Papers

Let $f: X\rightarrow S$ be a faithful affine map. In this article we construct a group homomorphism $\theta_{{fppf}}: \text{Br}(f) \longrightarrow H^1_{{fppf}}\!\left(S, (f_*\mathcal{O}_X^\times / \mathcal{O}_S^\times)_{{fppf}}\right)$, extending the formerly defined morphism over an étale site $\theta_{et}$. In doing so, we eliminate the characteristic constraint previously found in the relative version of Kummer's exact sequence. Furthermore, given a finite subintegral extension of noetherian rings $A \hookrightarrow B$, we show that all the $fppf$ cohomology groups, $H^i_{fppf}(Spec(A),\mu^f_n); \;\forall i\geq 0$ vanish whenever $n$ does not divide the characteristic of $A$.

Bicategorical shadows provide a categorical framework that encompasses Hochschild homology and topological Hochschild homology (THH), encodes their Morita invariance, and extends the notion of trace from symmetric monoidal categories to bicategories. Certain equivariant variants, such as $C_n$-twisted THH, do not fit into the ordinary shadow framework. We introduce bicategories with $G$-twisting data and show that every ordinary shadow on such a bicategory induces, for each $g\in G$, a $g$-twisted shadow on the associated bicategory of $G$-twists. These $g$-twisted shadows are invariant under $G$-Morita equivalence; examples include $C_n$-twisted THH and twisted Hochschild homology of $C_n$-Green functors. We further define a $g$-twisted bicategorical trace that recovers the $g$-twisted Hattori–Stallings trace. For $C_n$-Green functors, the orbitwise $g$-twisted Hattori–Stallings traces assemble into a morphism of $C_n$-Mackey functors that, at the orbit $C_n/C_n$, recovers the degree-zero twisted Dennis trace.

In this paper we apply our work on polylogarithmic cocycles representing the Borel classes to establish that the comparison map from bounded continuous cohomology to continuous cohomology is surjective for connected semisimple Lie groups with finite centre. This proves conjectures of Dupont and Monod.

We modify the definition of eventually homological isomorphisms so that it becomes more natural and fruitful. We introduce singular equivalences of $n$-adjoint type which can be viewed as standard singular equivalences of different levels and are more convenient for reducing various homological conjectures and transferring homological properties. We develop the theories of singular equivalences of $n$-adjoint type for $n\ge 2$ and standard eventually homological isomorphisms between derived categories by giving their characterizations in terms of $\otimes$-(co)perfect complexes of bimodules, two constructions via opposite algebras and tensor product algebras, examples induced by homomorphisms of algebras and idempotents, and applications to the reduction of homological conjectures such as the finitistic dimension conjecture, Gorenstein symmetry conjecture, Auslander-Reiten conjecture and Gorenstein projective conjecture, as well as to the preservation of homological properties such as Gorensteinness, the Fg condition, Happel's property and Han's property. Moreover, we show that standard eventually homological isomorphisms correspond to fully faithful standard triangle functors between singularity categories with canonical left adjoint, and essentially surjective standard eventually homological isomorphisms between derived categories correspond to singular equivalences of 3-adjoint type, thereby incorporating essentially surjective standard eventually homological isomorphisms into the framework of singular equivalences of 3-adjoint type.

Let $F$ be the fraction field of a discrete valuation ring. We give an explicit matrix in $\mathrm{SL}_2(F[X^{\pm1},Y^{\pm1}])\setminus \mathrm{E}_2(F[X^{\pm1},Y^{\pm1}])$. The proof of non-elementarity is based on techniques developed by Peter Abramenko in his preprint arXiv:0808.1095v1.

