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Daniel Krashen

Publications and source records attributed to Daniel Krashen.

At least 19 recordsLinked to original sources
arXiv2025

A Looming of phantoms

Following Krah's method, we construct new examples of phantom categories as semiorthogonal components of the derived categories of two types of rational surfaces: the blowup of the plane at 11 points in general position, and the blowup of the second Hirzebruch surface at 9 points in general position. We also pose conjectures about the existence of phantom subcategories in the derived categories of other rational surfaces, obtained as the blowups of the other Hirzebruch surfaces.

math.AG↗
arXiv2025

On rational connectedness and parametrization of finite Galois extensions

Given two $G$-Galois extensions of $\mathbb Q$, is there an extension of $\mathbb Q(t)$ that specializes to both? The equivalence relation on $G$-Galois extension of $\mathbb Q$, induced by the above question, is called $R$-equivalence. The number of $R$-equivlance classes indicates how many rational spaces are required in order to parametrize all $G$-Galois extensions of $\mathbb Q$. We determine the $R$-equivalence classes for basic families of groups $G$, and consequently obtain parametrizations of the $G$-Galois extensions of $\mathbb Q$ in the absence of a generic extension for $G$.

math.NT↗
arXiv2025

Finiteness of formal pushforwards

Under mild hypotheses, given a scheme $U$ and an open subset $V$ whose complement has codimension at least two, the pushforward of a torsion-free coherent sheaf on $V$ is coherent on $U$, and in particular is finite. We prove an analog of this finiteness assertion in the context of formal schemes over a complete discrete valuation ring, but show that coherence does not always hold. We then relate this to the problem of gluing formal functions, where the patches do not cover the entire scheme.

math.AG↗
arXiv2024

Classes in $\mathrm H_{p^m}^{n+1}(F)$ of lower exponent

Let $F$ be a field of characteristic $p>0$. We prove that if a symbol $A=\omega \otimes \beta_1 \otimes \dots \otimes \beta_n$ in $H_{p^m}^{n+1}(F)$ is of exponent dividing $p^{m-1}$, then its symbol length in $H_{p^{m-1}}^{n+1}(F)$ is at most $p^n$. In the case $n=2$ we also prove that if $A= \omega_1\otimes \beta_1+\cdots+\omega_r\otimes \beta_r$ in $H_{p^{m}}^2(F)$ satisfies $\exp(A)|p^{m-1}$, then the symbol length of $A$ in $H_{p^{m-1}}^2(F)$ is at most $p^r+r-1$. We conclude by looking at the case $p=2$ and proving that if $A$ is a sum of two symbols in $H_{2^m}^{n+1}(F)$ and $\exp A |2^{m-1}$, then the symbol length of $A$ in $H_{2^{m-1}}^{n+1}(F)$ is at most $(2n+1)2^n$. Our results use norm conditions in characteristic $p$ in the same manner as Matrzi in his paper ``On the symbol length of symbols''.

math.RA↗
arXiv2024

Morita equivalences for Zhu's algebra

Through the introduction of new ideals, and with the assistance of the $d$-th mode transition algebras $\mathfrak{A}_d$, for $d\in \mathbb{N}$, we show how Zhu's associative algebra $\mathsf{A}$, conventionally valued for tracking information about the degree $0$ part of an $\mathbb{N}$-graded module over a vertex operator algebra $V$, also contains information about components of higher degree. As an application, equivalent conditions are given for rationality of $V$, and explicit presentations for higher-level Zhu algebras are given, including for a large class of non-rational VOAs.

math.RT↗
arXiv2023

Conformal blocks on smoothings via mode transition algebras

Here we define a series of associative algebras attached to a vertex operator algebra $V$, called mode transition algebras, showing they reflect both algebraic properties of $V$ and geometric constructions on moduli of curves. One can define sheaves of coinvariants on pointed coordinatized curves from $V$-modules. We show that if the mode transition algebras admit multiplicative identities with certain properties, these sheaves deform as wanted on families of curves with nodes (so $V$ satisfies smoothing). Consequently, coherent sheaves of coinvariants defined by vertex operator algebras that satisfy smoothing form vector bundles. We also show that mode transition algebras give information about higher level Zhu algebras and generalized Verma modules. As an application, we completely describe higher level Zhu algebras of the Heisenberg vertex algebra for all levels, proving a conjecture of Addabbo--Barron.

math.QA↗
arXiv2023

Brauertsch fields

We prove a local-to-global principle for Brauer classes: for any finite collection of non-trivial Brauer classes on a variety over a field of transcendence degree at least 3, there are infinitely many specializations where each class stays non-trivial. This is deduced from a Grothendieck--Lefschetz-type theorem for Brauer groups of certain smooth stacks. This also leads to the notion of a Brauertsch field.

