Coimbra Mathematical Texts
Volume 1 / 2021 to Volume 2 / 2024
Book Series
Volume 1 / 2021 to Volume 2 / 2024
Chapter
This unusually long introduction is intended to be both a short history of the considerations which, progressively, led to the development of Galois theory, but also a detailed overview of the content of this ...
Chapter
The Grothendieck approach to the Galois theorem extends the classical Galois bijection to an equivalence of categories. If K ⊆ L is a finite-dimensional Galois extension of fields, a finite-dimensional K-algebra
Chapter
In Chapter 3 we pointed out an equivalent definition of a finite-dimensional Galois extension in terms of a tensor product, instead of polynomials over a field. Th...
Chapter
A groupoid is a category in which every morphism is an isomorphism. A group G determines a groupoid \(\mathbb{G}\) ...
Chapter
A spectrum of a ring is a topological space associated with the ring which allows, up to an isomorphism, a representation of the ring as a ring of continuous functions on that spectrum. Many different spectra ...
Chapter
The “Spectrum functor” of a ring and its right adjoint are the key to generalizing to the case of rings the notion of split algebra encountered in the case of fields. The same functors make it possible to defi...
Chapter
This first chapter presents the classical Galois theorem for fields. A finite-dimensional extension of fields K ⊆ L is a Galois extension when every element l ∈ L is root of a polynomial p(X) ∈ K[X] which factors...
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The finite-dimensional Galois theorems of the previous chapters admit generalizations to arbitrary dimensions. This requires superposing topological structures on the algebraic ones. These topological aspects ...
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This chapter presents two essential and closely related notions: adjunctions and monads; each adjunction generates a monad, each monad generates an adjunction. Every module on a ring R is an additive group; but c...
Chapter
A morphism σ: R → of rings induces a pair of adjoint functors between the categories of R-modules and S-modules. Via this adjunction, the category of S-modules is always monadic over the category of R-modules: th...
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Article
We prove that the stable category associated with the category \(\mathsf {PreOrd}({\mathbb {C}})\) ...
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The real line is provided with a natural ordering. In affine real spaces, this allows defining an orientation of the space, but also to speak of the interior or the exterior of various geometric figures. This ...
Chapter
Affine complex spaces can be provided with an “Hermitian product” which yields corresponding notions of distance and perpendicularity. The similarities and differences with the Euclidean case are clearly under...
Chapter
Already two millenniums before Christ, a substantial geometric knowledge exists: results like the Pythagoras or the Thales intercept theorem are known of the Egyptians or Mesopotamians, long before the Greek m...
Chapter
We present a whole bunch of historically important plane curves and list their major characteristics and properties.