We develop a spectrum-level graded $K$-theory for real super Banach algebras. Our construction is categorical and homotopy theoretic, in the style of algebraic $K$-theory: the graded $K$-theory spectrum is obtained by a (co)fiber sequence from the $(\infty,1)$-categorical group completion of topological groupoids of finitely generated projective graded modules, rather than from spaces of Fredholm operators or Kasparov cycles. We define a connective spectrum $k^{\mathrm{ABS}}_A$ refining the Atiyah–Bott–Shapiro construction as a cofiber, together with its periodification $K^{\mathrm{gr}}_A$, and show that both are lax symmetric monoidal and functorial in bimodules, not merely in homomorphisms. The failure of graded $A$-modules to present all cocycles in $K^{\mathrm{gr}}_0(A)$ is shown to be purely a $\pi_0$-phenomenon on $k^{\mathrm{ABS}}_A$. We obtain a natural equivalence $K^{\mathrm{gr}}_{A\widehat{\otimes} \mathrm{Cl}_{p,q}} \simeq \Sigma^{p-q}K^{\mathrm{gr}}_A$, which links topological Bott periodicity with the Morita equivalence between $\mathrm{Cl}_8$ and $\mathbb{R}$. Restricting to invertible finite-dimensional semisimple super algebras yields a symmetric monoidal functor $\operatorname{Pic}(\operatorname{Bim}(\mathrm{sBan}_{\mathbb{R}})^{\mathrm{fd}}) \to \operatorname{Pic}(\mathrm{Mod}(KO))$ which splits off the bottom three Postnikov layers of $\operatorname{Pic}(\mathrm{Mod}(KO))$, giving a direct link between super division algebras and invertible $KO$-modules. We also give spectral refinements of Karoubi's and van Daele's graded $K$-groups, with explicit comparison equivalences, therefore connecting to $KK$-theory. We also provide an extensive general treatment for $K$-theory of ungraded topological rings that might be of independent interest. In particular, we characterize connective topological $K$-theory of ungraded Banach algebras by a universal property.

Recently, Kupers–Rudenko–Sierra introduced the Goncharov Lie coalgebra. They gave a presentation using correlator generators and four types of relations: homogeneity, cyclic symmetry, shuffle relations, and decomposition relations. We show that the first three families of relations follow from the decomposition relations.

We study the algebraic K-theory of quotients of even ring spectra. Building on Efimov's work, we prove that the limit of the K-theories of these quotients is the continuous K-theory of nuclear modules. For quotients by elements in the height $n$ chromatic ideal, we show that this agrees with the K-theory of the ring spectrum, after chromatic localization at heights $\geq n{+}1$.

We study a family of fibre-cofibre sequences in the $\infty$-category of exact $\infty$-categories, which we call Keller sequences, generalising the notion of Verdier sequences of stable $\infty$-categories. We show that algebraic K-theory is localising with respect to this family, and use this to give a categorical proof of Quillen's resolution theorem. Our arguments hinge upon a mapping space characterisation of special inclusions as studied by Keller, from which we deduce another proof of Keller's theorem that such inclusions induce fully faithful functors on stable envelopes.

Let $R\ltimes_{n}M$ be the $n$-trivial extension of a ring $R$ by an $R$-bimodule $M$. We first characterize projective, injective, flat and finitely generated modules over $R\ltimes_{n}M$. Then we describe when the ring $R\ltimes_{n}M$ is left perfect (Noetherian, coherent, Artinian, hereditary, Kasch). Finally, we establish some homological formulas of $n$-trivial extensions of rings.

We study the $K$-theory classes of higher Kazhdan projections of a discrete group under a spectral-gap assumption and recall how they are obtained, via the Baum–Connes assembly map, from equivariant Euler classes. After reviewing the case of virtually free groups, where the universal space for proper actions has a one-dimensional model, we prove the main new result: an explicit description of the $K$-theory class of the first higher Kazhdan projection for finitely generated infinite one-relator groups whose first Laplacian has a spectral gap. If moreover the one-relator group is hyperbolic, this also gives corresponding formulas for delocalised $\ell^2$-Betti numbers.

Let $G$ be a compact connected Lie group and $K$ its connected Lie subgroup. Using the $K$-theoretic version of equivariant formality developed in [F2] which involves analysis of vector bundles over $G/K$, we find two characterizations of equivariant formality of the isotropy action of $K$ on $G/K$. The first one equates equivariant formality with a "smoothness" condition of the restriction map of the representation ring of $G$ to that of $K$. The second characterization asserts that equivariant formality amounts to the equivariant $K$-theoretic index pairing of two certain special $K$-theory classes being nontrivial in some sense. By applying these characterizations, we are able to give a representation theoretic criterion for equivariant formality of the isotropy action by a circle subgroup, as well as an invariant theory criterion for the case where $K$ is a torus of dimension one less than the rank of $G$, culminating in a complete classification of equivariant formality where $G$ is further assumed to be simple.

We develop a purely algebraic framework for index-type theorems based on the Rees construction for filtered differential graded algebras (FDGAs). Alongside the classical Rees module we introduce a smooth variant $C^\omega_{\mathcal{R}}A$, adapted to analytic arguments, and we study traces and their pointwise and coefficient-wise extensions to Rees algebras. The main result (Corollary 3.11) is an abstract index theorem: given a compatible datum of filtered differential graded associative algebras with traces $A$, $B$, $G$ with a morphism $\phi\colon A\to B$, an action $\rho\colon G\otimes A\to A$ and a morphism $i\colon G\to B$, together with a graded-central element, one has $\mathrm{tr}_B(e^{f_0+f_1}) = \mathrm{tr}_{\widehat{\mathrm{gr}}A}(e^{\gamma}h)$, where the left-hand side is the trace in $B$ and the right-hand side the trace in the Laurent series associated graded $\widehat{\mathrm{gr}}A$ of $A$. Here $f_0+f_1$ is a curvature-type element, i.e., an element of the form $d_B\beta+\beta^2$ for some odd-degree element $\beta$, while $\gamma$ and $h$ are certain elements in $\widehat{\mathrm{gr}}A$. The formalism is modelled on the Getzler rescaling technique and on the derivation of the localization formula for the loop space Chern character by Ludewig and Yi.

