Stefan Schroeer
Using Maruyama's theory of elementary transformations, I show that the Brauer group surjects onto the cohomological Brauer group for separated geometrically normal algebraic surfaces. As an application, I infer the existence of nonfree vector bundles on proper normal algebraic surfaces.
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We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if $A$ and $A'$ are Morita equivalent Azumaya algebras of degree $r$ then $2r$ divides $c_2(A) - c_2(A')$. In particular this implies that if $A$ is an Azumaya algebra on a K3 surface and $c_2(A)$ is within $2r$ of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non empty, is a proper algebraic space.
We use the definition of differential operators on noncommutative rings given by V.Lunts and A.Rosenberg to find the differential operators on Azumaya algebras and the Heisenberg algebras.
Pinching Azumaya algebras
本文证明在有限多个闭点处做有限修改的概形态射下，上同调 Brauer 类若拉回后可由 Azumaya 代数表示，则原类也可表示，并将 Ferrand 的捏合结果推广到 Azumaya 代数。
Using formal-local methods, we prove that a separated and normal tame Artin surface has the resolution property. By proving that normal tame Artin stacks can be rigidified, we ultimately reduce our analysis to establishing the existence of Azumaya algebras. Our construction passes through the case of tame Artin gerbes, tame Artin curves, and algebraic space surfaces, each of which we establish independently.
In this short note we want to give a definition of a generalized Deligne pairing for modules over an Azumaya algebra on an arithmetic surface $X$. We do this by defining Hermitian metrics on the Azumaya algebra and on the modules in question. Then we go on and define the determinant of the cohomology for a pair of modules over an Azumaya algebra. Using this we give the definition of a generalized Deligne pairing and study some of its properties.
A note on Azumaya algebras and one-forms
本文证明若光滑簇X存在非闭的整体一次微分形式，则其Frobenius扭转X'上由晶体微分算子诱导的非分裂Azumaya代数在X'的某个一次覆盖上仍不分裂，回答了Sasha Petrov的问题。
We show that, in general, over a regular integral noetherian affine scheme X of dimension at least 6, there exist Brauer classes on X for which the associated division algebras over the generic point have no Azumaya maximal orders over X. Despite the algebraic nature of the result, our proof relies on the topology of classifying spaces of algebraic groups.
We compute arithmetic Chern classes of sheaves on an arithmetic surface X associated to a Hermitian Azumaya algebra.
Let $A$ be a sheaf of Azumaya algebras over a Noetherian base $S$. In this paper we describe using generalized Severi-Brauer varieties, a quasi-projective moduli space parametrizing sheaves of ètale subalgebras of $A$. In the case that $S$ is the spectrum of a field, we study the geometry of this moduli space, and show that in certain cases it is a rational variety.
For a smooth toric variety X over a field of positive characteristic, a T-equivariant étale cover Y \rightarrow T^*X^{(1)} trivializing the sheaf of crystalline differential operators on X is constructed. This trivialization is used to show that the sheaf of differential operators is a trivial Azumaya algebra along the fibers of the moment map. This result is then extended to certain Azumaya algebras on hypertoric varieties, whose global sections are central reductions of the hypertoric enveloping algebra in positive characteristic. A criteria for a derived Beilinson-Bernstein localization the…
Azumaya Algebras and Obstructions to Quadratic Pairs over a Scheme
本文研究了基概形S上带对合的Azumaya代数的二次对，描述了带正交对合的Azumaya代数承认二次对的同调阻碍，并构造了非平凡阻碍的显式例子。
Let $A$ be a ring equipped with a derivation $δ$. We study differential Azumaya $A$ algebras, that is, Azumaya $A$ algebras equipped with a derivation that extends $δ$. We calculate the differential automorphism group of the trivial differential algebra, $M_{n}(A)$ with coordinatewise differentiation. We introduce the $δ$-flat Grothendieck topology to show that any differential Azumaya $A$ algebra is locally isomorphic to a trivial one and then construct, as in the non-differential setting, the embedding of the differential Brauer group into $H^{2}(A_{δ-pl},G_{m,δ}) $. We conclude by showing t…
We study moduli spaces of maximal orders in a ramified division algebra over the function field of a smooth projective surface. As in the case of moduli of stable commutative surfaces, we show that there is a Kollár-type condition giving a better moduli problem with the same geometric points: the stack of blt Azumaya algebras. One virtue of this refined moduli problem is that it admits a compactification with a virtual fundamental class.
