Some explicit counter-examples to Weibel's conjecture
We give two related but different methods for constructing rings $R$ of Krull dimension 1 and $K_{-d}(R) \neq 0$. The first works for $d = 2$, and the second works for any $d \geq 2$.
math.AG↗
arXiv subjects
Shane Kelly
Publications and source records attributed to Shane Kelly.
Log homotopy types are homotopy types with modulus
We show that the category of log homotopy types is a full subcategory of a category of homotopy types with modulus.
math.AT↗
A pro-cdh topology on formal schemes
We introduce a pro-cdh topology on formal schemes and prove that the $\infty$-topos of pro-cdh sheaves of spaces has an optimal bound of homotopy dimension. This remedies a defect for a pro-cdh topology on schemes introduced in [KS23]. As an application, we give a topos-theoretic interpretation of Weibel's vanishing of negative K-theory and motivic cohomology of Elmanto and Morrow.
math.AG↗
On pro-cdh descent on derived schemes
Grothendieck's formal functions theorem states that the coherent cohomology of a Noetherian scheme can be recovered from that of a blowup and the infinitesimal thickenings of the center and of the exceptional divisor of the blowup. In this article, we prove an analogous descent result, called ``pro-cdh descent'', for certain cohomological invariants of arbitrary quasi-compact, quasi-separated derived schemes. Our results in particular apply to algebraic $K$-theory, topological Hochschild and cyclic homology, and the cotangent complex. As an application, we deduce that $K_n(X) = 0$ when $n < -d$ for quasi-compact, quasi-separated derived schemes $X$ of valuative dimension $d$. This generalises Weibel's conjecture, which was originally stated for Noetherian (non-derived) $X$ of Krull dimension $d$, and proved in this form in 2018 by Kerz, Strunk, and the third author.
math.KT↗
Two new motivic complexes for non-smooth schemes
These are expanded notes from a talk at the RIMS Workshop, Algebraic Number Theory and Related Topics, December 13th, 2023. We discussed Elmanto-Morrow's motivic complex, the procdh sheafification of the classical motivic complex, and their comparison. The procdh topology and the comparison is joint work with Shuji Saito. The comparison was obtained through joint discussion with Morrow, and its proof relies heavily on the main results of [EM23].
math.AG↗
A procdh topology
In this article we propose a definition of a procdh topos. We show that it encodes procdh excision, has bounded homotopy dimension and therefore is hypercomplete and admits a conservative family of fibre functors. We also describe the local rings. As an application, we show that nonconnective $K$-theory is the procdh sheafification of connective $K$-theory, and that the motivic cohomology recently proposed by Elmanto and Morrow is the procdh sheafification of Voevodsky's motivic cohomology.
math.AG↗
Non-reduced valuation rings and descent for smooth blowup squares
We consider a class of non-reduced valuation rings, known in the literature as chain rings. We observe that the Grothendieck topology generated by the Zariski topology and smooth blowup squares is exactly the topology which has chain rings for its local rings, and that sheaves for this topology are \emph{not} characterised by excision for smooth blowup squares.
math.AG↗
Hodge cohomology with a ramification filtration, I
We consider a filtration on the cohomology of the structure sheaf indexed by (not necessarily reduced) divisors ``at infinity''. We show that the filtered pieces have transfers morphisms, fpqc descent, and are so called cube invariant. In the presence of resolution of singularities and weak factorisation they are invariant under blowup ``at infinity''. As such, they lead to a realisation functor from Kahn, Miyazaki, Saito and Yamazaki's category of motives with modulus over a characteristic zero base field.
math.AG↗
Hodge cohomology with a ramification filtration, II
As a sequel of Part I, we consider a filtration of Hodge cohomology groups indexed by divisors "at infinity", and prove that they are represented in the category of motives with modulus. In particular, we obtain a realisation functor of the Hodge cohomology groups.
math.AG↗
Modulus sheaves with transfers
We generalise Kahn, Miyazaki, Saito, Yamazaki's theory of modulus pairs to pairs $(X, D)$ consisting of a qcqs scheme $X$ equipped with an effective Cartier divisor $D$ representing a ramification bound. We develop theories of sheaves on such pairs for modulus versions of the Zariski, Nisnevich, \'etale, fppf, and qfh-topologies. We extend the Suslin-Voevodsky theory of correspondances to modulus pairs, under the assumption that the interior $U = X \setminus D$ is Noetherian. The resulting point of view highlights connections to (Raynaud-style) rigid geometry, and potentially provides a setting where wild ramification can be compared with irregular singularities. This framework leads to a homotopy theory of modulus pairs $\underline{M}H(X,D)$ and a theory of motives with modulus $\underline{M}DM^{eff}(X,D)$ over a general base $(X, D)$. For example, the case where $X$ is the spectrum of a rank one valuation ring (of mixed or equal characteristic) equipped with a choice $D$ of pseudo-uniformiser is allowed.
math.AG↗
Milnor excision for motivic spectra
We prove that the $\infty$-category of motivic spectra satisfies Milnor excision: if $A\to B$ is a morphism of commutative rings sending an ideal $I\subset A$ isomorphically onto an ideal of $B$, then a motivic spectrum over $A$ is equivalent to a pair of motivic spectra over $B$ and $A/I$ that are identified over $B/IB$. Consequently, any cohomology theory represented by a motivic spectrum satisfies Milnor excision. We also prove Milnor excision for Ayoub's \'etale motives over schemes of finite virtual cohomological dimension.
