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wiki_24_chunk_79 | American Revolution | Historians in the early 20th century such as J. Franklin Jameson examined the class composition of the Patriot cause, looking for evidence of a class war inside the revolution. More recent historians have largely abandoned that interpretation, emphasizing instead the high level of ideological unity. Both Loyalists and ... | wikipedia |
wiki_24_chunk_80 | American Revolution | King George III The war became a personal issue for the king, fueled by his growing belief that British leniency would be taken as weakness by the Americans. He also sincerely believed that he was defending Britain's constitution against usurpers, rather than opposing patriots fighting for their natural rights. | wikipedia |
wiki_24_chunk_81 | American Revolution | Although Prime Minister Lord North was not an ideal war leader, George III managed to give Parliament a sense of purpose to fight, and Lord North was able to keep his cabinet together. Lord North's cabinet ministers, the Earl of Sandwich, First Lord of the Admiralty, and Lord George Germain, Secretary of State for the ... | wikipedia |
wiki_24_chunk_82 | American Revolution | George III is often accused of obstinately trying to keep Great Britain at war with the revolutionaries in America, despite the opinions of his own ministers. In the words of the British historian George Otto Trevelyan, the King was determined "never to acknowledge the independence of the Americans, and to punish their... | wikipedia |
wiki_24_chunk_83 | American Revolution | With the setbacks in America, Lord North asked to transfer power to Lord Chatham, whom he thought more capable, but George refused to do so; he suggested instead that Chatham serve as a subordinate minister in North's administration, but Chatham refused. He died later in the same year. Lord North was allied to the "Kin... | wikipedia |
wiki_24_chunk_84 | American Revolution | As late as the Siege of Charleston in 1780, Loyalists could still believe in their eventual victory, as British troops inflicted defeats on the Continental forces at the Battle of Camden and the Battle of Guilford Court House. In late 1781, the news of Cornwallis's surrender at the siege of Yorktown reached London; Lor... | wikipedia |
wiki_24_chunk_85 | American Revolution | When John Adams was appointed American Minister to London in 1785, George had become resigned to the new relationship between his country and the former colonies. He told Adams, "I was the last to consent to the separation; but the separation having been made and having become inevitable, I have always said, as I say n... | wikipedia |
wiki_24_chunk_86 | American Revolution | Those who fought for independence were called "Revolutionaries" "Continentals", "Rebels", "Patriots", "Whigs", "Congress-men", or "Americans" during and after the war. They included a full range of social and economic classes but were unanimous regarding the need to defend the rights of Americans and uphold the princip... | wikipedia |
wiki_24_chunk_87 | American Revolution | According to historian Robert Calhoon, 40– to 45-percent of the white population in the Thirteen Colonies supported the Patriots' cause, 15– to 20-percent supported the Loyalists, and the remainder were neutral or kept a low profile. Mark Lender analyzes why ordinary people became insurgents against the British, even i... | wikipedia |
wiki_24_chunk_88 | American Revolution | Thomas Paine published his pamphlet Common Sense in January 1776, after the Revolution had started. It was widely distributed and often read aloud in taverns, contributing significantly to concurrently spreading the ideas of republicanism and liberalism, bolstering enthusiasm for separation from Great Britain and encou... | wikipedia |
wiki_24_chunk_89 | American Revolution | The consensus of scholars is that about 15– to 20-percent of the white population remained loyal to the British Crown. Those who actively supported the king were known at the time as "Loyalists", "Tories", or "King's men". The Loyalists never controlled territory unless the British Army occupied it. They were typically... | wikipedia |
wiki_24_chunk_90 | American Revolution | The revolution could divide families, such as William Franklin, son of Benjamin Franklin and royal governor of the Province of New Jersey who remained loyal to the Crown throughout the war. He and his father never spoke again. Recent immigrants who had not been fully Americanized were also inclined to support the King,... | wikipedia |
