Related papers: A note on secondary K-theory

Making use of topological periodic cyclic homology, we extend Grothendieck's standard conjectures of type C and D (with respect to crystalline cohomology theory) from smooth projective schemes to smooth proper dg categories in the sense of…

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…

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…

In this paper we describe how Grothendieck groups of coherent sheaves and locally free sheaves can be used to describe type II D-branes, in the case that all D-branes are wrapped on complex varieties and all connections are holomorphic. Our…

We prove that geometric intersections between Weinstein handles induce algebraic relations in the wrapped Fukaya category, which we use to study the Grothendieck group. We produce a surjective map from middle-dimensional singular cohomology…

Let $\mc G$ be a reductive group over an algebraically closed field of characteristic $p>0$. We study homogeneous $\mc G$-spaces that are induced from the $G\times G$-space $G$, $G$ a suitable reductive group, along a parabolic subgroup of…

We study basic properties of the category of smooth representations of a p-adic group G with coefficients in any commutative ring R in which p is invertible. Our main purpose is to prove that Hecke algebras are noetherian whenever R is ; a…

Let $R$ be a commutative ring. An Azumaya coring consists of a couple $(S,\Cc)$, with $S$ a faithfully flat commutative $R$-algebra, and an $S$-coring $\Cc$ satisfying certain properties. If $S$ is faithfully projective, then the dual of…

In the present paper we investigate the question about the injectivity of the map F(R) --> F(K) induced by the canonical inclusion of a local regular ring of geometric type R to its field of fractions K for a homotopy invariant functor F…

We define the derived category of a concrete category in a way which extends the usual definition of the derived category of a ring, and we prove that the bounded-below derived category of $\Spec \mathbb{M}_0$ (an approximation, used by…

We prove some new cases of the Grothendieck-Serre conjecture for classical groups. This is based on a new construction of the Gersten-Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, with explicit…

Let k be an infinite field. Let R be the semi-local ring of a finite family of closed points on a k-smooth affine irreducible variety, let K be the fraction field of R, and let G be a reductive simple simply connected R-group scheme…

We construct an equivariant coarse homology theory arising from the algebraic $K$-theory of spherical group rings and use this theory to derive split injectivity results for associated assembly maps. On the way, we prove that the…

Let T be a torus (not assumed to be split) over a field F, and denote by $_n{H^{2}_{et}(X,Gm)}$ the subgroup of elements of exponent dividing n in the cohomological Brauer group of a scheme X over the field F. We provide conditions on X and…

The Grothendieck-Serre conjecture predicts that every generically trivial torsor under a reductive group $G$ over a regular semilocal ring $R$ is trivial. We establish this for unramified $R$ granted that $G^{\mathrm{ad}}$ is totally…

Let k be a field, X a smooth, projective k-variety. If X is geometrically rational, there is an injective map from the quotient of Brauer groups Br(X)/Br(k) into the first Galois cohomology group of the lattice given by the geometric Picard…

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part…

We study commutative algebras with Gorenstein duality, i.e. algebras $A$ equipped with a non-degenerate bilinear pairing such that $\langle ac,b\rangle=\langle a,bc\rangle$ for any $a,b,c\in A$. If an algebra $A$ is Artinian, such pairing…

Let R be an unramified regular local ring of mixed characteristic, D an Azumaya R-algebra, K the fraction field of R, Nrd the reduced norm homomorphism for the Azumaya R-algebra D. Let a be a unit in R. It is proved the following: suppose…

In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck type theorem is obtained which states that there is a canonical correspondence between…