arXiv subjects
Publications and source records attributed to Noah Olander.
We prove that a homomorphism of rings of prime characteristic is b-nil formally unramified if and only if its relative Frobenius is an epimorphism. We specialize this result under different finiteness conditions such as (relative) $F$-finiteness and Noetherianity. We give an example where the absolute Frobenius is an epimorphism but is not surjective, as well as an example of a formally unramified homomorphism that is not b-nil formally unramified.
We prove that the Kuznetsov component of a flat family of even-dimensional quadrics of corank at most 2 is equivalent to the twisted derived category of an algebraic space whenever: (i) the open subset of the base over which the quadrics has corank at most 1 is scheme-theoretically dense; and (ii) a certain \'etale double cover of the closed complement admits a section. This provides the first general geometricity result for Kuznetsov components of higher dimensional quadrics, thereby generalizing works of Kapranov, Bondal, Orlov, Kuznetsov, Moschetti, Xie, and others. Our main tool is the moduli stack of spinor sheaves on a family of quadrics, which we define and study in detail. In the situation of our main result, we produce an open substack which is a $\mathbf{G}_m$-gerbe, and show that the associated twisted derived category is equivalent to the Kuznetsov component of the family of quadrics, thereby providing a geometric interpretation of the Brauer classes appearing in previous works.
Rouquier proved that for a smooth projective variety $X$, the group scheme $\operatorname{Pic}^0_X \rtimes \operatorname{Aut}^0_X$ is an invariant of the derived category of $X$. This was generalized to the twisted case by Olsson, who associated a group algebraic space $\operatorname{Aut}^0_{\mathcal{X}}$ to a $\mathbf{G}_m$-gerbe $\mathcal{X} \to X$ and proved it to be a twisted derived invariant. In characteristic zero, Olsson showed that $\operatorname{Aut}^0_{\mathcal{X}}$ is an extension of $\operatorname{Aut}^0_X$ by a subgroup scheme of $\operatorname{Pic}_X$. It is important in applications to compute $\operatorname{Aut}^0_{\mathcal{X}}$ in positive characteristic as well, which is the problem we consider here. We use deformation theory and representability results of Bragg and Olsson to prove that in many cases of interest, Olsson's description of $\operatorname{Aut}^0_{\mathcal{X}}$ as an extension also holds in characteristic $p$. As a corollary, we show that twisted derived equivalent abelian varieties are isogenous, simultaneously generalizing results of Honigs and Lane. We also provide an example showing that Olsson's extension description does not hold in general. This uses a result of Illusie on crystalline and flat cohomology and a result of Yang on the Brauer group of an ordinary variety, which we generalize to cohomology in arbitrary degree.
A field extension $L/K$ of characteristic $p > 0$ is formally \'etale if and only if the relative Frobenius of $L/K$ is an isomorphism. Inspired by this classical result, we explore whether the formally \'etale property for a map $R \to S$ of $\mathbf{F}_p$-algebras is characterized by isomorphism of the relative Frobenius $F_{S/R}$. While $F_{S/R}$ being an isomorphism implies $R \to S$ is formally \'etale, the converse fails in the non-Noetherian setting. Thus, following Morrow, we introduce an enhancement of the formally \'etale property that we call b-nil (bounded nil) formally \'etale, and we show that $F_{S/R}$ is an isomorphism precisely when $R \to S$ is b-nil formally \'etale. We prove this result by first establishing several structural properties of b-nil formally smooth maps, which are defined analogously to the formally smooth case. Our structural results reveal that the b-nil formally smooth (resp. \'etale) property is quite different from the formally smooth (resp. \'etale) property. For instance, we show that any b-nil formally smooth algebra over an $F$-pure ring is reduced, whereas non-reduced formally \'etale algebras exist over $\mathbf{F}_p$ by a construction of Bhatt. We also show that the b-nil formally \'etale property neither implies nor is implied by having a trivial cotangent complex. We explore when formally smooth (resp. \'etale) implies b-nil formally smooth (resp. \'etale) in prime characteristic. A satisfactory picture emerges for ideal adic completions.
We prove that smooth, separated Deligne--Mumford stacks in mixed characteristic with quasi-projective coarse moduli space are global quotient stacks and satisfy the resolution property. This builds on work of Kresch and Vistoli and of Bragg, Hall, and Matthur which proves the case when the stack is over a base field, as well as work of Gabber and de Jong which proves the same holds for a $\mu_n$-gerbe over a scheme with an ample line bundle. The key technical input is a result that gerbes banded by so-called trigonalizable group schemes admit faithful vector bundles and are quotient stacks.
We give a complete characterization of the line bundles on a proper variety whose tensor powers generate the derived category, answering a 2010 question of Chris Brav. The condition is analogous to the Nakai--Moishezon criterion and can be stated purely in terms of classical notions of positivity of line bundles. There is also a generalization which works for all Noetherian schemes. We use our criterion to prove basic properties of such line bundles and provide non-trivial examples of them. As an application, we give new examples of varieties which can be reconstructed from their derived categories in the sense of the Bondal--Orlov Reconstruction Theorem.
This work explores bounds on the Rouquier dimension in the bounded derived category of coherent sheaves on Noetherian schemes. By utilizing approximations, we exhibit that Rouquier dimension is inherently characterized by the number of cones required to build all perfect complexes. We use this to prove sharper bounds on Rouquier dimension of singular schemes. Firstly, we show Rouquier dimension doesn't go up along \'{e}tale extensions and is invariant under \'{e}tale covers of affine schemes admitting a dualizing complex. Secondly, we demonstrate that the Rouquier dimension of the bounded derived category for a curve, with a delta invariant of at most one at closed points, is no larger than two. Thirdly, we bound the Rouquier dimension for the bounded derived category of a (birational) derived splinter variety by that of a resolution of singularities.
We show that the weakly étale morphisms, used to define the pro-étale site of a scheme, are characterized by a lifting property similar to the one which characterizes formally étale morphisms. In order to prove this, we prove a theorem called Henselian descent which is a "Henselized version" of the fact that a scheme defines a sheaf for the fpqc topology. Finally, we show that weakly étale algebras over regular rings arising in geometry are ind-étale and that weakly étale algebras do not always lift along surjective ring homomorphisms.
We define the countable Rouquier dimension of a triangulated category and use this notion together with Theorem 2 of [Ola21] to prove that if there is a fully faithful embedding $D^b_{coh}(X) \subset D^b_{coh}(Y)$ with $X, Y$ smooth proper varieties, then $\mathrm{dim}(X) \leq \mathrm{dim}(Y)$.
We prove that for $X$ a regular quasi-affine scheme of dimension $d$, $\mathcal{O}_X$ is a $d$-step generator of $D^b_{coh}(X)$, establishing Orlov's conjecture in this case. We prove something weaker in the projective case. The main techniques are a spectral sequence argument borrowed from topology and the converse ghost lemma, both suitably adapted to work in this setting. Along the way we prove that on a regular scheme $X$ of dimension $d < \infty$ any composition of $d+1$ morphisms of $D^b_{coh}(X)$ which are zero on cohomology sheaves is zero.
We prove Conjecture 4.16 of the paper [EL21] of Elagin and Lunts; namely, that a smooth projective curve of genus at least 1 over a field has diagonal dimension 2.
We extend Orlov's result that certain functors between derived categories of smooth projective varieties are Fourier--Mukai transforms to the case when those varieties are smooth and proper.