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail. [ABSTRACT FROM AUTHOR]

In numerous generalizations of the original theory of fuzzy sets proposed by Zadeh, the considered membership degrees are often taken from lattices isomorphic to the lattice L I of closed subintervals of the unit interval [ 0 , 1 ]. This is, for example, the case of intuitionistic values, Pythagorean values or q -rung orthopair values. The mentioned isomorphisms allow to transfer the results obtained for the lattice L I directly to the other mentioned lattices. In particular, basic connectives in Zadeh's fuzzy set theory, i.e., special functions on the lattice [ 0 , 1 ] , can be extended to the interval-valued connectives, i.e., to special functions on the lattice L I , and then to the connectives on the lattices L ⁎ of intuitionistic values, P of Pythagorean values, and also on the lattice L τ q of q -rung orthopair values. We give several examples of such connectives, in particular, of those which are related to strict t-norms. For all these connectives we show their link to an additive generator f of the considered strict t-norm T. Based on our approach, many results discussed in numerous papers can be treated in a unique framework, and the same is valid for possible newly proposed types of connectives based on strict t-norms. Due to this approach, a lot of tedious proofs of the properties of the proposed connectives could be avoided, which gives researchers more space for presented applications. [ABSTRACT FROM AUTHOR]

We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds |$G/P$|⁠. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way, we advertise many properties of the left and right divided difference operators in cohomology and K-theory and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K-theory, generating Schubert classes and satisfying a Leibniz rule compatible with the quantum product. [ABSTRACT FROM AUTHOR]

We use the analogy between class groups and even K-groups of the ring of integers of a number field and "Higher Stickelberger" ideals within K-theory to prove an index formula for these ideals in a finite abelian extension of real number fields, which is similar to the classic Stickelberger ideal index formula proved by Iwasawa. [ABSTRACT FROM AUTHOR]

Using a result of Vdovina, we may associate to each complete connected bipartite graph $\kappa $ a two-dimensional square complex, which we call a tile complex, whose link at each vertex is $\kappa $. We regard the tile complex in two different ways, each having a different structure as a $2$ -rank graph. To each $2$ -rank graph is associated a universal $C^{\star }$ -algebra, for which we compute the K-theory, thus providing a new infinite collection of $2$ -rank graph algebras with explicit K-groups. We determine the homology of the tile complexes and give generalisations of the procedures to complexes and systems consisting of polygons with a higher number of sides. [ABSTRACT FROM AUTHOR]

Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S. When s = 1, the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of the tautological bundles for all ranks s over all K-trivial surfaces. Furthermore, in rank s = 2, the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler characteristics over the Hilbert schemes of points. [ABSTRACT FROM AUTHOR]

Summary: The cooling/heating performance improvement of supercritical carbon dioxide (sCO2) in heat exchange tube with changing cross‐section is significant and crucial for solar‐driven advanced thermal systems' application and development. The study introduced and constructed straight, diverging, and converging horizontal tubes with the changing cross‐section. The analyzation and evaluation of the sCO2 cooling performance used the combination of SST k‐ω turbulence model‐based numerical method and the field synergy theory, including the influence of tube cross‐sectional shape, inlet pressure and temperature on the cooling performance. As these simulation results indicate, the converging tube can enhance the flow field's synergy and increase the heat transfer ability by 13.15% under cooling conditions. Nevertheless, the heat transfer ability of the sCO2 decreases in the diverging tube under cooling conditions. Besides, the total heat transfer rate rises and the surface heat transfer coefficient has the opposite trends when the inlet temperature increases. Meanwhile, when inlet pressure varies from 11 to 12 MPa, the total heat exchange and surface convective heat transfer coefficient are the maximum. In conclusion, a converging horizontal tube and a suitable inlet pressure can effectively improve the cooling performance of the sCO2. The converging cross‐section design will provide an alternative for its application in advanced thermal systems. [ABSTRACT FROM AUTHOR]

Green-Griffiths observed that we could eliminate obstructions to deforming divisors. Motivated by recent work of Bloch-Esnault-Kerz on deformation of algebraic cycle classes, we use Chern character to generalize Green-Griffiths' observation and to show how to eliminate obstructions to deforming cycles of codimension p. [ABSTRACT FROM AUTHOR]

We discuss various aspects of the noncommutative geometry of a smooth subalgebra of the Toeplitz algebra. In particular, we study the structure of derivations on this subalgebra. [ABSTRACT FROM AUTHOR]

