Serre made some progress towards a solution in 1957 when he proved that every finitely generated projective module over a polynomial ring over a field was stably free, meaning that after forming its direct sum with a finitely generated free module, it became free.

Serre did not provide a complete proof and what was missing became known as the epsilon conjecture or ε-conjecture.

In coherent sheaf theory, pushing to the limit of what could be done with Serre duality without the assumption of a non-singular scheme, the need to take a whole complex of sheaves in place of a single dualizing sheaf became apparent.

Prince Boris moved to the Moika palace in St. Petersburg ( Also known as Yusupov Palace ) with his second wife, Zenaida Ivanovna Narishkina, ( who later became Comtesse de Chauveau, Marquise de Serre through her second marriage ) ( May 18, 1810-February 26, 1893 ) ( daughter of Ivan Dimitrievitch Narishkin April 17, 1776-April 15, 1840, Marshal of the Sytchev Nobility in 1829 and later a Chamberlain, and a relative of Peter the Great's mother, and Varvara Ivanovna Narishkina, née Ladomirsky May 17, 1785-November 26, 1840 ), and their only son Nikolai.

This expedition included the naturalist and writer Antoine-Joseph Pernety ( 1716 – 1801 ) known as Dom Pernety, the priest and chronicler accompanying the expedition, together with the engineer and geographer Lhuillier de la Serre.

Jean-Pierre Serre, in his 1955 paper " Faisceaux algébriques cohérents ", remarked that the equivalent question was not known for algebraic vector bundles: " It is not known if there exist projective A-modules of finite type which are not free.

The 4th Division attacked between the Serre and Beaumont Hamel and managed to capture the German strongpoint known as Quadrilateral Redoubt.

Influenced by the French school of Henri Cartan and Jean-Pierre Serre, he reformulated and strengthened their method of killing homotopy groups in spectral sequence terms, creating the basic tool of stable homotopy theory now known as the Adams spectral sequence.

As shown by Serre and Ribet, the Taniyama – Shimura conjecture ( whose status was unresolved at the time ) and the epsilon conjecture together imply that Fermat's Last Theorem is true.

* 1926 – Jean-Pierre Serre, French mathematician

* September 15 – Jean-Pierre Serre, French mathematician

Serre's thesis concerned the Leray – Serre spectral sequence associated to a fibration.

Together with Cartan, Serre established the technique of using Eilenberg – MacLane spaces for computing homotopy groups of spheres, which at that time was considered as the major problem in topology.

* Bass – Serre theory

* Serre – Tate theorem

* Grothendieck – Serre Correspondence ( 2003 ), bilingual edition, edited with Pierre Colmez

* Serre – Tate theorem

* 13 November – Claude Serre, French cartoonist ( b. 1938 )

* Claude Serre ( 1938 – 1998 ), a French cartoonist

The Serre duality is also a key ingredient in the proof of the Riemann – Roch theorem.

Alexandre was born at Versailles to Lieutenant-Colonel Jean Baptiste Berthier ( 1721 – 1804 ), an officer in the Corps of Topographical Engineers, and first wife ( married in 1746 ) Marie Françoise L ' Huillier de La Serre.

Free products are also important in Bass – Serre theory, the study of groups acting by automorphisms on trees.

Free products with amalgamation and a closely related notion of HNN extension are basic building blocks in Bass – Serre theory of groups acting on trees.

In the mathematical fields of topology and K-theory, the Serre – Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: " projective modules over commutative rings are like vector bundles on compact spaces ".

This can be made precise for the ring of continuous real-valued functions on a compact Hausdorff space, as well as for the ring of smooth functions on a smooth manifold ( see Serre – Swan theorem that says a finitely generated projective module over the space of smooth functions on a compact manifold is the space of smooth sections of a smooth vector bundle ).

Other precursors of geometric group theory include small cancellation theory and Bass – Serre theory.

Bass – Serre theory, introduced in the 1977 book of Serre, derives structural algebraic information about groups by studying group actions on simplicial trees.

* Development of Bass – Serre theory, particularly various accessibility results and the theory of tree lattices.

Generalizations of Bass – Serre theory such as the theory of complexes of groups.

