Roger Godement (ɡɔdmɑ̃; 1 October 1921 – 21 July 2016) was a French mathematician, known for his work in functional analysis as well as his expository books.
Godement started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan. He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan. Work on the abstract theory of spherical functions published in 1952 proved very influential in subsequent work, particularly that of Harish-Chandra. The isolation of the concept of square-integrable representation is attributed to him. The Godement compactness criterion in the theory of arithmetic groups was a conjecture of his. He later worked with Jacquet on the zeta function of a simple algebra.
He was an active member of the Bourbaki group in the early 1950s, and subsequently gave a number of significant Bourbaki seminars. He also took part in the Cartan seminar.
His book Topologie Algébrique et Théorie des Faisceaux from 1958 was, as he said, a very unoriginal idea for the time (that is, to write an exposition of sheaf theory); as a non-specialist, he managed to write an enduring classic. It introduced the technical method of flasque resolutions, nowadays called Godement resolutions. It has also been credited as the place in which a comonad can first be discerned.
He also wrote texts on Lie groups, abstract algebra and mathematical analysis.
Topologie algébrique et théorie des faisceaux. Hermann 1958, 1960.
Cours d ́algèbre. Hermann 1963, 1966.
Algebra. Hermann 1968.
Analyse mathématique. 4 vols., Springer-Verlag 1998–2001.
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In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria.
Alexander Grothendieck (ˈgroʊtəndiːk; ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk; ɡʁɔtɛndik; 28 March 1928 – 13 November 2014) was a French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century.
Covers injective modules, Ox-modules, and their relevance in algebraic structures, emphasizing their importance in resolving acyclic resolutions and computing cohomology.