wiki_24_chunk_91 | American Revolution | After the war, the most of the approximately 500,000 Loyalists remained in America and resumed normal lives. Some became prominent American leaders, such as Samuel Seabury. Approximately 46,000 Loyalists relocated to Canada; others moved to Britain (7,000), Florida, or the West Indies (9,000). The exiles represented ap... | wikipedia |
wiki_24_chunk_92 | American Revolution | A minority of uncertain size tried to stay neutral in the war. Most kept a low profile, but the Quakers were the most important group to speak out for neutrality, especially in Pennsylvania. The Quakers continued to do business with the British even after the war began, and they were accused of supporting British rule,... | wikipedia |
wiki_24_chunk_93 | American Revolution | Women contributed to the American Revolution in many ways and were involved on both sides. Formal politics did not include women, but ordinary domestic behaviors became charged with political significance as Patriot women confronted a war which permeated all aspects of political, civil, and domestic life. They particip... | wikipedia |
wiki_24_chunk_94 | American Revolution | American women were integral to the success of the boycott of British goods, as the boycotted items were largely household articles such as tea and cloth. Women had to return to knitting goods and to spinning and weaving their own cloth—skills that had fallen into disuse. In 1769, the women of Boston produced 40,000 sk... | wikipedia |
wiki_24_chunk_95 | American Revolution | Other participants France and Spain In early 1776, France set up a major program of aid to the Americans, and the Spanish secretly added funds. Each country spent one million "livres tournaises" to buy munitions. A dummy corporation run by Pierre Beaumarchais concealed their activities. American Patriots obtained some ... | wikipedia |
wiki_24_chunk_96 | American Revolution | In 1777, Charles François Adrien le Paulmier, Chevalier d’Annemours, acting as a secret agent for France, made sure General George Washington was privy to his mission. He followed Congress around for the next two years, reporting what he observed back to France. | wikipedia |
wiki_24_chunk_97 | American Revolution | Spain did not officially recognize the United States, but it was a French ally and it separately declared war on Britain on June 21, 1779. Bernardo de Gálvez y Madrid, general of the Spanish forces in New Spain, also served as governor of Louisiana. He led an expedition of colonial troops to capture Florida from the Br... | wikipedia |
wiki_24_chunk_98 | American Revolution | Ethnic Germans served on both sides of the American Revolutionary War. As George III was also the Elector of Hanover, many supported the Loyalist cause and served as allies of the Kingdom of Great Britain; most notably rented auxiliary troops from German states such as the Landgraviate of Hessen-Kassel. | wikipedia |
wiki_24_chunk_99 | American Revolution | American Patriots tended to represent such troops as mercenaries in propaganda against the British Crown. Even American historians followed suit, in spite of Colonial-era jurists drawing a distinction between auxiliaries and mercenaries, with auxiliaries serving their prince when sent to the aid of another prince, and ... | wikipedia |
wiki_24_chunk_100 | American Revolution | Other German individuals came to assist the American revolutionaries, most notably Friedrich Wilhelm von Steuben, who served as a general in the Continental Army and is credited with professionalizing that force, but most who served were already colonists. Von Steuben's native Prussia joined the League of Armed Neutral... | wikipedia |
wiki_24_chunk_101 | American Revolution | However, when the War of the Bavarian Succession erupted, Frederick II became much more cautious with Prussian/British relations. US ships were denied access to Prussian ports, and Frederick refused to officially recognize the United States until they had signed the Treaty of Paris. Even after the war, Frederick II pre... | wikipedia |
wiki_24_chunk_102 | American Revolution | Most indigenous people rejected pleas that they remain neutral and instead supported the British Crown. The great majority of the 200,000 indigenous people east of the Mississippi distrusted the colonists and supported the British cause, hoping to forestall continued expansion of settlement into their territories. Thos... | wikipedia |
wiki_24_chunk_103 | American Revolution | The great majority of indigenous people did not participate directly in the war, with the notable exceptions of warriors and bands associated with four of the Iroquois tribes in New York and Pennsylvania which allied with the British, and the Oneida and Tuscarora tribes among the Iroquois of central and western New Yor... | wikipedia |
