Related papers: Associative Schemes

The lower central series invariants M_k of an associative algebra A are the two-sided ideals generated by k-fold iterated commutators; the M_k provide a filtration of A. We study the relationship between the geometry of X = Spec A_ab and…

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let $Y\ra X$ be a covariant morphism…

This paper gives insight into intriguing connections between two apparently unrelated theories: the theory of skein modules of 3-manifolds and the theory of representations of groups into special linear groups of 2 by 2 matrices. Let R be a…

Let K be the field of fractions of a Henselian discrete valuation ring O_K. Let X_K/K be a smooth proper geometrically connected scheme admitting a regular model X/O_K. We show that the index \delta(X_K/K) of X_K/K can be explicitly…

We construct a quasi-coherent sheaf of associative algebras which controls a category of $AV$-modules over a smooth quasi-projective variety. We establish a local structure theorem, proving that in \'etale charts these associative algebras…

We propose a simple approach to formal deformations of associative algebras. It exploits the machinery of multiplicative coresolutions of an associative algebra A in the category of A-bimodules. Specifically, we show that certain…

In this this paper we introduce entanglement among the points in a non-commutative scheme, in addition to the tangent directions. A diagram of $A$-modules is a pair $\uc=(|\uc|,\Gamma)$ where $|\uc|={V_1,...,V_r}$ is a set of $A$-modules,…

Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide…

Let k be a commutative ring with unit. We endow the categories of filtered complexes and of bicomplexes of k-modules, with cofibrantly generated model structures, where the class of weak equivalences is given by those morphisms inducing a…

In Kapranov, M. {\it Noncommutative geometry based on commutator expansions,} J. reine angew. Math {\bf 505} (1998) 73-118, a theory of noncommutative algebraic varieties was proposed. Here we prove a structure theorem for the…

Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R = K[[h]]). We consider associative (resp. Poisson) R-deformations of the structure sheaf O_X. The set of…

Let $X/K$ be a variety over a field, and $A/K$ an abelian variety. A regular homomorphism to $A$ (in codimension $i$) induces, for every smooth geometrically connected pointed $K$-scheme $(T,t_0)$ and every cycle class $Z \in CH^i(T\times…

In this paper, we introduce an algebraic-topological invariant for commutative pm-rings, termed the spectral fundamental group, which is denoted by $\pi_{k}^{alg}(A)$. This group is defined via homotopy classes of loops within the space of…

We study families of algebraic varieties parametrized by topological spaces and generalize some classical results such as Hilbert Nullstellensatz and primary decomposition of commutative rings. We show that there is an equivalence between…

We prove that for every reductive algebraic group $H$ with centre of positive dimension and every integer $K$ there is a smooth and projective variety $X$ and an algebraic $H$-torsor $P \to X$ such that the classifying map $X \to \Bclass H$…

Let k be a perfect field and A a finite dimensional k-algebra of finite global dimension (e.g. the path algebra of a finite quiver without oriented cycles). Making use of the recent theory of noncommutative motives, we prove that the value…

We review the theory of non-commutative deformations of sheaves and describe a versal deformation by using an A-infinity algebra and the change of differentials of an injective resolution. We give some explicit non-trivial examples.

Let $\k$ be an algebraically closed field, let $\A$ be a finite dimensional $\k$-algebra and let $V$ be a $\A$-module with stable endomorphism ring isomorphic to $\k$. If $\A$ is self-injective then $V$ has a universal deformation ring…

Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-like versions of usual deformations. We prove that…

For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism \rho_X: C_X --> \pi_1^\ab(X), which is surjective and whose kernel is the connected component of the identity. The (topological)…