Let $p$ be an odd prime and let $K/\mathbb{Q}_p$ be a finite unramified extension. From a finite presentation by roots of unity of order prime to $p$, we construct a localized Iwasawa pseudomeasure on $\mathbb{Z}_p^\times$. Although the pseudomeasure depends on the chosen presentation, its image modulo bounded measures depends only on the associated Bloch class: it is the Frobenius-depleted Coleman regulator of that class multiplied by a universal half-shifted zeta principal part. Consequently, every nonexceptional weight component is bounded, while the exceptional component has at most a simple pole with explicitly determined residue. For $p>3$ and $\mathbb{Z}_p$-valued coefficients, vanishing of the principal part is equivalent to vanishing of the corresponding class in $K_3(K;\mathbb{Z}_p)$. Cyclotomic refinements preserve the Bloch class and act on the associated pseudomeasures by explicit Iwasawa multipliers. Normalized finite linear combinations of refinements interpolate arbitrary finite jets of the bounded weight-space data, subject only to the normalization at the exceptional point. We also establish a half-shifted complex Mellin factorization, compare the construction with the GSWZ germ family, and derive, at simple degree-one places above primes $p>3$, a finite-polylogarithm criterion for the local $K_3$-class of the knot $5_2$.

We study how motivic and étale motivic cohomology of a smooth and proper variety changes under extensions of algebraically closed base fields $k$. With finite coefficients, motivic cohomology is independent of $k$, and étale motivic cohomology is independent of $k$ away from the characteristic, but depends on $k$ at the characteristic. We then consider the fiber of $F-1$ for $F$ the Frobenius, on motivic cohomology and étale motivic cohomology for varieties defined over finite fields. We show that this fiber is independent of $k$ with finite coefficients. In addition, we give some results and examples with integral coefficients.

Recent work by Wanchun Shen shows that $K_2$-regular affine local complete intersections of any dimension in characteristic zero are smooth. In this note, we show that the same holds for affine toric varieties.

Poincaré duality is a classical theorem relating the homology and cohomology of closed oriented manifolds. This theorem has been extended to more general settings and to generalized (co)homology theories, including $K$-theory. In this paper, we construct an infinite-dimensional analogue of the $K$-theoretic Poincaré duality homomorphism. More precisely, for an infinite-dimensional Hilbert manifold $\mathcal{M}$, we construct a homomorphism $$K^{geo}_*(\mathcal{M})\to K_*(\mathcal{A}(\mathcal{M})),$$ where $K^{geo}_*(\mathcal{M})$ denotes the geometric $K$-homology of Baum and Douglas, and $\mathcal{A}(\mathcal{M})$ is a $C^*$-algebra associated to $\mathcal{M}$, based on a construction of Gong, Wu, and Yu. We also prove that the constructed homomorphism is non-trivial in certain cases.

This paper relates algebraic K-theory of fields to polylogarithms via general linear groups. We focus on the case of number fields and prove that the motivic realisation map from the Goncharov Lie coalgebra to the motivic Lie coalgebra is an isomorphism. This implies the Goncharov universality conjecture and a structural result for special values of Dedekind zeta functions. We also construct explicit polylogarithmic cocycles representing nonzero multiples of the Borel classes.

Let \(D\in\{\C,\Hh,\Oct\}\) be a real composition division algebra of dimension \(d\in\{2,4,8\}\). We construct an explicit rank-\(d\) real vector bundle $ E_D\longrightarrow\RP^d$ by deleting one homogeneous coordinate on each standard affine chart, interpreting the remaining coordinates as an element of \(D\), and using the corresponding left-multiplication matrices as transition data. The bundle admits a matrix-determined trivialization away from the \(d+1\) coordinate points. Relative to this trivialization, each deleted point has local clutching map $S^{d-1}\longrightarrow\SO(d),\; u\longmapsto L_u$, which is respectively the complex, quaternionic, or octonionic Hopf clutching map. A section arising from the same matrices has exactly the coordinate points as nondegenerate zeros. Consequently, $w(E_D)=1+x^d$, and real \(K\)-theory together with Euler-class cancellation gives $E_D\cong\gamma^{\oplus d}$. Thus the construction supplies explicit framed-defect realizations of these familiar bundles. In particular, it gives a semialgebraic octonionic realization of \(\gamma^{\oplus8}\) on \(\RP^8\) with nine specified local Hopf defects. We also describe the associated Pfister quadratic bundle over arbitrary fields.