wiki_24_chunk_104 | American Revolution | In 1776, Cherokee war parties attacked American Colonists all along the southern Quebec frontier of the uplands throughout the Washington District, North Carolina (now Tennessee) and the Kentucky wilderness area. The Chickamauga Cherokee under Dragging Canoe allied themselves closely with the British, and fought on for... | wikipedia |
wiki_24_chunk_105 | American Revolution | Joseph Brant (also Thayendanegea) of the powerful Mohawk tribe in New York was the most prominent indigenous leader against the Patriot forces. In 1778 and 1780, he led 300 Iroquois warriors and 100 white Loyalists in multiple attacks on small frontier settlements in New York and Pennsylvania, killing many settlers and... | wikipedia |
wiki_24_chunk_106 | American Revolution | In 1779, the Continental Army forced the hostile indigenous people out of upstate New York when Washington sent an army under John Sullivan which destroyed 40 evacuated Iroquois villages in central and western New York. Sullivan systematically burned the empty villages and destroyed about 160,000 bushels of corn that c... | wikipedia |
wiki_24_chunk_107 | American Revolution | At the peace conference following the war, the British ceded lands which they did not really control, and which they did not consult about with their indigenous allies during the treaty negotiations. They transferred control to the United States of all the land south of the Great Lakes east of the Mississippi and north... | wikipedia |
wiki_24_chunk_108 | American Revolution | Free blacks in the New England Colonies and Middle Colonies in the North as well as Southern Colonies fought on both sides of the War, but the majority fought for the Patriots. Gary Nash reports that there were about 9,000 black veteran Patriots, counting the Continental Army and Navy, state militia units, privateers, ... | wikipedia |
wiki_24_chunk_109 | American Revolution | The effects of the War were more dramatic in the South. Tens of thousands of slaves escaped to British lines throughout the South, causing dramatic losses to slaveholders and disrupting cultivation and harvesting of crops. For instance, South Carolina was estimated to have lost about 25,000 slaves to flight, migration,... | wikipedia |
wiki_24_chunk_110 | American Revolution | During the War, the British commanders attempted to weaken the Patriots by issuing proclamations of freedom to their slaves. In the November 1775 document known as Dunmore's Proclamation, royal Virginia, governor Lord Dunmore recruited black men into the British forces with the promise of freedom, protection for their ... | wikipedia |
wiki_24_chunk_111 | American Revolution | Davis underscores the British dilemma: "Britain, when confronted by the rebellious American colonists, hoped to exploit their fear of slave revolts while also reassuring the large number of slave-holding Loyalists and wealthy Caribbean planters and merchants that their slave property would be secure". The Americans, ho... | wikipedia |
wiki_24_chunk_112 | American Revolution | The existence of slavery in the American colonies had attracted criticism from both sides of the Atlantic as many could not reconcile the existence of the institution with the egalitarian ideals espoused by leaders of the Revolution. British writer Samuel Johnson wrote "how is it we hear the loudest yelps for liberty a... | wikipedia |
wiki_24_chunk_113 | American Revolution | The 1779 Philipsburg Proclamation expanded the promise of freedom for black men who enlisted in the British military to all the colonies in rebellion. British forces gave transportation to 10,000 slaves when they evacuated Savannah and Charleston, carrying through on their promise. They evacuated and resettled more tha... | wikipedia |
wiki_24_chunk_114 | American Revolution | Effects of the Revolution | wikipedia |
wiki_24_chunk_115 | American Revolution | After the Revolution, genuinely democratic politics became possible in the former American colonies. The rights of the people were incorporated into state constitutions. Concepts of liberty, individual rights, equality among men and hostility toward corruption became incorporated as core values of liberal republicanism... | wikipedia |
wiki_24_chunk_116 | American Revolution | Interpretations