We describe the derived Picard group of an Azumaya algebra A on an affine scheme X in terms of global sections of the constant sheaf of integers on X, the Picard group of X, and the stabilizer of the Brauer class of A under the action of Aut(X). In particular, we find that the derived Picard group of an Azumaya algebra is generally not isomorphic to that of the underlying scheme. In the case of the trivial Azumaya algebra, our result refines Yekutieli's description of the derived Picard group of a commutative algebra. We also get, as a corollary, an alternate proof of a result of Antieau which…
On deformations of Azumaya algebras with quadratic pair
本文研究带二次对的Azumaya代数的形变理论，构造了切-阻碍理论，证明在特定条件下其形变可归结为底层Azumaya代数的形变，并在特征2情形下构造了有阻碍形变的反例，同时引入了一个介于强弱阻碍之间的新中间阻碍。
Let A be an Azumaya algebra over a smooth projective variety X or more generally, a torsion free coherent sheaf of algebras over X whose generic fiber is a central simple algebra. We show that generically simple torsion free A-module sheaves have a projective coarse moduli scheme; it is smooth and even symplectic if X is an abelian or K3 surface and A is Azumaya. We explain a relation to the classical theory of the Brandt groupoid.
Let X be a cubic surface over a local number field k. Given an Azumaya algebra on X, we describe the local evaluation map X(k) -> Q/Z in two cases, showing a sharp dependence on the geometry of the reduction of X. We show that a suitably generic cubic surface over a number field, whose reduction at some prime is a cone, has no Brauer-Manin obstruction. This extends results of Colliot-Thélène, Kanevsky and Sansuc.
Group schemes over LG-rings and applications to cancellation theorems and Azumaya algebras
本文证明LG-环上约化群概形的极大环面存在性与抛物子群共轭性，推广SGA3半局部环结果，并应用于建立埃尔米特与二次型消去定理及Azumaya代数Brauer类的唯一代表与分解。
We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson-Lichtenbaum conjecture, we apply our results to show the vanishing of SK_2(A) for a central simple algebra A of square-free index.
Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra $A$ over a ring represents a $2$-torsion class in the Brauer group if and only if there is an algebra $A'$ in the Brauer class of $A$ admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose $A'$ such that $\mathrm{deg}\, A'=2\mathrm{deg}\, A$. We show that $2\mathrm{deg}\, A$ is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra $A$ of degree $4$ and period $2$ such that the degree of any algebra $A'$ in the Brauer class of $A$ admitt…
In this note we prove that the moduli space of torsion-free modules of rank one over an Azumaya algebra on a K3-surface is an irreducible symplectic variety deformation equivalent to a Hilbert scheme of points on the K3-surface.
In arXiv:1606.07885, many different Grothendieck topologies were introduced on the category of Azumaya algebras. Here we give a classification in terms of sets of supernatural numbers. Then we discuss the associated categories of sheaves and their topos-theoretic points, which are related to UHF-algebras. The sheaf toposes that correspond to a single supernatural number have an alternative description, involving actions of the associated projective general linear group.
We lay down an elementary yet fundamental lemma concerning a finite algebraicness property of a smooth map from an Azumaya/matrix manifold with a fundamental module to a smooth manifold. This gives us a starting point to build a synthetic (synonymously, $C^{\infty}$-algebraic) symplectic geometry and calibrated geometry that are both tailored to and guided by D-brane phenomena in string theory and along the line of our previous works D(11.1) (arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]).
