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Technique from algebraic geometry
Faithfully flat descent or flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully
Faithfully_flat_descent
Mathematical concept that extends the intuitive idea of gluing in topology
category Beck's monadicity theorem Cohomological descent Faithfully flat descent Monadic descent Descent data for quasi-coherent sheaves, Stacks Project
Descent_(mathematics)
the theory of descent (faithfully flat descent). The term flat here comes from flat modules. There are several slightly different flat topologies, the
Flat_topology
homomorphism is faithfully flat, the Amitsur complex is exact (thus determining a resolution), which is the basis of the theory of faithfully flat descent. The notion
Amitsur_complex
Scheme theory concept
of schemes is called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: flatness is a generic property;
Flat_morphism
Theorem in category theory
with descent data, so the fact that T is comonadic is equivalent to the main theorem of faithfully flat descent, saying that B-modules with descent are
Beck's_monadicity_theorem
Direct summand of a free module (mathematics)
projective. In other words, the property of being projective satisfies faithfully flat descent. Submodules of projective modules need not be projective; a ring
Projective_module
Construct in algebraic geometry
the cotangent complex satisfies (derived) faithfully flat descent. In other words, for any faithfully flat morphism f : A → B of R-algebras, one has an
Cotangent_complex
Operation in algebra and mathematics
theorem, if and only if B is faithfully flat as an A-module. It thus allows to descend B-modules, equipped with a descent datum (i.e., an action of the
Monad_(category_theory)
Construction of a ring of fractions
is a faithfully flat module when the direct sum is taken over all prime ideals (or over all maximal ideals of R). See also Faithfully flat descent. A property
Localization (commutative algebra)
Localization_(commutative_algebra)
that the flat topology has been considered the 'right' foundational topos for scheme theory goes back to the fact of faithfully-flat descent, the discovery
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Faithfully flat morphism of schemes
morphism, both variations of faithfully flat morphisms of schemes. Sometimes an fpqc morphism means one that is faithfully flat and quasi-compact. This is
Fpqc_morphism
Construction in algebraic geometry
include flatness, smoothness, properness, and many other classes of morphisms. These results form part of Grothendieck's theory of faithfully flat descent. Example:
Fiber_product_of_schemes
American mathematician (born 1937)
harder", given "descent machinery", a reference to the role descent morphisms play. This background explains why faithfully flat descent is introduced in
David_Mumford
Israeli mathematician (1921–1994)
homomorphism is faithfully flat, the Amitsur complex is exact, a fact that provides the algebraic foundation for the theory of faithfully flat descent. The construction
Shimshon_Amitsur
American country singer (1954–1989)
Hill Cemetery outside Nashville. His gravestone reads, "Forever yours faithfully" (part one) and "His being was my reason" (part two). [citation needed]
Keith_Whitley
V\subset W} and a map Spec W → X lifting v. Examples of v-covers include faithfully flat maps, proper surjective maps. In particular, any Zariski covering is
V-topology
Basilica under construction since 1882 in Barcelona, Spain
[translated] Objectives: Construction, conservation and restoration of a temple, faithfully executing Antoni Gaudí's project. Gómez Gimeno, María José (2006). La
Sagrada_Família
Algebraic geometry analog of a principal bundle in algebraic topology
the spectrum of a field) and f : X → S {\displaystyle f:X\to S} a faithfully flat morphism, locally of finite type. Assume f {\displaystyle f} has a
Torsor_(algebraic_geometry)
Mode in Arabic music
however such instruments cannot faithfully reproduce the microtonal details of the maqam scale. Maqamat can be faithfully performed either on fretless instruments
Arabic_maqam
Mathematical structure
the fpqc topology, any faithfully flat and quasi-compact morphism is a cover. These topologies are closely related to descent. The fpqc topology is finer
Grothendieck_topology
American pianist (1946–2023)
pianist and composer Franz Liszt, making it clear that Liszt practiced faithfully. Watts found inspiration in Liszt, adopting his theatrical playing style
André_Watts
English singer and actress (1946–2025)
Archived from the original on 9 July 2023. Retrieved 9 February 2023. "Faithfully Speaking". Advocate.com. 27 March 2009. Archived from the original on
Marianne_Faithfull
American writer (1925–2011)
illustrated by Lillian Hoban, whose drawn renditions of these characters were faithfully replicated by the Muppet creators. The story tells of a poor otter mother
Russell_Hoban
Generalization of algebraic spaces or schemes
R ] {\displaystyle [U/R]} . It turns out using the fppf-topology (faithfully flat and locally of finite presentation) on ( S c h / S ) {\displaystyle
Algebraic_stack
Town in New York, United States
