arXiv subjects

Ilia Pirashvili

Publications and source records attributed to Ilia Pirashvili.

16 recordsLinked to original sources

arXiv2026

Deformation Theory of Monoid Schemes II: Precartesian Coextensions of Commutative Monoids

This paper is a continuation of arXiv:2606.17088, where I studied precartesian coextensions of commutative monoids by systems of abelian groups. In this paper, we generalise it to study coextensions by systems of commutative monoids. Among other things, we showcase that this version is able to simultaneously generalise Leech's version of coextensions and Redei's version, also called Schreier coextensions. We show that what going from systems of abelian groups to systems of commutative monoids costs us is quasi-inverses in $\mathsf{Pcoex}(M, \mathcal{L})$. Specifically, instead of a symmetric categorical group, they now only become symmetric monoidal groupoids. Moreover, though not explicitly stated in the body of the paper, another core difference is that for a monoid scheme $X$, regarded as a monoid functor, a precartesian coextension no longer need to be a monoid scheme. Thus, $\mathsf{Pcoex}(X, \mathcal{L})$ is, as stated, ineffective at studying monoid scheme coextensions. The rest of the theory goes through, but requires developing a new cohomological approach.

math.CT↗

arXiv2026

Deformation Theory of Monoid Schemes I

The aim of this paper is to develop a deformation theory of monoid schemes, generalising the approach developed by Grillet. The core idea of this approach is to introduce the notion of a system of abelian groups, as the naive approach to exactness does not work for monoids. We first study the case of monoid sheaves (functors over a poset into the category of monoids) and prove a classification theorem in this setting, showing that the coextensions of a monoid functor with a system of abelian groups is a symmetric categorical group and equivalent to the one obtained by the abelian group homomorphism $[\mathcal{C}^0 \to \mathsf{ker}\partial^1]$, thereby linking with cohomology of certain types of complexes, as expected. We then move towards monoid schemes, which are a type of a monoid sheaf, but where localisations now allow us to develop our most noteworthy result: We show that coextensions can be seen in a natural way as a stack of symmetric categorical groups. We will mention a few mild implications of this, but leave the deeper uses of stack theory in this setting for later papers.

math.CT↗

arXiv2026

n-ary elliptic groups, rings, and primes in arithmetic progressions

I introduced the notion of an elliptic group in [Elliptic groups and rings. Beiträge zur Algebra und Geometrie 66(2), 497-529]. It is a quasi-group based on the tangent-chord law of elliptic curves and thus, becomes an abelian group upon singling out an element. This close proximity to abelian groups is reflected in the theory, and among other things, we can define elliptic rings, which are monoidal objects in elliptic groups. An other way of expressing this is to say that they are commutative monoids with an elliptic group structure that distributes over them. In this paper, we generalise this theory from the binary elliptic group structure to the $n$-ary structure, which we call $n$-ary elliptic groups and $n$-ary elliptic rings. The latter are once again (binary) commutative monoids with an $n$-ary operation that distributes over the monoidal structure in an $n$-ary sense. The key interest of these objects for us is their arithmetic properties, which are surprisingly pleasant. The key result is that Dirichlet's famous theorem on arithmetic progressions becomes simply Euclid's theorem in these $n$-ary rings, at least for progressions of the form $an + 1$. Motivated by the hope to eventually prove this $n$-ary Euclidean theorem purely algebraically using the theory of $n$-ary rings (and thus give an alternative and purely algebraic proof of Dirichlet's theorem), we start by exploring the first arithmetic facts of these objects, including introducing the $n$-ary class group and showing that it indeed captures the unique $n$-ary factorisation. We also obtain a type of Dedekinds theorem for our main $n$-ary ring of interest: $\mathsf{nEl}(\mathbb{Z})$.

math.RA↗

arXiv2025

Grothendieck's theory of fibred categories for monoids

Grothendieck's theory of fibred categories establishes an equivalence between fibred categories and pseudo functors. It plays a major role in algebraic geometry and categorical logic. This paper aims to show that fibrations are also very important in monoid theory. Among other things, we generalise Grothendieck's result slightly and show that there exists an equivalence between prefibrations (also known as Schreier extension in the monoidal world) and lax functors. We also construct two exact sequences which involve various automorphism groups arising from a given fibration. This exact sequence was previously only known for group extensions.