Various constructions for quantum groups have been generalized to ι-quantum groups. Such a generalization is called an ι-program. In this paper, we fill one of the parts in the ι-program. Namely, we provide an equivariant K-theory approach to ι-quantum groups, which is the Langlands dual picture of that constructed in Bao et al. (Transform. Groups 23 (2018), 329-389), where a geometric realization of ι-quantum groups is provided by using perverse sheaves. As an application of the main results, we prove Li's conjecture (Li, Represent. Theory 23 (2019), 1-56) for special cases. [ABSTRACT FROM AUTHOR]

We build on previous work on multirings ([17]) that provides generalizations of the available abstract quadratic forms theories (special groups and real semigroups) to the context of multirings ([10], [14]). Here we raise one step in this generalization, introducing the concept of pre-special hyperfields and expand a fundamental tool in quadratic forms theory to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor's K-theory ([11]) and Special Groups K-theory, developed by Dickmann-Miraglia ([5]). We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in [2] in order to provide a solution of Marshall's Signature Conjecture. [ABSTRACT FROM AUTHOR]

We prove that the graph C*-algebra C*(E) of a trimmable graph E is U(1)-equivariantly isomorphic to a pullback C*-algebra of a subgraph C*-algebra C*(E") and the C*-algebra of functions on a circle tensored with another subgraph C*-algebra C* (E'). This allows us to unravel the structure and K-theory of the fixed-point subalgebra C* (E)U(1) through the (typically simpler) C*-algebras C* (E'), C* (E") and C* (E")U(1). As examples of trimmable graphs, we consider one-loop extensions of the standard graphs encoding respectively the Cuntz algebra O2 and the Toeplitz algebra T. Then we analyze equivariant pullback structures of trimmable graphs yielding the C*-algebras of the Vaksman-Soibelman quantum sphere Sq2n+1 and the quantum lens space Lqµq(l;1,l), respectively. [ABSTRACT FROM AUTHOR]

Let q be a non-negative integer. We prove that a perfect field K has cohomological dimension at most q + 1 if, and only if, for any finite extension L of K and for any homogeneous space Z under a smooth linear connected algebraic group over L, the q-th Milnor K-theory group of L is spanned by the images of the norms coming from finite extensions of L over which Z has a rational point. We also prove a variant of this result for imperfect fields. [ABSTRACT FROM AUTHOR]

In this work we study the E_{\infty }-ring \operatorname {THH}(\mathbb {F}_p) as a graded spectrum. Following an identification at the level of E_2-algebras with \mathbb {F}_p[\Omega S^3], the group ring of the E_1-group \Omega S^3 over \mathbb {F}_p, we show that the grading on \operatorname {THH}(\mathbb {F}_p) arises from decomposition on the cyclic bar construction of the pointed monoid \Omega S^3. This allows us to use trace methods to compute the algebraic K-theory of \operatorname {THH}(\mathbb {F}_p). We also show that as an E_2 H\mathbb {F}_p-ring, \operatorname {THH}(\mathbb {F}_p) is uniquely determined by its homotopy groups. These results hold in fact for \operatorname {THH}(k), where k is any perfect field of characteristic p. Along the way we expand on some of the methods used by Hesselholt-Madsen and later by Speirs to develop certain tools to study the THH of graded ring spectra and the algebraic K-theory of formal DGAs. [ABSTRACT FROM AUTHOR]

We develop a local index theory for a class of operators associated with non-proper and non-isometric actions of Lie groupoids on smooth submersions. Such actions imply the existence of a short exact sequence of algebras, relating these operators to their non-commutative symbol. We then compute the connecting map induced by this extension on periodic cyclic cohomology. When cyclic cohomology is localized at appropriate isotropic submanifolds of the groupoid in question, we find that the connecting map is expressed in terms of an explicit Wodzicki-type residue formula, which involves the jets of non-commutative symbols at the fixed-point set of the action. [ABSTRACT FROM AUTHOR]

In this paper, the authors build on their previous work to show that periodic rational G$G$‐equivariant topological K$K$‐theory has a unique genuine‐commutative ring structure for G$G$ a finite abelian group. This means that every genuine‐commutative ring spectrum whose homotopy groups are those of KUQ,G$KU_{\mathbb {Q},G}$ is weakly equivalent, as a genuine‐commutative ring spectrum, to KUQ,G$KU_{\mathbb {Q},G}$. In contrast, the connective rational equivariant K$K$‐theory spectrum does not have this type of uniqueness of genuine‐commutative ring structure. [ABSTRACT FROM AUTHOR]

