arXiv subjects

Arvid Siqveland

Publications and source records attributed to Arvid Siqveland.

14 recordsLinked to original sources

arXiv2026

Sites and Grothendieck Topologies, Sites and Sheaves

We give the minimum of category theory necessary for understanding the definition and applications of Grothendieck topos. We state the basic properties of sites and sheaves, and give applications to the theory of moduli. We prove that for categories $\mathbf C$ with explicit stated properties, we can construct a category of schemes of objects in $\mathbf C$ which is a proper category rather than a $2$-category as in the case of stacks.

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arXiv2026

Connections on Associative Varieties

We have embedded the category of smooth real $n$-manifolds faithfully into the category of associative varieties, \cite{S2603}. Many constructions on smooth manifolds can then be generalized to associative varieties. Here we will generalize the well known interpretation of connections on manifolds to connections on associative varieties. This eventually proves the existence of geodesics in an associative variety.

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arXiv2026

Riemannian Geometry on Associative Varieties

We prove that the classical algebraic varieties over algebraically closed fields can be defined over arbitrary fields $k.$ Then we prove that for associative algebras $A$, there exist local representing objects $A_M$ for simple modules $M.$ Replacing the localization in maximal ideals in the commutative situation with the local representations in simple modules in the associative, we define an associative generalization of varieties. Now we realize that replacing $\mathbb R[x_1,\dots,x_n]=\mathbb R[n]$ with $C^\infty(\mathbb R^n),$ we can do differential geometry for associative $\mathbb R[n]$-algebras. This says that we can define a Riemannian geometry on associative varieties. This gives us the definition of connections and algebraic geodesic curves, introducing real geometry into associative algebras.

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arXiv2025

Associative Schemes and Subschemes

In the preprint arXiv:2511.07900 we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\rightarrow B$ we define the contraction of a simple $B$-module to $A.$ Then we define the set of aprime right $A$-modules ${\rm aSpec} A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, ${\rm aSpec} A = {\rm Spec} A$. and we define a topology on ${\rm aSpec} A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\mathcal O_X$ on ${\rm aSpec} A,$ agreeing with the usual sheaf of rings on ${\rm Spec} A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\subseteq{\rm aSpec}$. In particular, this proves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\mathbb R\subseteq\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.

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arXiv2025

Localization in Associative Rings

In arXiv:2511.04191 we constructed schemes of objects in small categories which contained a set of basepoints with local representing (localizing) objects. Here we prove that the category $\cat{Rings}$ of associative rings with unit has a certain set of basepoints for which localizing rings exist. We take the set of base points $B$ to be the set of rings on the form $\enm_{\mathbb Z}(M)$ where $M$ is a simple right $A$-module for some associative ring $A.$ The set of base-points in the associative ring $A$ is defined as $\pts_B(A)=\{\mor_{\cat{Rings}}(A,\enm_{\mathbb Z}(M))\}.$ For any finite subset $M\subseteq\pts_B(A)$ we prove that the localizing ring $A_M$ exists. and so the construction from arXiv:2511.04191 gives a definition of schemes of associative algebras. Defining a topology on $\pts_B(A)$ such that when $A$ is commutative it is the Zariski topology, we get the ordinary definition of schemes when we consider the category of commutative rings. This article is in line with the philosophy that a scheme is a moduli of its base-points.

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arXiv2025

Schemes of Objects in Abelian Categories

In the article Categorical Construction of Schemes, arXiv:2511.03433 we gave a natural definition of ordinary schemes based on the fact that the localization of a ring in a maximal ideal is a local representation of the corresponding function field. In this text, we replace the category of rings with a general locally small category $\cat C$, we consider a subcategory $\cat B\subset C$ of base-points, and assume that each $X\in\ob\cat C$ that contains $P\in\ob\cat B,$ i.e. there is a morphism $P\rightarrow X,$ there exists a local representing object $X_P.$ Assuming that coproducts exists, we can use the construction of ordinary schemes to construct schemes of objects in any such category.

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arXiv2025

Categorical construction of Schemes

In the authors book, Associative Algebraic Geometry, 2023, and the following article Shemes of Associative Algebras,\\ https://doi.org/10.48550/arXiv.2410.17703,2024, we use an algebraization of the semi-local formal moduli of simple modules to construct associative schemes. Here, we consider a commutative ring for which we can use the localization in maximal ideals as local moduli. This gives a categorical definition of schemes that is equivalent to the definition in Hartshorne's book, Algebraic Geometry, 1977. The definition includes a construction of the sheaf associated to a presheaf using projective limits, and this makes the basic results in scheme theory more natural.

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arXiv2024

Schemes of Associative Algebras

We give a definition of associative schemes, schemes of associative rings, over a field $k,$ using the definition of completion of an associative $k$-algebra in a finite set of simple modules. We start by giving a weaker but sufficient definition of ordinary schemes of commutative rings which can be generalized to associative rings.

