Abstractly constructed prime spectra
| Autor: | Alberto Facchini, Carmelo Antonio Finocchiaro, George Janelidze |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Algebra and Number Theory
Commutator Complete lattice Lattice of congruences Lattice of ideals Multiplicative lattice Prime spectrum Sober space Spectral space Mathematics - Category Theory 06F99 13A15 14A05 06D05 16Y60 20M12 16D25 18E13 06D22 54D30 08A30 08B99 08B10 16Y30 FOS: Mathematics Category Theory (math.CT) |
| DOI: | 10.48550/arxiv.2104.09840 |
| Popis: | The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum $\mathrm{Spec}(R)$ of a unital commutative ring $R$ is always a spectral (=coherent) topological space. In this generalization, which includes several other known ones, the role of ideals of $R$ is played by elements of an abstract complete lattice $L$ equipped with binary multiplication with $xy\leqslant x\wedge y$ for all $x,y\in L$. In fact when no further conditions on $L$ are required, the resulting space can be and is only shown to be sober, and we discuss further conditions sufficient to make it spectral. This discussion involves establishing various comparison theorems on so-called prime, radical, solvable, and locally solvable elements of $L$; we also make short additional remarks on semiprime elements. We consider categorical and universal-algebraic applications involving general theory of commutators, and an application to ideals in what we call the commutative world. The cases of groups and of non-commutative rings are briefly considered separately. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|