The algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and the automorphism groups of free groups. In this paper, we fully address the question of how Morita equivalence classes of Lawvere theories interact with algebraic K-theory. On the one hand, we show that the higher algebraic K-theory is invariant under passage to matrix theories. On the other hand, we show that the higher algebraic K-theory is not fully Morita invariant because of the behavior of idempotents in non-additive contexts: We compute the K-theory of all Lawvere theories Morita equivalent to the theory of Boolean algebras.
more »
« less
A Shadow Perspective on Equivariant Hochschild Homologies
Abstract Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $C_n$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $K$-theory to twisted topological Hochschild homology.
more »
« less
- PAR ID:
- 10381468
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday–Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other homology theories associated with spaces (such as Pontryagin algebras, ${{\mathbb{S}}}^1$-equivariant homology of the free loop space, and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces, and some 3-dimensional manifolds, such as link complements in ${\mathbb{R}}^3$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in ${\mathbb{R}}^3$.more » « less
-
Grothendieck duality theory assigns to essentially finite-type maps $f$ of noetherian schemes a pseudofunctor $f^{\times }$ right-adjoint to $\mathsf{R}f_{\ast }$ , and a pseudofunctor $f^{!}$ agreeing with $f^{\times }$ when $f$ is proper, but equal to the usual inverse image $f^{\ast }$ when $f$ is étale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by ‘compactly supported’ versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes.more » « less
-
null (Ed.)The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for non-hyperelliptic curves is in general not easy. We present a "combinatorialization" of this problem, explaining how to define a Ceresa class for a tropical algebraic curve, and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over ℂ((t)), and show that the Ceresa class in each of these settings is torsion.more » « less
-
We describe a construction of the cyclotomic structure on topological Hochschild homology (THH) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bökstedt coherence machinery. We are also able to define two relative versions of topological cyclic homology (TC) and TR-theory: one starting with a ring C_n-spectrum and one starting with an algebra over a cyclotomic commutative ring spectrum A. We describe spectral sequences computing the relative theory over A in terms of TR over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on TR and TC.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnac250