For an étale correspondence Ω : G → H of étale groupoids, we construct an induction functor Ind Ω : KK H → KK G between equivariant Kasparov categories. We introduce the crossed product of an H -equivariant correspondence by Ω, and use this to build a natural transformation α Ω : K ⁎ (G ⋉ Ind Ω −) ⇒ K ⁎ (H ⋉ −). When Ω is proper these constructions naturally sit above an induced map in K-theory K ⁎ (C ⁎ (G)) → K ⁎ (C ⁎ (H)). [ABSTRACT FROM AUTHOR]

Grothendieck polynomials Gw\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathfrak {G}_w$$\end{document} of permutations w∈Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$w\in S_n$$\end{document} were introduced by Lascoux and Schützenberger (C R Acad Sci Paris Sér I Math 295(11):629–633, 1982) as a set of distinguished representatives for the K-theoretic classes of Schubert cycles in the K-theory of the flag variety of Cn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}^n$$\end{document}. We conjecture that the exponents of nonzero terms of the Grothendieck polynomial Gw\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathfrak {G}_w$$\end{document} form a poset under componentwise comparison that is isomorphic to an induced subposet of Zn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {Z}^n$$\end{document}. When w∈Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$w\in S_n$$\end{document} avoids a certain set of patterns, we conjecturally connect the coefficients of Gw\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathfrak {G}_w$$\end{document} with the Möbius function values of the aforementioned poset with 0^\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hat{0}$$\end{document} appended. We prove special cases of our conjectures for Grassmannian and fireworks permutations [ABSTRACT FROM AUTHOR]

We use a determinant functor to describe a map from Grothendieck group to local cohomology. This map is closely related with negative K-groups and generalizes a method of Green and Griffiths. Moreover, we obtain a natural transformation from the local Hilbert functor to the first K-theoretic cycle groups, which are variants of Balmer's tensor triangular Chow groups. [ABSTRACT FROM AUTHOR]

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric. [ABSTRACT FROM AUTHOR]

For two unital Kirchberg algebras with finitely generated K-groups, we introduce a property, called reciprocality, which is proved to be closely related to the homotopy theory of Kirchberg algebras. We show the Spanier–Whitehead duality for bundles of separable nuclear UCT C*-algebras with finitely generated K-groups and conclude that two reciprocal Kirchberg algebras share the same structure of their bundles. [ABSTRACT FROM AUTHOR]

Let G be a split connected reductive algebraic group. Let E⟶B$\mathcal {E}\longrightarrow \mathcal {B}$ be a G×G$G\times G$‐torsor over a smooth base scheme B$\mathcal {B}$ and X be a regular compactification of G. We describe the Grothendieck ring of the associated fibre bundle E(X):=E×G×GX$\mathcal {E}(X):=\mathcal {E}\times _{G\times G} X$, as an algebra over the Grothendieck ring of a canonical toric bundle over a flag bundle over B$\mathcal {B}$. These are relative versions of the corresponding results on the Grothendieck ring of X in the case when B$\mathcal {B}$ is a point, and generalize the classical results on the Grothendieck rings of projective bundles, toric bundles and flag bundles. [ABSTRACT FROM AUTHOR]

We determine the mod |$p$| cohomological invariants for several affine group schemes |$G$| in characteristic |$p$|⁠. These are invariants of |$G$| -torsors with values in étale motivic cohomology, or equivalently in Kato's version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod |$p$| étale motivic cohomology of fields, extending Vial's computation of the operations on the mod |$p$| Milnor K-theory of fields. [ABSTRACT FROM AUTHOR]

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call "fiberwise essentially minimal", that is a class that includes essentially minimal systems and systems in which every orbit is minimal. We prove that the associated crossed product C ∗ -algebra of such a system is an A -algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product C ∗ -algebra has real rank zero, which tells us that such C ∗ -algebras are classifiable by K -theory. The associated crossed product C ∗ -algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case). [ABSTRACT FROM AUTHOR]

We show that S n ∨ S m is Z / p r -hyperbolic for all primes p and all r ∈ Z + , provided n , m ≥ 2 , and consequently that various spaces containing S n ∨ S m as a p-local retract are Z / p r -hyperbolic. We then give a K-theory criterion for a suspension Σ X to be p-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian Σ G r k , n is p-hyperbolic for all odd primes p when n ≥ 3 and 0 < k < n . We obtain similar results for some related spaces. [ABSTRACT FROM AUTHOR]