math.AG↗
Cdh descent, cdarc descent, and Milnor excision
We give necessary and sufficient conditions for a cdh sheaf to satisfy Milnor excision, following ideas of Bhatt and Mathew. Along the way, we show that the cdh infinity-topos of a quasi-compact quasi-separated scheme of finite valuative dimension is hypercomplete, extending a theorem of Voevodsky to nonnoetherian schemes. As an application, we show that if E is a motivic spectrum over a field k which is n-torsion for some n invertible in k, then the cohomology theory on k-schemes defined by E satisfies Milnor excision.
math.AG↗
Smooth blowup square for motives with modulus
In this self-contained paper we prove that Voevodsky's smooth blowup triangle of motives generalises to a smooth blowup triangle of motives with modulus.
math.AG↗
$K$-theory of valuation rings
We prove several results showing that the algebraic $K$-theory of valuation rings behave as though such rings were regular Noetherian, in particular an analogue of the Geisser--Levine theorem. We also give some new proofs of known results concerning cdh descent of algebraic $K$-theory.
math.KT↗
A better comparison of cdh- and ldh-cohomologies
In order to work with non-Nagata rings which are Nagata "up-to-completely-decomposed-universal-homeomorphism", specifically finite rank hensel valuation rings, we introduce the notions of pseudo-integral closure and pseudo-normalisation. We use this notion to give a much more direct and shorter proof that $H^n_{cdh}(X, F) = H^n_{ldh}(X, F)$ for homotopy sheaves $F$ of modules over the $\mathbb{Z}_{(l)}$-linear motivic Eilenberg-Maclane spectrum. This comparison is an alternative to the first half of the authors volume Ast\'erisque 391, whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky's motivic cohomology (with $\mathbb{Z}[1/p]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.
math.AG↗
Weak three-dimensional mediators of two-dimensional triplet pairing
Recent experiments demonstrate the ability to construct cold atom mixtures with species selective optical lattices. This allows for the possibility of a mixed-dimension system, where one fermionic atomic species is confined to a two dimensional lattice, while another species is confined to a three dimensional lattice that contains the two-dimensional one. We show that by tuning the density of an arbitrary number of three-dimensional atomic species, we can engineer an arbitrary, rotationally-symmetric, density-density, effective interaction for the two-dimensional particles. This possibility allows for an effective interaction that favours triplet pairing for two-dimensional, $SU(2)$ symmetric particles. Using a functional renormalization-group analysis for the two-dimensional particles, we derive and numerically confirm that the critical temperature for triplet pairing depends exponentially on the effective interaction strength. We then analyse how the stability of this phase is affected by the particle densities and the fine tuning of interaction parameters. We conclude by briefly discussing experimental considerations and the potential to study triplet pairing physics, including Majorana fermions and spin textures, with cold atoms on optical lattices.
cond-mat.quant-gas↗
Differential forms in positive characteristic II: cdh-descent via functorial Riemann-Zariski spaces
This paper continues our study of the sheaf associated to K\"ahler differentials in the cdh-topology and its cousins, in positive characteristic, without assuming resolution of singularities. The picture for the sheaves themselves is now fairly complete. We give a calculation $\mathcal{O}_{cdh}(X) \cong \mathcal{O}(X^{sn})$ in terms of the seminormalisation. We observe that the category of representable cdh-sheaves is equivalent to the category of seminormal varieties. We conclude by proposing some possible connections to Berkovich spaces, and $F$-singularities in the last section. The tools developed for the case of differential forms also apply in other contexts and should be of independent interest.
math.AG↗
Mixed Motives and Geometric Representation Theory in Equal Characteristic
Let $\mathbb{k}$ be a field of characteristic $p$. We introduce a formalism of mixed sheaves with coefficients in $\mathbb{k}$ and showcase its use in representation theory. More precisely, we construct for all quasi-projective schemes $X$ over an algebraic closure of $\mathbb{F}_p$ a $\mathbb{k}$-linear triangulated category of motives on $X$. Using work of Ayoub (2007), Cisinski-Deglise (2012) and Geisser-Levine (2000), we show that this system of categories has a six functors formalism and computes higher Chow groups. Indeed, it behaves similarly to other categories of sheaves that one is used to. We attempt to make its construction also accessible to non-experts. We then consider the subcategory of stratified mixed Tate motives defined for affinely stratified varieties $X$, discuss perverse and parity motives and prove formality results. As an example, we combine these results and Soergel (2000) to construct a geometric and graded version of Soergel's modular category $\mathscr O(G)$, consisting of rational representations of a split semisimple group $G/\mathbb{k}$, and thereby equip it with a full six functor formalism (see Riche-Soergel-Williamson (2014) and Achar-Riche (2016) for other approaches). The main idea of using motives in geometric representation theory in this way as well as many results about stratified mixed Tate motives are directly borrowed from Soergel and Wendt, who tell the story in characteristic zero.
math.RT↗