There is a method due to Jean-Pierre Serre which allows one, at least theoretically, to compute homotopy groups of spaces using a spectral sequence for special fibrations with Eilenberg – MacLane spaces for fibers.

Amongst his most original contributions were: his " Conjecture II " ( still open ) on Galois cohomology ; his use of group actions on Trees ( with H. Bass ); the Borel-Serre compactification ; results on the number of points of curves over finite fields ; Galois representations in ℓ-adic cohomology and the proof that these representations have often a " large " image ; the concept of p-adic modular form ; and the Serre conjecture ( now a theorem ) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.

Already in 1955, Jean-Pierre Serre had used the analogy of vector bundles with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over a polynomial ring is free ; this assertion is correct, but was not settled until 20 years later.

) In 1959, Serre formed the Grothendieck group construction for rings, and used it to prove a weak form of the conjecture.

As a corollary he proved the celebrated Ramanujan-Petersson conjecture for modular forms of weight greater than one ; weight one was proved in his work with Serre.

Quillen was awarded the Fields Medal in 1978 in part for his proof of the Serre conjecture.

Its development in the 1960s and 1970s was linked to attempts to solve a conjecture of Serre on projective modules that now is the Quillen-Suslin theorem ; numerous other connections with classical algebraic problems were found in this era.

# REDIRECT Serre conjecture

Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre and others, after sheaves had been defined by Jean Leray.

It appeared in print in a paper written by Armand Borel with Serre.

The Pals ' first day of action, Saturday 1 July 1916, took place in Serre in the north of France.

The German defences in Serre were supposed to have been obliterated by sustained, heavy, British shelling during the preceding week ; however, as the battalion advanced it met with fierce resistance.

* Serre, Jean-Pierre.

On his return in 1545, he assisted the prominent physician Louis Serre in his fight against a major plague outbreak in Marseille, and then tackled further outbreaks of disease on his own in Salon-de-Provence and in the regional capital, Aix-en-Provence.

This generalizes an earlier analogous result of Jean-Pierre Serre for topologically finitely-generated pro-p groups.

Some mathematicians, therefore, do not recognize blackboard bold as a separate style from bold: Jean-Pierre Serre, for example, has publicly inveighed against the use of " blackboard bold " anywhere other than on a blackboard, and uses double-struck letters when writing bold on the blackboard, whereas his published works consistently use ordinary bold for the same symbols.

The climate is influenced by the mountainous and hilly relief of the region: cold in the area of Monte Pollino, temperate with a very limited temperature range in the area of Aspromonte, while the Sila and Serre massifs ensure greater humidity on the Tyrrhenian coast and a drier climate on the Ionian coast.

The 31st Division had attacked Serre on 1 July and four and a half months later, was called on to do it again ; the results were similar.

South of Serre, the British, with the benefit of their hard-earned experience, succeeded in capturing most of their objectives.

François Francoeur et François Rebel composed Pirame et Thisbée, a liric tragedy in 5 acts and a prologue, with libretto by Jean-Louis-Ignace de la Serre ; it was played at the Académie royale de musique, on October 17, 1726.

In the early 10th century, economic recovery saw the building of many Romanesque churches in the region including Ailhon, Mercuer, St Julien du Serre, Balazuc, Niègles and Rochecolombe.

The Neolithic site of Serre Paradis reveals the presence of semi-nomadic cultivators in the period 4000 to 3500 BC on the future site of Nîmes.

Other notable participants in later days were Hyman Bass, Laurent Schwartz, Jean-Pierre Serre, Alexander Grothendieck, Jean-Louis Koszul, Samuel Eilenberg, Serge Lang and Roger Godement.

In particular Serre has often championed greater attention to problem-solving, within number theory especially, not an area treated in the main Bourbaki texts.

The leaders of the party, beside Royer-Collard, were Guizot, PFH de Serre, Camille Jordan and Charles de Rémusat.

In 1820 Royer-Collard was excluded from the council of state by a decree signed by his former ally Serre.

Jean-Pierre Serre (; born 15 September 1926 ) is an influential French mathematician.

Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris.

In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl praised Serre in seemingly extravagant terms, and also made the point that the award was for the first time awarded to an algebraist.

Serre subsequently changed his research focus.