Let $\mathbb{Z}[1/n] := \{a/n^r: a \in \mathbb{Z},r \in \mathbb{Z} \geq 0\}$ be the euclidean domain obtained from the ring of integers $\mathbb{Z}$ by localizing at $n$. Moreover, let $\text{SL}_2(\mathbb{Z}[1/n])$ be the group of all invertible 2-by-2 matrices of determinant one with entries in $\mathbb{Z}[1/n]$. The homology groups $H_k(\text{SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ are of interest in Geometric Group Theory, Algebraic Number Theory and Algebraic K-theory. In this dissertation, we study these groups via a spectral sequence derived by the action of $\text{SL}_2(\mathbb{Z}[1/n])$ on the product of the trees $B_p$ associated with the $p$-adic valuation on $\mathbb{Q}$ where $p$ is a prime factor of $n$.

We show that the operadic structure on the cohomology of the poset of noncrossing 2-partitions is isomorphic to the Brace operad.

We study the relation between the Gorenstein homological and the Gorenstein cohomological dimension of groups. We prove that for every module of type $FP_\infty$ over any ring the Gorenstein projective and the Gorenstein flat dimensions are always equal to each other. In particular, we have an equality $\mathrm{Gcd}_kG=\mathrm{Ghd}_kG$ for every group $G$ of type $FP_\infty$ over a commutative coefficient ring $k$. For non-locally-finite groups in Kropholler's class ${\scriptstyle\mathbf{LH}}\mathfrak F$, we characterize the groups of Gorenstein (co)homological dimension one, in terms of actions on trees with finite stabilizers. The Hirsch length of a virtually soluble group $G$ determines its Gorenstein (co)homological dimension, even when $G$ has torsion: We show that $\mathrm{Ghd}_\mathbb{Z}G =h(G)$ and, if $G$ is countable, $h(G)\leq\mathrm{Gcd}_\mathbb{Z}G\leq h(G)+1$. Some consequences for elementary amenable groups of finite Hirsch length are also obtained. Finally, we investigate the Gorenstein dimension of modules over certain algebras of groups with torsion and discuss applications to classifying spaces for proper actions.

In this paper, for various pairs of topological spaces $X$ and $Y$, we introduce reflection functors between the categories $\mathfrak{KK}(X)$ and $\mathfrak{KK}(Y)$ of Kirchberg's ideal-related KK-theory for separable C*-algebras over $X$ and $Y$. We prove that these functors induce an equivalence $\mathfrak{KK}(X)_{\mathrm{loc}} \simeq \mathfrak{KK}(Y)_{\mathrm{loc}}$ between the localizing subcategories introduced by Meyer and Nest, and that this equivalence restricts to an equivalence $\mathcal{B}(X) \simeq \mathcal{B}(Y)$ between the bootstrap categories. Combining these equivalences with a combinatorial argument due to Bernstein, Gelfand, and Ponomarev from the representation theory of quivers, we show that, for a finite $T_0$-space $X$ whose Hasse diagram is an orientation of a tree, the categories $\mathfrak{KK}(X)_{\mathrm{loc}}$ and $\mathcal{B}(X)$ depend only on the underlying tree and not on its orientation. We also prove the analogous results for Dadarlat and Meyer's ideal-related E-theory. Moreover, we prove a rearrangement property of reflection functors, which yields an analogue of Coxeter functors. Finally, we apply reflection functors to prove that filtrated K-theory satisfies the universal coefficient theorem for C*-algebras over finite $T_0$-spaces whose Hasse diagrams are orientations of Dynkin diagrams of type A.

Let $H$ be a Hopf algebra with a bijective antipode over a field $\mathbf k$ and $B/A$ be a flat right $H$-Galois extension. Ştefan constructed a spectral sequence converging to the Hochschild cohomology $\mathrm{HH}^{p+q}(B, N)$ with $\mathrm{E}_2^{p,q} = \mathrm{H}^p(H, \mathrm{HH}^q(A, N))$. We show that when $B/A$ is faithfully flat, the Ştefan spectral sequence is multiplicative. More precisely, a new formalism of functorial multiplicative spectral sequences via lax monoidal functors over monoidal categories is introduced. A functorial multiplicative spectral sequence is constructed for any faithfully flat Hopf Galois extension. Identification with Ştefan's spectral sequence from the $E_2$-page follows from Künzer's comparison theorems. Applications include strongly graded algebras, smash products, crossed products, group and Lie algebra extensions, with new multiplicativity results in several cases.