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arXiv2024

Associative Schemes

We state results from noncommutative deformation theory of modules over an associative $k$-algebra $A,$ $k$ a field, necessary for this work. We define a set of $A$-modules $\operatorname{aSpec}A$ containing the simple modules, whose elements we call spectral, for which there exists a topology where the simple modules are the closed points. Applying results from deformation theory we prove that there exists a sheaf of associative rings $\mathcal O_X$ on the topological space $X=\operatorname{aSpec}A$ giving it the structure of a pointed ringed space. In general, an associative variety $X$ is a ringed space with an open covering $\{U_i=\operatorname{aSpec}{A_i}\}_{i\in I}.$ When $A$ is a commutative $k$-algebra, $\operatorname{aSpec}A\simeq\spec A,$ and so the category $\cat{aVar}_k$ of associative varieties is an extension of the category of varieties $\cat{Var}_k,$ i.e. there exists a faithfully full functor $I:\cat{Var}_k\rightarrow\cat{aVar}_k.$ Our main result says that any associative variety $X$ is $\operatorname{aSpec}(\mathcal O_X(X))$ for the $k$-algebra $\mathcal O_X(X),$ and so any study of varieties can be reduced to the study of the associative algebra $\mathcal O_X(X).$

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arXiv2024

Countably Generated Matrix Algebras

We define the completion of an associative algebra $A$ in a set $M=\{M_1,\dots,M_r\}$ of $r$ right $A$-modules in such a way that if $\mathfrak a\subseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the (right) module $A/\mathfrak a$ is $\hat A^M\simeq \hat A^{\mathfrak a}.$ This works by defining $\hat A^M$ as a formal algebra determined up to a computation in a category called GMMP-algebras. From deformation theory we get that the computation results in a formal algebra which is the prorepresenting hull of the noncommutative deformation functor, and this hull is unique up to isomorphism.

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arXiv2024

Associative Local Function Rings

We prove that for an arbitrary field $k,$ a complete, associative $k^r$-algebra $\hat H$ augmented over $k^r$ has exactly $r$ maximal two-sided ideals and deserves the name $r$-pointed. If $A$ is any $k$-algebra, $M=\{M_i\}_{i=1}^r$ is a family of simple right $A$-modules with a countable $k$-basis, and there is a homomorphism $ρ_A:A\rightarrow\enm_{\hat H}(H\hat{\otimes}_{k^r}(\oplus_{i=1}^r M_i))=:\hat O(M)$ then $\hat O(M)$ is $r$-pointed and $M$ is contained in the set of right simple $\hat O(M)$-modules. Our main result is that the subalgebra generated $ρ_A(A)$ and all $ρ_A(a)^{-1}$ whenever $ρ_A(a)$ is a unit, is a natural substitute for the localization $A(M)$ of the $k$-algebra $A$ in $M$ which only exists when the Ore condition is fulfilled.

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arXiv2019

The algebra of observables in noncommutative deformation theory

We consider the algebra $\mathcal O(\mathsf M)$ of observables and the (formally) versal morphism $η: A \to \mathcal O(\mathsf M)$ defined by the noncommutative deformation functor $\mathsf{Def}_{\mathsf M}$ of a family $\mathsf M = \{ M_1, \dots, M_r \}$ of right modules over an associative $k$-algebra $A$. By the Generalized Burnside Theorem, due to Laudal, $η$ is an isomorphism when $A$ is finite dimensional, $\mathsf M$ is the family of simple $A$-modules, and $k$ is an algebraically closed field. The purpose of this paper is twofold: First, we prove a form of the Generalized Burnside Theorem that is more general, where there is no assumption on the field $k$. Secondly, we prove that the $\mathcal O$-construction is a closure operation when $A$ is any finitely generated $k$-algebra and $\mathsf M$ is any family of finite dimensional $A$-modules, in the sense that $η_B: B \to \mathcal O^B(\mathsf M)$ is an isomorphism when $B = \mathcal O(\mathsf M)$ and $\mathsf M$ is considered as a family of $B$-modules.

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arXiv2012

Deformation of diagrams

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 $\uc=(|\uc|,Γ)$ where $|\uc|={V_1,...,V_r}$ is a set of $A$-modules, and $Γ=\{γ_{ij}(l)\}$ is a set of $A$-module homomorphisms $γ_{ij}(l):V_i\rightarrow V_j$, seen as the 0'th order tangent directions. This concludes the discussion on non-commutative schemes by defining the deformation theory for diagrams, making these the fundamental points of the non-commutative algebraic geometry, which means that the construction of non-commutative schemes is a closure operation. Two simple examples of the theory are given: The space of a line and a point, which is a non-commutative but untangled example, and the space of a line and a point on the line, in which the condition of the point on the line gives an entanglement between the point and the line.

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arXiv2006

Generalized Matric Massey Products for Graded Modules

The theory of generalized matric Massey products has been applied for some time to $A$-modules $M$, $A$ a $k$-algebra. The main application is to compute the local formal moduli $\hat{H}_M$, isomorphic to the local ring of the moduli of $A$-modules. This theory is also generalized to $\mathcal{O}_X$-modules $\mathcal{M}$, $X$ a $k$- scheme. In these notes we consider the definition of generalized Massey products and the relation algebra in any obstruction situation (a differential graded $k$-algebra with certain properties), and prove that this theory applies to the case of graded $R$-modules, $R$ a graded $k$-algebra, $k$ algebraically closed. When the relation algebra is algebraizable, that is the relations are polynomials rather than power series, this gives a combinatorial way to compute open (étale) subsets of the moduli of graded $R$-modules. This also gives a sufficient condition for the corresponding point in the moduli of $\mathcal{O}_{\Proj(R)}$-modules to be singular. The computations are straight forward, algorithmic, and an example on the postulation Hilbert scheme is given.

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