We develop Azumaya geometry, which is an extension of classical affine geometry to the world of Azumaya algebras, and package the information contained in all quotient stacks $[\mathrm{rep}_n R\,/\,\mathrm{PGL}_n]$ into a presheaf $\mathrm{Rep}_R$ on it. We show that the classical étale and Zariski topologies extend to Grothendieck topologies on Azumaya geometry in uncountably many ways, and prove that $\mathrm{Rep}_R$ is a sheaf for all of them. The restriction to a specific Azumaya algebra $A$ with center $C$ gives us a sheaf in the étale topology which is represented by an affine $C$-scheme…
We consider the general circumstance of an Azumaya algebra $A$ of degree $n$ over a locally ringed topos $(\mathbf{X}, {\mathcal{O}}_{\mathbf{ X}})$ where the latter carries a (possibly trivial) involution, denoted $λ$. This generalizes the usual notion of involutions of Azumaya algebras over schemes with involution, which in turn generalizes the notion of involutions of central simple algebras. We provide a criterion to determine whether two Azumaya algebras with involutions extending $λ$ are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the questi…
Localizing motives of Azumaya algebras
本文证明 qcqs 概形的导出 Brauer 群到其 $X$-线性动机 Picard 群的自然映射为单射，扩展了已有相关结果。
Azumaya algebras over unramified extensions of function fields
本文利用Milnor-Quillen $K$-群解释函数域的étale上同调挠部分，证明了在范数映射的温和条件下，光滑簇上存在可由Azumaya代数表示的上同调Brauer类，从而部分回答了Grothendieck的问题。
Azumaya algebras with involution and classical semisimple group schemes
本文通过平展下降证明了一个从带对合Azumaya代数到经典伴随群概型的函子等价，从而在一般概型上分类了经典型伴随群概型，并推广了域上已知结果，且给出了特殊化定理与积分模型唯一性等应用。
We construct a relative version of topological $K$-theory of dg categories over an arbitrary quasi-compact, quasi-separated $\mathbb{C}$-scheme $X$. This has as input a $\text{Perf}(X)$-linear stable $\infty$-category and output a sheaf of spectra on $X(\mathbb{C})$, the space of complex points of $X$. We then characterize the values of this functor on inputs of the form $Mod_{A}^ω$, for $A$ a derived Azumaya algebra over $X$. In such cases we show that this coincides with the $α$-twisted topological $K$-theory of $X(\mathbb{C})$ for some appropriately defined twist of $K$-theory. We use this …
We introduce a notion of derived Azumaya's algebras over rings and schemes. We prove that any such algebra $B$ on a scheme $X$ provides a class $ϕ(B)$ in $H^{1}_{et}(X,\mathbb{Z})\times H^{2}_{et}(X,\mathbb{G}_{m})$. We prove that for $X$ a quasi-compact and quasi-separated scheme $ϕ$ defines a bijective correspondence, and in particular that any class in $H^{2}_{et}(X,\mathbb{G}_{m})$, torsion or not, can be represented by a derived Azumaya's algebra on $X$. Our result is a consequence of a more general theorem about the existence of compact generators in \emph{twisted derived categories, wit…
Recent work of Chinburg, Reid, and Stover has shown that certain arithmetic and algebro-geometric properties of the character variety of a hyperbolic knot complement in the 3-sphere $M=S^3\setminus K$ yields topological and number theoretic information about Dehn fillings of M. Specifically, they show how the study of a certain extension problem for quaternion Azumaya algebras is related to topological invariants associated to these fillings. In this paper, we extend their work to the setting of hyperbolic once punctured torus bundles. Along the way, we exhibit new phenomenon in the relevant e…
Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry
本文研究具有正交对合的 Azumaya 代数，证明若存在非正常等距则 Brauer 类平凡，并指出该假设不足以推出矩阵代数结构。
Let $M$ be a compact 3-manifold and $Γ=π_1(M)$. Work of Thurston and Culler--Shalen established the $\mathrm{SL}_2(\mathbb{C})$ character variety $X(Γ)$ as fundamental tool in the study of the geometry and topology of $M$. This is particularly the case when $M$ is the exterior of a hyperbolic knot $K$ in $S^3$. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of $X(Γ)$, as well as distinguished points on the canonical component, when $Γ$ is a knot group. In…