Interpretations vary concerning the effect of the Revolution. Historians such as Bernard Bailyn, Gordon Wood, and Edmund Morgan view it as a unique and radical event which produced deep changes and had a profound effect on world affairs, such as an increasing belief in the principles of the Enlightenmen... | wikipedia |
wiki_24_chunk_117 | American Revolution | Gordon Wood states:
The American Revolution was integral to the changes occurring in American society, politics and culture .... These changes were radical, and they were extensive .... The Revolution not only radically changed the personal and social relationships of people, including the position of women, but also d... | wikipedia |
wiki_24_chunk_118 | American Revolution | Edmund Morgan has argued that, in terms of long-term impact on American society and values:
The Revolution did revolutionize social relations. It did displace the deference, the patronage, the social divisions that had determined the way people viewed one another for centuries and still view one another in much of the ... | wikipedia |
wiki_24_chunk_119 | American Revolution | Inspiring other independence movements and revolutions The first shot of the American Revolution at the Battle of Lexington and Concord is referred to as the "shot heard ‘round the world" due to its historical and global significance. The ensuing Revolutionary War not only established the United States as the first mod... | wikipedia |
wiki_24_chunk_120 | American Revolution | The U.S. Constitution, drafted shortly after independence, remains the world's oldest written constitution, and has been emulated by other countries, in some cases verbatim. Some historians and scholars aruge that the subsequent wave of independence and revolutionary movements has contributed to the continued expansion... | wikipedia |
wiki_24_chunk_121 | American Revolution | The Dutch Republic, also at war with Britain, was the next country after France to sign a treaty with the United States, on October 8, 1782. On April 3, 1783, Ambassador Extraordinary Gustaf Philip Creutz, representing King Gustav III of Sweden, and Benjamin Franklin, signed a Treaty of Amity and Commerce with the U.S. | wikipedia |
wiki_24_chunk_122 | American Revolution | The Revolution had a strong, immediate influence in Great Britain, Ireland, the Netherlands, and France. Many British and Irish Whigs in Parliament spoke glowingly in favor of the American cause. In Ireland, the Protestant minority who controlled Ireland demanded self-rule. Under the leadership of Henry Grattan, the Ir... | wikipedia |
wiki_24_chunk_123 | American Revolution | The Revolution, along with the Dutch Revolt (end of the 16th century) and the 17th century English Civil War, was among the examples of overthrowing an old regime for many Europeans who later were active during the era of the French Revolution, such as the Marquis de Lafayette. The American Declaration of Independence ... | wikipedia |
wiki_24_chunk_124 | American Revolution | Status of African Americans | wikipedia |
wiki_24_chunk_125 | American Revolution | During the revolution, the contradiction between the Patriots' professed ideals of liberty and the institution of slavery generated increased scrutiny of the latter. As early as 1764, the Boston Patriot leader James Otis, Jr. declared that all men, "white or black", were "by the law of nature" born free. Anti-slavery c... | wikipedia |
wiki_24_chunk_126 | American Revolution | In the late 1760s and early 1770s, a number of colonies, including Massachusetts and Virginia, attempted to restrict the slave trade, but were prevented from doing so by royally appointed governors. In 1774, as part of a broader non-importation movement aimed at Britain, the Continental Congress called on all the colon... | wikipedia |
wiki_24_chunk_127 | American Revolution | In the first two decades after the American Revolution, state legislatures and individuals took actions to free slaves, in part based on revolutionary ideals. Northern states passed new constitutions that contained language about equal rights or specifically abolished slavery; some states, such as New York and New Jers... | wikipedia |
wiki_24_chunk_128 | American Revolution | No southern state abolished slavery, but for a period individual owners could free their slaves by personal decision, often providing for manumission in wills but sometimes filing deeds or court papers to free individuals. Numerous slaveholders who freed their slaves cited revolutionary ideals in their documents; other... | wikipedia |