math.CT↗

arXiv2024

On internal categories and crossed objects in the category of monoids

It is a well-known fact that the category $\mathsf{Cat}(\mathbf{C})$ of internal categories in a category $\mathbf{C}$ has a description in terms of crossed modules, when $\mathbf{C}=\mathbf{Gr}$ is the category of groups. The proof of this result heavily uses the fact that any split epimorphism decomposes as a semi-direct product. An equivalent statement does not hold in the category $\mathbf{Mon}$ of monoids. In a previous work on quadratic algebras, I constructed an internal category in the category of monoids, see Section 6. Based on this construction, this paper will introduce the notion of a crossed semi-bimodule and show that it gives rise to an object in $\mathsf{Cat}(\mathbf{Mon})$. I will also relate this new notion to the crossed semi-modules introduced earlier by A. Patchkoria.

math.CT↗

arXiv2023

On the equvialence of colimits and 2-colimits

We compare the colimit and 2-colimit of strict 2-functors in the 2-category of groupoids, over a certain type of posets. These posets are of special importance, as they correspond to coverings of a topological space. The main result of this paper gives conditions on the 2-functor $\mathfrak{F}$, for which $\mathsf{colim}\mathfrak{F}\simeq2\mathsf{colim}\mathfrak{F}$. One can easily see that any 2-functor $\mathfrak{F}$ can be deformed to a 2-functor $\mathfrak{F}'$, which satisfied the conditions of the theorem. At last, we also optimise our conditions, reducing from exponential to polynomial complexity.

math.CT↗

arXiv2022

Elliptic groups and rings

As it is well known, one can define an abelian group on the points of an elliptic curve, using the so called chord-tangent law \cite{dale}, and a chosen point. However, that very chord-tangent law allows us to define a rather more obscure algebraic structure, which we call an elliptic group, on the points of an elliptic curve. In the cases when our curve has a so called flex point (intersection number with the tangent is $3$), the classical abelian group and the elliptic group carry the same information. However, if our curve does not have such a point (which often happens over $\mathbb{Q}$), the abelian group is not enough to recover the elliptic group. The aim of this paper is to study this algebraic structure in more detail, its connections to abelian groups and at the very end even introduce the notion of an elliptic ring (a monoid object in the category of elliptic groups).

math.AG↗

arXiv2020

Idempotents and the points of the topos of M-sets

The aim of this paper is to study the points and localising subcategories of the topos of $M$-sets, for a finite monoid $M$. We show that the points of this topos can be fully classified using the idempotents of $M$. We introduce a topology on the iso-classes of these points, which differs from the classical topology introduced in SGA4. Likewise, the localised subcategories of the topos $M$-sets correspond to the set of all two-sided idempotent Ideals of $M$.

math.CT↗

arXiv2020

Reconstruction theorem for monoid schemes

We aim to reconstruct a monoid scheme $X$ from the category of quasi-coherent sheaves over it. This is much in the vein of Gabriel's original reconstruction theorem. Under some finiteness condition on a monoid schemes $X$, we show that the localising subcategories of the topos $\mathfrak{Qc}(X)$ of quasi-coherent sheaves on $X$ is in a one-to-one correspondence with open subsets of $X$, while the elements of $X$ correspond to the topos points of $\mathfrak{Qc}(X)$. This allows us to reconstruct $X$ from $\mathfrak{Qc}(X)$.

math.CT↗

arXiv2019

The fundamental group of binoid varieties

Binoid schemes generalise monoid schemes, which in turn enable us to generalise toric varieties. Let $X$ be a binoid scheme. The aim of this paper is to calculate the topological fundamental group of $KX$, where $K=\mathbb{C}$ or $\mathbb{R}$. For the latter, we will give an explicit way of calculating the fundamental group using methods from 2-category theory. Indeed, we will calculate the more general fundamental groupoid. As a specialisation, we will also look at the Stanley Reisner Rings. Our method simplifies in this case, allowing us to describe the fundamental groupoid in terms of the simplicial complex directly.