Given an Azumaya algebra with involution $(A,σ)$ over a commutative ring $R$ and some auxiliary data, we construct an $8$-periodic chain complex involving the Witt groups of $(A,σ)$ and other algebras with involution, and prove it is exact when $R$ is semilocal. When $R$ is a field, this recovers an $8$-periodic exact sequence of Witt groups of Grenier-Boley and Mahmoudi, which in turn generalizes exact sequences of Parimala--Sridharan--Suresh and Lewis. We apply this result in several ways: We establish the Grothendieck--Serre conjecture on principal homogeneous bundles and the local purity c…
Let $(A,σ)$ be an Azumaya algebra with involution over a regular ring $R$. We prove that the Gersten-Witt complex of $(A,σ)$ defined by Gille is isomorphic to the Gersten-Witt complex of $(A,σ)$ defined by Bayer-Fluckiger, Parimala and the author. Advantages of both constructions are used to show that the Gersten-Witt complex is exact when $\dim R\leq 3$, $\mathrm{ind}\, A\leq 2$ and $σ$ is orthogonal or symplectic. This means that the Grothendieck-Serre conjecture holds for the group $R$-scheme of $σ$-unitary elements in $A$ under the same hypotheses; $R$ is not required to contain a field.
We attach to any commutative ring R a subgroup of the Brauer group of R, called the Brauer-Galois group of R. Its elements are the classes of the Azumaya R-algebras which can be represented, via Brauer equivalence, by a Galois extension of R. We compute this group for some particular commutative fields.
A class of noncommutative spaces, named `soft noncommutative schemes via toric geometry', are constructed and the mathematical model for (dynamical/nonsolitonic, complex algebraic) D-branes on such a noncommutative space, following arXiv:0709.1515 [math.AG] (D(1)), is given. Any algebraic Calabi-Yau space that arises from a complete intersection in a smooth toric variety can embed as a commutative closed subscheme of some soft noncommutative scheme. Along the study, the notion of `soft noncommutative toric schemes' associated to a (simplicial, maximal cone of index $1$) fan, `invertible sheave…
Let k be a base commutative ring, R a commutative ring of coefficients, X a quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of rank r, and Hmo(R) the category of noncommutative motives with R-coefficients. Assume that 1/r belongs to R. Under this assumption, we prove that the noncommutative motives with R-coefficients of X and A are isomorphic. As an application, we show that all the R-linear additive invariants of X and A are exactly the same. Examples include (nonconnective) algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homolo…
Using twisted and formal-local methods, we prove that every separated algebraic space which is the (open/flat) pushout of affine schemes has enough Azumaya algebras. As a corollary we show that, under mild hypothesis, every cohomological Brauer class is representable by an Azumaya algebra away from a closed subset of codimension $\geq 3$, generalizing an early result of Grothendieck. Next, we show that $\mathrm{Br}(X)=\mathrm{Br}'(X)$ when $X$ is an algebraic space obtained from a quasi-projective scheme by contracting a curve. This is the first method of descending an Azumaya algebra along a …
Spaces of Generators for Azumaya Algebras with Unitary Involution
本文研究带酉对合的Azumaya代数的生成元空间，确定了特定带对合矩阵代数对应的非生成元r元组闭子簇的维数与不可约分支，推导了有限生成K-环上带酉对合Azumaya代数的生成元个数上下界，且方法可推广至满足弱假设的K代数。
We study the nontrivial elements in the Brauer group of a bielliptic surface and show that they can be realized as Azumaya algebras with a simple structure at the generic point of the surface. We go on to study some properties of the noncommutative Picard scheme associated to such an Azumaya algebra.