wiki_24_chunk_129 | American Revolution | Thousands of free Blacks in the northern states fought in the state militias and Continental Army. In the south, both sides offered freedom to slaves who would perform military service. Roughly 20,000 slaves fought in the American Revolution. Status of American women The democratic ideals of the Revolution inspired cha... | wikipedia |
wiki_24_chunk_130 | American Revolution | The concept of republican motherhood was inspired by this period and reflects the importance of revolutionary republicanism as the dominant American ideology. It assumed that a successful republic rested upon the virtue of its citizens. Women were considered to have the essential role of instilling their children with ... | wikipedia |
wiki_24_chunk_131 | American Revolution | The traditional constraints gave way to more liberal conditions for women. Patriarchy faded as an ideal; young people had more freedom to choose their spouses and more often used birth control to regulate the size of their families. Society emphasized the role of mothers in child rearing, especially the patriotic goal ... | wikipedia |
wiki_24_chunk_132 | American Revolution | Whatever gains they had made, however, women still found themselves subordinated, legally and socially, to their husbands, disfranchised and usually with only the role of mother open to them. But, some women earned livelihoods as midwives and in other roles in the community not originally recognized as significant by m... | wikipedia |
wiki_24_chunk_133 | American Revolution | The Revolution sparked a discussion on the rights of woman and an environment favorable to women's participation in politics. Briefly the possibilities for women's rights were highly favorable, but a backlash led to a greater rigidity that excluded women from politics. For more than thirty years, however, the 1776 New ... | wikipedia |
wiki_24_chunk_134 | American Revolution | Loyalist expatriation | wikipedia |
wiki_24_chunk_135 | American Revolution | Tens of thousands of Loyalists left the United States following the war, and Maya Jasanoff estimates as many as 70,000. Some migrated to Britain, but the great majority received land and subsidies for resettlement in British colonies in North America, especially Quebec (concentrating in the Eastern Townships), Prince E... | wikipedia |
wiki_24_chunk_136 | American Revolution | Commemorations The American Revolution has a central place in the American memory as the story of the nation's founding. It is covered in the schools, memorialized by a national holiday, and commemorated in innumerable monuments. George Washington's estate at Mount Vernon was one of the first national pilgrimages for t... | wikipedia |
wiki_24_chunk_137 | American Revolution | The Revolution became a matter of contention in the 1850s in the debates leading to the American Civil War (1861–1865), as spokesmen of both the Northern United States and the Southern United States claimed that their region was the true custodian of the legacy of 1776. The United States Bicentennial in 1976 came a yea... | wikipedia |
wiki_24_chunk_138 | American Revolution | Today, more than 100 battlefields and historic sites of the American Revolution are protected and maintained by the government. The National Park Service alone owns and maintains more than 50 battlefield parks and many other sites such as Independence Hall that are related to the Revolution, as well as the residences, ... | wikipedia |
wiki_24_chunk_139 | American Revolution | See also
Founding Fathers of the United States
List of George Washington articles
List of plays and films about the American Revolution
List of television series and miniseries about the American Revolution
Museum of the American Revolution
Timeline of the American Revolution Notes References General sources Bibl... | wikipedia |
wiki_24_chunk_140 | American Revolution | Reference works
Barnes, Ian, and Charles Royster. The Historical Atlas of the American Revolution (2000), maps and commentary excerpt and text search
Cappon, Lester J. Atlas of Early American History: The Revolutionary Era, 1760–1790 (1976)
Fremont-Barnes, Gregory, and Richard A. Ryerson, eds. The Encyclopedia o... | wikipedia |
wiki_24_chunk_141 | American Revolution | Surveys of the era
Alden, John R. A history of the American Revolution (1966) 644pp online free to borrow, A scholarly general survey
Allison, Robert. The American Revolution: A Concise History (2011) 128 pp excerpt and text search
Atkinson, Rick. The British Are Coming: The War for America, Lexington to Princeton, ... | wikipedia |
wiki_24_chunk_142 | American Revolution | Specialized studies
Bailyn, Bernard. The Ideological Origins of the American Revolution. (Harvard University Press, 1967).