math.AG↗

arXiv2018

Topos points of quasi-coherent sheaves over monoid schemes

Let $X$ be a monoid scheme. We will show that the stalk at any point of $X$ defines a point of the topos $\Qc(X)$ of quasi-coherent sheaves over $X$. As it turns out, every topos point of $\Qc(X)$ is of this form if $X$ satisfies some finiteness conditions. In particular, it suffices for $M/M^\times$ to be finitely generated when $X$ is affine, where $M^\times$ is the group of invertible elements. This allows us to prove that two quasi-projective monoid schemes $X$ and $Y$ are isomorphic if and only if $\Qc(X)$ and $\Qc(Y)$ are equivalent. The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani \cite{cc1}. We will study the topos points of free commutative monoids and show that already for $\mathbb{N}^\infty$, there are `hidden' points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting `geometry of monoids'.

math.CT↗

arXiv2016

The Étale Fundamental Groupoid as a Terminal Costack

Let $X$ be a noetherian scheme. We denote by $Π_1(X)$ the fundamental groupoid. In this paper we prove that the assignments $U\mapstoΠ_1(U)$ is the 2-terminal costack over the site of étale coverings of $X$.

math.AG↗

arXiv2016

On the group of separable quadratic algebras and stacks

The aim of this paper is to study the group of isomorphism classes of torsors of finite flat group schemes of rank 2 over a commutative ring $R$. This, in particular, generalises the group of quadratic algebras (free or projective), which is especially well studied. Our approach however, yields new results even in this case.

math.AG↗

arXiv2014

The fundamental groupoid as a terminal costack

Let $X$ be a topological space. We denote by $π_0(X)$ the set of connected components of $X$ and by $Π_1(U)$ the fundamental groupoid. In this paper we prove that for good topological spaces the assignments $U\mapstoπ_0(U)$ and $U\mapstoΠ_1(U)$ are the terminal cosheaf and costack respectively.

math.AT↗

arXiv2014

On Cohomology and vector bundles over monoid schmes

The aim of this paper is to study the cohomology theory of monoid schemes in general and apply it to vector and line bundles. We will prove that over separated monoid schemes, any vector bundle is a coproduct of line bundles and then go on to study the line bundles in more detail. Amongst other things, we prove that over separated monoid schemes, ${\sf Pic}$ respects finite products. Next we will introduce the notion of $s$-cancellative monoids. They are monoids for which $ax=ay$ implies that $(xy)^nx=(xy)^ny, n\in\mathbb{N}$. This class is important since it is the biggest class of monoids for which $M^*_{\mathfrak{p}}$ maps injectively into its group of fractions for every prime ideal ${\mathfrak{p}}$. As we will see in section 6, this will enable us to embed $\mathcal{O}^*_X$ injectively in a constant sheaf provided $X$ is locally $s$-cancellative. We develop the theory of $s$-divisors and we prove that for an $s$-cancellative monoid scheme $X$, the group ${\sf Pic}(X)$ can be described in terms of $s$-divisors. For cancellative monoid schemes, $s$-divisors agree with the Cartier divisors. We then introduce the notion of $s$-smooth monoid schemes, which generalise smooth monoids schemes, and prove that for them $H^i(X,\mathcal{O}^*_X)=0$ for all $i\geq 2$. Furthermore we show that it is a local property and respects finite products. Finally we investigate the relationship between line bundles over a monoid scheme $X$ and over its geometric realisation $X_k$, where $k$ is a commutative ring. We prove that if $k$ is an integral domain (resp. principal ideal domain) and $X$ is a cancellative and torsion free (resp. seminormal and torsion-free) monoid scheme, then the induced map ${\sf Pic}(X)\to {\sf Pic}(X_k)$ is a monomorphism (resp. isomorphism).

math.AG↗

arXiv2012

On the spectrum of monoids and semilattices

Our main observation is that the contravariant functor Spec on the category of commutative monoids is representable. We discuss a few consequences of this fact. In particular, we give an efficient way of calculating the Spec(M) of a finitely generated monoid explicitly.

math.AG↗