Let $W$ be a henselian valuation ring with fraction field $L$, residue field $k$, and value group $\Gamma_W$. Let $N=\ell^\nu$ be invertible in $W$. Choose an ordered $\mathbf Z/N$-basis $B$ of $\Gamma_W/N\Gamma_W$. Products of suitable classes define an equivalence of complete filtered spectra $$ \bigoplus_{\substack{J\subseteq B\\J\text{ finite}}} \Sigma^{|J|}\operatorname{Fil}_{\mathrm{mot}}^{\bullet-|J|} K(k;\mathbf Z/N) \xrightarrow{\simeq} \operatorname{Fil}_{\mathrm{mot}}^{\bullet} K(L;\mathbf Z/N) $$ The summand indexed by $\varnothing$ is the generic restriction map, after the rigidity equivalence $K(W;\mathbf Z/N)\simeq K(k;\mathbf Z/N)$, and is therefore split injective. Independently of this splitting, excision yields a lifting theorem for regular henselian pairs. In particular, this yields finite-coefficient Gersten injectivity for noetherian henselian regular local rings. Applications to completions along regular primes give relative and sometimes nonhenselian examples. More generally, if $P$ is a Prüfer ring and $R$ is a henselian local ind-smooth $P$-algebra, then $R$ is a domain and $K_n(R;\mathbf Z/N)\to K_n(\operatorname{Frac}(R);\mathbf Z/N)$ is injective for every $n$, provided $N\in R^\times$. These injectivity consequences extend from prime-power to arbitrary finite invertible coefficients by primary decomposition.

We study the $\mathbf{I}^*$-cohomology of a smooth real algebraic curve in terms of its real locus and its geometric genus. We notably extend results of Monnier to the twisted case, which is crucial to the understanding of proper pushforwards of Witt groups. We also perform some computations related to transfers along the finite étale extension $\mathbb{C}/\mathbb{R}$. We further describe how to compute twisted Witt groups of surfaces, extending work of Sujatha, and the image of the global signature homomorphism following Monnier. As an application of the main methods of the paper, we describe the shifted and twisted Witt groups of smooth anisotropic quadrics over $\mathbb{R}$ of dimension $\leq 3$.

We study Suslin's cancellation conjecture on smooth real affine varieties whose real locus is empty or, more generally, of small cohomological dimension.

For Rota-Baxter Lie algebras, a homotopy cooperad is exhibited, whose cobar construction is shown to be the minimal model of the operad of Rota-Baxter Lie algebras by using algebraic Morse theory. The deformation complex of Rota-Baxter Lie algebras as well as the $L_\infty$-algebra structure on this complex are deduced from the minimal model and the notion of homotopy Rota-Baxter Lie algebras is given as a consequence.

In this work, we compute the first integral homology, or abelianization, of the congruence subgroups $\Gamma(A, \mathfrak{m}_A), \Gamma_1(A, \mathfrak{m}_A)$, and $\Gamma_0(A, \mathfrak{m}_A)$ for a local ring $A$ with maximal ideal $\mathfrak{m}_A$, showing that $H_1(\Gamma(A, \mathfrak{m}_A), \mathbb{Z})$ is isomorphic to the additive group of $\mathfrak{sl}_2(\mathfrak{m}_A/\mathfrak{m}_A^2)$. We then use these results to determine the structure of the groups $H_1(\Gamma(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$, $H_1(\Gamma_1(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$ and $H_1(\Gamma_0(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$, where $\mathcal{O}_{K,S}$ is a Dedekind domain of arithmetic type, not totally imaginary, $|S| \geq 2$, and $\mathfrak{p}$ is a nonzero prime ideal. The computations are given in terms of the residue field $\kappa(\mathfrak{p})$ and the known $H_1(\mathrm{SL}_2(\mathcal{O}_{K, S}), \mathbb{Z})$. As a consequence, we also obtain the torsion subgroup of their second integral cohomology. These results will be of paramount importance for a forthcoming work concerning $H_2(\mathrm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$.

It is shown that the trivial extension of the Xu–Snashall algebra is a symmetric counterexample to the Snashall–Solberg conjecture, which states that the Hochschild cohomology ring of a finite-dimensional algebra modulo nilpotence is a finitely generated algebra. To the best of our knowledge, this is the first known selfinjective counterexample.