math.AG↗
arXiv2022

Factorization presentations

Modules over a vertex operator algebra V give rise to sheaves of coinvariants on moduli of stable pointed curves. If V satisfies finiteness and semi-simplicity conditions, these sheaves are vector bundles. This relies on factorization, an isomorphism of spaces of coinvariants at a nodal curve with a finite sum of analogous spaces on the normalization of the curve. Here we introduce the notion of a factorization presentation, and using this, we show that finiteness conditions on V imply the sheaves of coinvariants are coherent on moduli spaces of pointed stable curves without any assumption of semisimplicity.

math.AG↗
arXiv2022

Transcendental splitting fields of division algebras

We examine when division algebras can share common splitting fields of certain types. In particular, we show that one can find fields for which one has infinitely many Brauer classes of the same index and period at least 3, all nonisomorphic and having the same set of finite splitting fields as well as the same splitting fields of transcendence degree $1$ and genus at most $1$. On the other hand, we show that one fixes any division algebra over a field, then any division algebras sharing the same splitting fields of transcendence degree at most 3 must generate the same cyclic subgroup of the Brauer group. In particular, there are only a finite number of such division algebras. We also show that a similar finiteness statement holds for splitting fields of transcendence degree at most $2$.

math.RA↗
arXiv2022

Bounding cohomology classes over semiglobal fields

We provide a uniform bound for the index of cohomology classes in $H^i(F, \mu_\ell^{\otimes i-1})$ when $F$ is a semiglobal field (i.e., a one-variable function field over a complete discretely valued field $K$). The bound is given in terms of the analogous data for the residue field of $K$ and its finitely generated extensions of transcendence degree at most one. We also obtain analogous bounds for collections of cohomology classes. Our results provide recursive formulas for function fields over higher rank complete discretely valued fields, and explicit bounds in some cases when the information on the residue field is known. In the process, we prove a splitting result for cohomology classes of degree 3 in the context of surfaces over finite fields.

math.NT↗
arXiv2021

Local-global principles for constant reductive groups over semi-global fields

We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; i.e., over one variable function fields over complete discretely valued fields. We provide conditions on the group and the semiglobal field under which the local-global principle holds, and we compute the obstruction to the local-global principle in certain classes of examples. Using our description of the obstruction, we give the first example of a semisimple simply connected group over a semi-global field where the local-global principle fails. Our methods include patching and R-equivalence.

math.AG↗
arXiv2020

Local-global principles for tori over arithmetic curves

In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the underlying discrete valuation ring, the obstruction to a local-global principle with respect to discrete valuations can be computed using methods coming from patching. We give a sufficient condition for the vanishing of the obstruction, as well as examples were the obstruction is nontrivial or even infinite. A major tool is the notion of a flasque resolution of a torus.

math.AG↗
arXiv2020

A Tannakian approach to patching

We use Tannakian methods to show that patching for coherent sheaves implies patching for objects in any Noetherian algebraic stack with affine stabilizers. Among other things, this gives a straightforward way to prove patching for torsors under linear algebraic groups, as well as patching for sheaves and torsors on proper algebraic spaces.

math.AG↗
arXiv2019

Period-index bounds for arithmetic threefolds

The standard period-index conjecture for the Brauer group of a field of transcendence degree 2 over a $p$-adic field predicts that the index divides the cube of the period. Using Gabber's theory of prime-to-$\ell$ alterations and the deformation theory of twisted sheaves, we prove that the index divides the fourth power of the period for every Brauer class whose period is prime to $6p$, giving the first uniform period-index bounds over such fields.

math.AG↗
arXiv2018

Local-global Galois theory of arithmetic function fields

We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a function field. In this context we also study algebraic versions of van Kampen's theorem, describing the global absolute Galois group as a pushout of local absolute Galois groups.

math.RA↗
arXiv2018

Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves

We describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n >= 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. We do not obtain a working protocol because of a missing step that is currently an open mathematical problem. What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian variety. Our framework builds a cryptographic invariant map, which is a new primitive closely related to a cryptographic multilinear map, but whose range does not necessarily have a group structure. Nevertheless, we show that a cryptographic invariant map can be used to build several cryptographic primitives, including NIKE, that were previously constructed from multilinear maps and indistinguishability obfuscation.

cs.CR↗
arXiv2018

Refinements to patching and applications to field invariants

We introduce a notion of refinements in the context of patching, in order to obtain new results about local-global principles and field invariants in the context of quadratic forms and central simple algebras. The fields we consider are finite extensions of the fraction fields of two-dimensional complete domains that need not be local. Our results in particular give the u-invariant and period-index bound for these fields, as consequences of more general abstract results.

math.RA↗