Becker, Carl. The Declaration of Independence: A Study on the History of Political Ideas (1922)
Becker, Frank: The American Revolution as a European Media Event, European History Online, Mainz:... | wikipedia |
wiki_24_chunk_143 | American Revolution | Historiography
Allison, David, and Larrie D. Ferreiro, eds. The American Revolution: A World War (Smithsonian, 2018) excerpt
Breen, Timothy H. "Ideology and nationalism on the eve of the American Revolution: Revisions once more in need of revising." Journal of American History (1997): 13–39. in JSTOR
Countrymen, Edw... | wikipedia |
wiki_24_chunk_144 | American Revolution | Primary sources
The American Revolution: Writings from the War of Independence (2001), Library of America, 880 pp
Dann, John C., ed. The Revolution Remembered: Eyewitness Accounts of the War for Independence (1999) excerpt and text search, recollections by ordinary soldiers
Humphrey, Carol Sue ed. The Revolution... | wikipedia |
wiki_24_chunk_145 | American Revolution | Contemporaneous sources: Annual Register
Murdoch, David H. ed. Rebellion in America: A Contemporary British Viewpoint, 1769–1783 (1979), 900+ pp of annotated excerpts from Annual Register Annual Register 1773, British compendium of speeches and reports
Annual Register 1774
Annual Register 1775
Annual Register 1776
... | wikipedia |
wiki_24_chunk_146 | American Revolution | External links
American Revolution, US National Park Service website portal
American Independence (Teaching with Historic Places) Teaching with Historic Places (TwHP) uses historic places in National Parks and in the National Park Service's National Register of Historic Places to enliven history, social studies, geog... | wikipedia |
wiki_24_chunk_147 | American Revolution | 1770s conflicts
1780s conflicts
1770s in the United States
1780s in the United States
18th-century rebellions
18th-century revolutions
Legal history of the United States
Age of Enlightenment
1760s conflicts | wikipedia |
wiki_25_chunk_0 | Algebraic geometry | Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. | wikipedia |
wiki_25_chunk_1 | Algebraic geometry | The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubi... | wikipedia |
wiki_25_chunk_2 | Algebraic geometry | Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving l... | wikipedia |
wiki_25_chunk_3 | Algebraic geometry | In the 20th century, algebraic geometry split into several subareas.
The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.
Real algebraic geometry is the study of the real points... | wikipedia |
wiki_25_chunk_4 | Algebraic geometry | Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this pa... | wikipedia |
wiki_25_chunk_5 | Algebraic geometry | Basic notions Zeros of simultaneous polynomials In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-di... | wikipedia |
wiki_25_chunk_6 | Algebraic geometry | First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed. We consider the affine space of dimension n over k, denoted An(k) (or more simply An, when k is clear from the context). Wh... | wikipedia |
wiki_25_chunk_7 | Algebraic geometry | A function f : An → A1 is said to be polynomial (or regular) if it can be written as a polynomial, that is, if there is a polynomial p in k[x1,...,xn] such that f(M) = p(t1,...,tn) for every point M with coordinates (t1,...,tn) in An. The property of a function to be polynomial (or regular) does not depend on the choic... | wikipedia |
wiki_25_chunk_8 | Algebraic geometry | When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the regular functions on An is a ring, which is denoted k[An]. We say that a polynomial vanishes at a point if evaluating it at that point ... | wikipedia |
wiki_25_chunk_9 | Algebraic geometry | A subset of An which is V(S), for some S, is called an algebraic set. The V stands for variety (a specific type of algebraic set to be defined below). | wikipedia |
wiki_25_chunk_10 | Algebraic geometry | Given a subset U of An, can one recover the set of polynomials which generate it? If U is any subset of An, define I(U) to be the set of all polynomials whose vanishing set contains U. The I stands for ideal: if two polynomials f and g both vanish on U, then f+g vanishes on U, and if h is any polynomial, then hf vanish... | wikipedia |
wiki_25_chunk_11 | Algebraic geometry | Two natural questions to ask are:
Given a subset U of An, when is U = V(I(U))?