country people. Its wheel is dripping and clattering still, doing duty as faithfully as of old. While the old miller who 'picked the stones and calked the
Lowville,_New_York
Irish guitarist (1948–1995)
broadcast by RTÉ as part of their Christmas concert series. The concert was faithfully recreated by the tribute band Sinnerboy on 29 October 2021. In February
Rory_Gallagher
Cast of the novels by George R. R. Martin
Tyrion's wife Sansa Stark. He becomes Brienne's squire and serves her as faithfully as he did Tyrion. He and Brienne are captured by the Brotherhood Without
List of A Song of Ice and Fire characters
List_of_A_Song_of_Ice_and_Fire_characters
1980 film by Stanley Kubrick
Shelley Duvall, Danny Lloyd and Scatman Crothers. The film presents the descent into insanity of a recovering alcoholic and aspiring novelist who takes
The_Shining_(film)
Vatican City military force and papal bodyguard
(German, Italian, and French): (English translation) I swear that I will faithfully, loyally and honourably serve the Supreme Pontiff (name of Pope) and his
Swiss_Guard
1st-century BCE king of Judea
either 25 or 28 years old (Greek original: "15 years of age"). There he faithfully farmed the taxes of that region for the Roman Senate, and he met with
Herod_the_Great
Process of creating circuit boards
conform to any slight bends in the material so all features are replicated faithfully. When milling PCBs however, any minute height variations encountered when
Printed_circuit_board_milling
his duties faithfully as a servant of the court. de fideli administratione of faithful administration Describes an oath taken to faithfully administer
List_of_Latin_phrases_(full)
Breton noble family
interest, sometimes carrying out the most important charges of the Duchy faithfully, sometimes rebelling, as John II of Rohan did in the last years of Breton
House_of_Rohan
American writer (1896–1940)
visited Hollywood in 1938. "She was the first girl I ever loved and I have faithfully avoided seeing her up to this moment to keep the illusion perfect," Fitzgerald
F._Scott_Fitzgerald
Anthem of the Austrian monarchy
What was wrought by the diligence of citizens, may the soldier's power faithfully protect! With cheery weapons of mind, Arts and Science may triumph! Blessings
Gott_erhalte_Franz_den_Kaiser
Cloth bearing the alleged image of Jesus
improbable; and their 'proofs' amount only to demonstrating (none too faithfully) that it was not totally impossible." Among other difficulties, Allen's
Shroud_of_Turin
Italian physicist and astronomer (1564–1642)
Throughout his trial, Galileo steadfastly maintained that since 1616 he had faithfully kept his promise not to hold any of the condemned opinions, and initially
Galileo_Galilei
Autonomous community of Spain
'98. Also of this generation were the Quintero brothers, dramatists who faithfully captured Andalusian dialects and idiosyncrasies. Also of note, 1956 Nobel
Andalusia
City in Michigan, United States
Joseph refused, the man left, only to return at dusk when Joseph, who faithfully performed the ritual of Angelus every day at that time, was in prayer
Grand_Rapids,_Michigan
South African politician (1825–1904)
divided. To admirers he was an astute reader of people, events and law who faithfully defended a maligned nation and became a tragic folk hero; to critics he
Paul_Kruger
French comics author (1938–2012)
Charlier, and was actually conceived as a television movie series, more faithfully adhering to the main comic series. That 1983 attempt petered out without
Jean_Giraud
American musician and recording engineer
performers performed. He found the best artists he could, signed them up and faithfully captured their performances on disc in the best possible sound available
Fred_Gaisberg
American actress (1909–2001)
Sweetheart (1931) Everybody's Welcome (1931) Of Thee I Sing (1932–1933) Faithfully Yours (1951) God Bless Our Bank (1963) The Solid Gold Cadillac (1965;
Ann_Sothern
Infantry regiment of the United States Army
you swear unqualified allegiance to the United States of America and faithfully defend the United States from any or all attack by foreign or domestic
442nd_Infantry_Regiment
German artist and printmaker (1497–1543)
Late Gothic school. Holbein was born in Augsburg to a family of Swiss descent but worked mainly in Basel as a young artist. At first, he painted murals
Hans_Holbein_the_Younger
Renaissance art in Florence
built from multiple vanishing points. The stories told are fluid and faithfully convey the human truth of the characters. Fra Filippo Lippi had a great
Florentine_Renaissance_art
Hungarian composer and pianist (1811–1886)
audience preferences that began in the 1830s, which started to favour a faithfully rendered memorised performance over the improvisation that Liszt employed
Franz_Liszt
French-Belgian comic series
abundant, enduring fan-mail from readers, who gave us their friendship by faithfully following his tumultuous adventures, told me that the Blueberry character
Blueberry_(comics)
Russian and French artist (1887–1985)
she joins me in this greeting, she who loved France and French art so faithfully. Her last joy was the liberation of Paris... Now, when Paris is liberated
Marc_Chagall
Region in County Galway, Ireland
