3 citations · 8 across the 13 of their papers we have counts for
16 papers
Deformation Theory of Monoid Schemes II: Precartesian Coextensions of Commutative Monoids
Ilia Pirashvili
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…
Deformation Theory of Monoid Schemes I
Ilia Pirashvili
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 notio…
n-ary elliptic groups, rings, and primes in arithmetic progressions
Ilia Pirashvili
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 o…
Grothendieck's theory of fibred categories for monoids
Ilia Pirashvili
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 log…
On internal categories and crossed objects in the category of monoids
Ilia Pirashvili
It is a well-known fact that the category of internal categories in a category has a description in terms of crossed modules, when $\mathbf{…
Elliptic groups and rings
Ilia Pirashvili
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 c…
Idempotents and the points of the topos of M-sets
Ilia Pirashvili
The aim of this paper is to study the points and localising subcategories of the topos of -sets, for a finite monoid . We show that the points of this topos can be fully clas…
Reconstruction theorem for monoid schemes
Ilia Pirashvili
We aim to reconstruct a monoid scheme from the category of quasi-coherent sheaves over it. This is much in the vein of Gabriel's original reconstruction theorem. Under some fin…
The fundamental group of binoid varieties
Holger Brenner, Ilia Pirashvili
Binoid schemes generalise monoid schemes, which in turn enable us to generalise toric varieties. Let be a binoid scheme. The aim of this paper is to calculate the topological f…
On the equvialence of colimits and 2-colimits
Ilia Pirashvili
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…
Topos points of quasi-coherent sheaves over monoid schemes
Ilia Pirashvili
Let be a monoid scheme. We will show that the stalk at any point of defines a point of the topos $\Qc(X)$ of quasi-coherent sheaves over . As it turns out, every topos p…
On the group of separable quadratic algebras and stacks
Ilia Pirashvili
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 . This, in particular, generalises t…
The Étale Fundamental Groupoid as a Terminal Costack
Ilia Pirashvili
Let be a noetherian scheme. We denote by the fundamental groupoid. In this paper we prove that the assignments is the 2-terminal costack over the site…
The fundamental groupoid as a terminal costack
Ilia Pirashvili
Let be a topological space. We denote by the set of connected components of and by the fundamental groupoid. In this paper we prove that for good topologi…
On Cohomology and vector bundles over monoid schmes
Ilia Pirashvili
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…
On the spectrum of monoids and semilattices
Ilia Pirashvili
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 g…