Given a set S of polynomials, when is S = I(V(S))? | wikipedia |
wiki_25_chunk_12 | Algebraic geometry | The answer to the first question is provided by introducing the Zariski topology, a topology on An whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of k[An]. Then U = V(I(U)) if and only if U is an algebraic set or equivalently a Zariski-closed set. The answer to the second ... | wikipedia |
wiki_25_chunk_13 | Algebraic geometry | For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U. Hilbert's basis theorem implies that ideals in k[An] are always finitely generated. | wikipedia |
wiki_25_chunk_14 | Algebraic geometry | An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the irreducible components of the algebraic set. An irreducible algebraic set is als... | wikipedia |
wiki_25_chunk_15 | Algebraic geometry | Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions or polynomial functions. A regular function on an algebraic set V contained in An is the rest... | wikipedia |
wiki_25_chunk_16 | Algebraic geometry | It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space. J... | wikipedia |
wiki_25_chunk_17 | Algebraic geometry | Since regular functions on V come from regular functions on An, there is a relationship between the coordinate rings. Specifically, if a regular function on V is the restriction of two functions f and g in k[An], then f − g is a polynomial function which is null on V and thus belongs to I(V). Thus k[V] may be identifie... | wikipedia |
wiki_25_chunk_18 | Algebraic geometry | Morphism of affine varieties
Using regular functions from an affine variety to A1, we can define regular maps from one affine variety to another. First we will define a regular map from a variety into affine space: Let V be a variety contained in An. Choose m regular functions on V, and call them f1, ..., fm. We defin... | wikipedia |
wiki_25_chunk_19 | Algebraic geometry | If V′ is a variety contained in Am, we say that f is a regular map from V to V′ if the range of f is contained in V′. The definition of the regular maps apply also to algebraic sets.
The regular maps are also called morphisms, as they make the collection of all affine algebraic sets into a category, where the objects a... | wikipedia |
wiki_25_chunk_20 | Algebraic geometry | Given a regular map g from V to V′ and a regular function f of k[V′], then . The map is a ring homomorphism from k[V′] to k[V]. Conversely, every ring homomorphism from k[V′] to k[V] defines a regular map from V to V′. This defines an equivalence of categories between the category of algebraic sets and the opposite ca... | wikipedia |
wiki_25_chunk_21 | Algebraic geometry | In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions. | wikipedia |
wiki_25_chunk_22 | Algebraic geometry | If V is an affine variety, its coordinate ring is an integral domain and has thus a field of fractions which is denoted k(V) and called the field of the rational functions on V or, shortly, the function field of V. Its elements are the restrictions to V of the rational functions over the affine space containing V. The ... | wikipedia |
wiki_25_chunk_23 | Algebraic geometry | As with regular maps, one may define a rational map from a variety V to a variety V'. As with the regular maps, the rational maps from V to V' may be identified to the field homomorphisms from k(V') to k(V). Two affine varieties are birationally equivalent if there are two rational functions between them which are inve... | wikipedia |
wiki_25_chunk_24 | Algebraic geometry | An affine variety is a rational variety if it is birationally equivalent to an affine space. This means that the variety admits a rational parameterization, that is a parametrization with rational functions. For example, the circle of equation is a rational curve, as it has the parametric equation which may also be vi... | wikipedia |
wiki_25_chunk_25 | Algebraic geometry | The problem of resolution of singularities is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular (see also smooth completion). It was solved in the affirmative in characteristic 0 by Heisuke Hironaka in 1964 and is yet unsolved in finite characteristic.... | wikipedia |
wiki_25_chunk_26 | Algebraic geometry | Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space. Whereas the comple... | wikipedia |
wiki_25_chunk_27 | Algebraic geometry | To see how this might come about, consider the variety . If we draw it, we get a parabola. As x goes to positive infinity, the slope of the line from the origin to the point (x, x2) also goes to positive infinity. As x goes to negative infinity, the slope of the same line goes to negative infinity. | wikipedia |
wiki_25_chunk_28 | Algebraic geometry | Compare this to the variety V(y − x3). This is a cubic curve. As x goes to positive infinity, the slope of the line from the origin to the point (x, x3) goes to positive infinity just as before. But unlike before, as x goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact op... | wikipedia |
wiki_25_chunk_29 | Algebraic geometry | The consideration of the projective completion of the two curves, which is their prolongation "at infinity" in the projective plane, allows us to quantify this difference: the point at infinity of the parabola is a regular point, whose tangent is the line at infinity, while the point at infinity of the cubic curve is a... | wikipedia |
wiki_25_chunk_30 | Algebraic geometry | Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry s... | wikipedia |
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