$K(1)$-local $K$-theory of Azumaya algebras
本文通过研究 Azumaya 代数的 $K(1)$-局部 $K$-理论，利用 Brauer 群计算严格 Picard 谱，并在特定条件下证明了 Künneth 公式。
Rational points in coarse moduli spaces and twisted representations
本文在一般设定下研究Azumaya代数中结合代数的Schur表示的模空间与模叠，阐明扭表示构成的模叠的结构，并通过非阿贝尔上同调解释经典粗模空间有理点的几何起源。
We describe an Azumaya algebra on the resolution of singularities of the double cover of a plane ramified along a nodal sextic associated to a non generic cubic fourfold containing a plane. We show that the derived category of such a resolution, twisted by the Azumaya algebra, is equivalent to the Kuznetsov component in the semiorthogonal decomposition of the derived category of the cubic fourfold.
We investigate a notion of Azumaya algebras in the context of structured ring spectra and give a definition of Brauer groups. We investigate their Galois theoretic properties, and discuss examples of Azumaya algebras arising from Galois descent and cyclic algebras. We construct examples that are related to topological Hochschild cohomology of group ring spectra and we present a K(n)-local variant of the notion of Brauer groups.
We prove that quantized multiplicative quiver varieties and quantum character varieties define sheaves of Azumaya algebras over the corresponding classical moduli spaces, and we prove that the Azumaya locus of the Kauffman bracket skein algebras contains the smooth locus, proving a strong form of the Unicity Conjecture of Bonahon and Wong. The proofs exploit a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as it arises in each setting. We therefore introduce the concepts of Frobenius quantum moment maps and their Hamiltonian reduction, and of …
Over a normal base scheme, we prove the generalized Theorem of the Cube for 1-motives and that a torsion class of the group H^2_ét(M,G_m)$ of a 1-motive M, whose pull-back via the unit section is zero, comes from an Azumaya algebra. In particular, we deduce that over an algebraically closed field of characteristic zero, all classes of H^2_ét(M,G_m) come from Azumaya algebras.
We give a geometrical criterion to determine when a quaternion algebra over the function field of a stable elliptic surface X is an Azumaya algebra over X.
Using depth of coherent sheaves on noetherian algebraic stacks, we construct non-Azumaya maximal orders in unramified central simple algebras over schemes of dimension at least $3$.
In this continuation of [L-Y1], [L-L-S-Y], [L-Y2], and [L-Y3] (arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], arXiv:0901.0342 [math.AG], arXiv:0907.0268 [math.AG]), we study D-branes in a target-space with a fixed $B$-field background $(Y,α_B)$ along the line of the Polchinski-Grothendieck Ansatz, explained in [L-Y1] and further extended in the current work. We focus first on the gauge-field-twist effect of $B$-field to the Chan-Paton module on D-branes. Basic properties of the moduli space of D-branes, as morphisms from Azumaya schemes with a twisted fundamental module to $(Y,α_B)$, ar…
We consider q-deformations of multiplicative hypertoric varieties, for q a non-zero element of an algebraically closed field of characteristic 0. We construct an algebra Dq of q-difference operators as a Heisenberg double in a braided monoidal category. We then focus on the case where q is specialized to a root of unity. In this setting, we use Dq to construct an Azumaya algebra on an l-twist of the multiplicative hypertoric variety, before showing that this algebra splits over the fibers of both the moment and resolution maps. Finally, we sketch a derived localization theorem for these Azumay…
In this paper, we study Azumaya algebras and Brauer groups in derived algebraic geometry. We establish various fundamental facts about Brauer groups in this setting, and we provide a computational tool, which we use to compute the Brauer group in several examples. In particular, we show that the Brauer group of the sphere spectrum vanishes, and we use this to prove two uniqueness theorems for the stable homotopy category. Our key technical results include the local geometricity, in the sense of Artin n-stacks, of the moduli space of perfect modules over a smooth and proper algebra, the étale l…
We construct quantizations of multiplicative hypertoric varieties using an algebra of q-difference operators on affine space, where q is a root of unity in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite étale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework.
Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras
研究通过几何框架探讨量子纠缠在阿祖马亚代数扭曲投影态空间中的表现，揭示子系统结构的全局化与拓扑约束的关系。
Moduli, Deformations and Algebraic K-Theory of Graded PI-Algebras
本文通过PI代数sheaves的范畴研究非交换几何，引入新的非交换几何结构，并证明了Morita-equivariant Betti/Riemann--Hilbert定理。
We prove a localization theorem for the type A rational Cherednik algebra H_c=H_{1,c} over an algebraic closure of the finite field F_p. In the most interesting special case where the parameter c takes values in F_p, we construct an Azumaya algebra A_c on Hilb^n, the Hilbert scheme of n points in the plane, such that the algebra of global sections of A_c is isomorphic to H_c. Our localisation theorem provides an equivalence between the bounded derived categories of H_c-modules and sheaves of coherent A_c-modules on the Hilbert scheme, respectively. Furthermore, we show that the Azumaya algebra…
In this sequel to works D(11.1) (arXiv:1406.0929 [math.DG]), D(11.2) (arXiv:1412.0771 [hep-th]), and D(11.3.1) (arXiv:1508.02347 [math.DG]), we re-examine --- and reformulate when in need --- several basic notions in super $C^{\infty}$-algebraic geometry as guided by the mathematical formulation of Ramond-Neveu-Schwarz fermionic strings and of Green-Schwarz fermionic strings from the viewpoint of Grothendieck on Algebraic Geometry. Two theorems that are the super counterpart of Theorem~3.1.1 and Theorem~3.2.1 of D(11.3.1) are proved. They unify the notion of "smooth maps from an Azumaya/matrix…
We prove that Toen's secondary Grothendieck ring is isomorphic to the Grothendieck ring of smooth proper pretriangulated dg categories previously introduced by Bondal, Larsen and Lunts. Along the way, we show that those short exact sequences of dg categories in which the first term is smooth proper and the second term is proper are necessarily split. As an application, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injective properties: in the case of a commutative ring of characteristic zero, it distinguishes between dg Azuma…
Outer Type Severi-Brauer Schemes
本文引入了降低的$\mathcal{O}$-模旗的概念，以定义与一般线性群$\mathbf{GL}_{1,\mathcal{A}}$的广义塞弗里-布拉乌尔方案相关的理想旗sheaf。进一步将该概念扩展到外类型$A_n$情况，并定义了与单位群相关的理想旗sheaf。当群适当分裂时，这些与向量丛或带有希伯特形式的向量丛的子模旗相关。还定义了与关联代数相关的idempotent元组sheaf，其与广义塞弗里-布拉乌尔方案的极子群对sheaf同构。展示了如何通过这些旗sheaf定义从极子群到Dynkin方案的类型映射。回顾了与阿祖马亚代数$\mathcal{A}$相关的塞弗里-布拉乌尔方案与该类型映射特定纤维的同构性，并将其推广到外情况以定义外塞弗里-布拉乌尔方案。提供了Quillen构造的新方法，从塞弗里-布拉乌尔方案产生阿祖马亚代数，并展示了外塞弗里-布拉乌尔方案的外Quillen构造也存在，能产生带有单位元的代数。
A natural question is to determine which algebraic stacks are qoutient stacks. In this paper we give some partial answers and relate it to the old question of whether, for a scheme X, the natural map from the Brauer goup (equivalence classes of Azumaya algebras) to the cohomological Brauer group (the torsion subgroup of $H^2(X,{\mathbb G}_m)$ is surjective.
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