Chronological account of Irish Events (collected from Very Ancient Documents faithfully compared with each other & supported by the Genealogical & Chronological
Connemara
Theoretical object in mathematics
equivalent to the category of Toën–Vaquié F1‑schemes. This category embeds faithfully, but not fully, into the category of blue schemes, and is a full subcategory
Field_with_one_element
Architecture in Brazil from 1500 to 1822
the former Jesuit church and college (known as Pátio do Colégio) was faithfully reconstructed based on ancient iconography. The facade shows the 16th
Colonial architecture of Brazil
Colonial_architecture_of_Brazil
Heart: (1922 & 1932) Faithful in My Fashion (1946) The Faithful Son (2017) Faithfully Yours: (1988 & 2022) Faithfulness (1965) Faithless: (1932 & 2000) Faithless
List_of_films:_F
Stone, bronze and iron ages of pre-history
that any savage tribe today is primitive, in the sense that its culture faithfully reflects that of much more ancient men is gratuitous. Childe concentrated
Three-age_system
Malaysian cartoonist (born 1951)
and capturing them in his works. Keluarga Si Mamat and The Kampung Boy faithfully depicted their characters' appearances and attitudes. Their narrations
Lat_(cartoonist)
Canadian pianist (1932–1982)
makes us assume that the situation of the man who wrote it accurately or faithfully reflects the situation of his time? ... What if the composer, as historian
Glenn_Gould
American historian and biographer (1915–1981)
had begun dating a classmate, Dilworth Jensen. They wrote to each other faithfully during Jensen's long absence on an LDS mission in Europe. In June 1935
Fawn_M._Brodie
Neighbourhood in Toronto, Ontario, Canada
Globe and Mail, page 17. "Williams performed his court duties 'well, faithfully,' hearing judge says", (December 16, 1977), by Brian Gory, Globe and Mail
Swansea,_Toronto
Mountain in Italy
Morphology: The geomorphological structure of the Peloritani villages faithfully mirrors that of the Ausoni and Aurunci Mountains. Hilltop towns such as
Peloritani
Masonic memorial in Alexandria, Virginia
left is the Alexandria-Washington Lodge No. 22 Replica Lodge Room, which faithfully duplicates the look of the lodge room at the time Washington presided
George Washington Masonic National Memorial
George_Washington_Masonic_National_Memorial
American sound designer and audio engineer
experience with very little dialogue. All of the songs would be presented as faithfully as possible to the Beatles' original versions, which posed a challenge
Abe_Jacob
or section of his town, obtaining from each an assurance that he will faithfully do his duty." Then, the town manager was instructed to compile lists and
History_of_Connecticut
Series of 80 prints by Francisco Goya
similar to the original sketches. The use of a burin allowed Goya to faithfully replicate all the features of the original sketches. Preparatory drawings
Los_caprichos
Horse breed originating in West Asia
the horses to return to him. Only five mares responded. Because they faithfully returned to their master, though desperate with thirst, these mares became
Arabian_horse
Thirteenth-century ruler of the Isle of Man and parts of the Hebrides
Crown. As a result, Óláfr pledged to secure the English and Irish coasts faithfully, and to supply the English with fifty galleys when needed: all for the
Olaf_the_Black
Set of musical compositions for organ by Johann Sebastian Bach
note. Some commentators have seen this falling figure as representing a descent to earth, but it could equally well reflect a repetition of the words "Nun
Orgelbüchlein
before it was disbanded. During the time the club was disbanded, she faithfully cleaned up the room in the hopes that the club would once again be revived
List_of_Saki_characters
Church
designated "a personal parish for those Catholics of the Latin Rite of Polish descent" in Cleveland. A personal parish is designated under Canon 518 of the 1983
St. Casimir Church (Cleveland)
St._Casimir_Church_(Cleveland)
use of textual criticism modern (critical) copies are considered to "faithfully represent the original", and our understanding of the original language
Criticism_of_Christianity
Aspect of military life
compassion within its pages to be a source of strength to you as you faithfully serve in our uniformed services. June 2012 news sources announced that
Religious symbolism in the United States military
Religious_symbolism_in_the_United_States_military
Social movement
Some, like Tong Bunian, stayed as diehard Ming loyalists, while others faithfully served the Qing after Qing's conquest of Liaodong. Qing enrolled the Tongs
De-Sinicization
Généralités. Descente par morphismes fidèlement plats (descent (category theory), faithfully flat morphisms) Jean-Louis Koszul, Travaux de B. Kostant sur
Séminaire Nicolas Bourbaki (1950–1959)
Séminaire_Nicolas_Bourbaki_(1950–1959)
according to his own original manuscripts, circumspectly examined, and faithfully transcribed by Edward Sparke (J.L. and Humphrey Moseley, London 1649)
Edward_Sparke
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FAITHFULLY FLAT-DESCENT
FAITHFULLY FLAT-DESCENT
FAITHFULLY FLAT-DESCENT
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