Chiral homology: Difference between revisions

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In algebraic geometry, a '''quotient stack''' is a [[stack (mathematics)|stack]] that generalizes a quotient of a scheme or a variety by a group.


It is defined as follows. Let ''G'' be an affine smooth group scheme over a scheme ''S'' and ''X'' a ''S''-scheme on which ''G'' acts. Let <math>[X/G]</math> be the category over the category of ''S''-schemes: an object over ''T'' is a principal ''G''-bundle ''E'' →''T'' together with equivariant map ''E'' →''X''; an arrow from ''E'' →''T'' to ''E<nowiki>'</nowiki>'' →''T<nowiki>'</nowiki>'' is a bundle map (i.e., forms a cartesian diagram) that is compatible with the equivariant maps ''E'' →''X'' and ''E<nowiki>'</nowiki>'' →''X''.


In general, <math>[X/G]</math> is an [[Artin stack]] (also called algebraic stack). If the stabilizers of the [[geometric point]]s are finite and reduced, then it is the [[Deligne–Mumford stack]]. If the stabilizers are trivial, then it is an [[algebraic space]].
Booker T. Washington<br><br>As you travel this great nation, it is no accident you will see a lot of schools given the name of Booker T. Washington.  That is because this great black educator and leader set the standard and carved out a new path in the years right after  [http://www.riversidemediagroup.com/uggboots.asp UGG Boots Outlet] the fall of slavery to lead his people to a better way. He showed his people a way of education, accomplishment, achievement and the prosperity that naturally comes with those goals.<br><br>In 1901, the biography of Booker T. Washington was published with the fitting title Up From Slavery.  Washington's struggle to rise up from the limitations of a slave's life  [http://www.riversidemediagroup.com/uggboots.asp http://www.riversidemediagroup.com/uggboots.asp] to be come one of the most respected black leaders in America is one of the reasons he is revered in black history as one of the greats who really made a difference for his people.<br><br>When Booker T. Washington's family was freed from slavery in Virginia, young Booker immediately began pursuing the path where he would make his mark, in education.  Achieving success at Hampton University and then at Wayland Seminary, he was soon to pioneer new achievements for African Americans in higher education, becoming one of the first leaders of the Tuskegee University in Alabama.  <br><br>But it was more than just academic success that marked Washington's career. He became prominent in many areas of leadership  [http://www.riversidemediagroup.com/uggboots.asp UGG Boots] becoming a spokesperson for post slavery black America to the powerful and influential in this country. Book T. Washington lived the concept that the pen was mightier than the sword and was an early voice for moderation and learning to excel within the institutions and customs of America rather than deal in violence.<br><br>One of Washington's great strengths was finding partnership and coalitions between leaders of many communities to improve the opportunities for education and excellence for the African American community of the time. One of the most influential speeches of black history was given by Washington and became known as the Atlanta Address of 1895 in which Washington, speaking to a largely white  [http://www.riversidemediagroup.com/uggboots.asp UGG Boots] audience instigated a profound change in way economic opportunity and hiring was done in [http://en.Wiktionary.org/wiki/America America] at its time.  In that one speech he...<br><br>*    Called up on the black community to become part of the economy and industry of America thus beginning the healing process that was so necessary at the time.<br><br>*   Stated without reservation that the south was the region of the country where there were the greatest opportunities for black employment.  By bringing together the strong black labor force with an economy in the midst of recovery from the civil war, Washington may have been one of the chief architects for the recovery of the south from the ravages of that war.<br><br>*    Introduced to the economic institutions predominantly run by  [http://www.riversidemediagroup.com/uggboots.asp http://www.riversidemediagroup.com/uggboots.asp] the white citizens of the country that it made more sense to take advantage of the large resident black population for reliable workers than to look to immigrants.  The outcome was a boom in employment for the black community that was a huge leap forward in the struggle to rise up out of slavery.<br><br>The Atlanta Address of 1895 propelled Booker T. Washington into national prominence becoming a healing voice and a powerful catalyst for change in this country.  Using his sophisticated network of supporters from every arena of leadership including political, academic and business leaders, Washington worked tirelessly to provide hope and new opportunities for black families trying to make their way in America.<br><br>His work ethic was profound and produced change at a rate that was phenomenal by any standard.  But it took a toll on Washington who died relatively young, at the age of 59 from exhaustion and overwork. But this too points out the tremendous drive and devotion this important black leader had to use all of his talents, his intellect and his contacts to better the lives of black people and speed the road to acceptance and integration across America. We all owe a Booker T. Washington a great deal of gratitude for being "the man of the hour" to lead all people forward, black and white, to find ways to work together in partnership rather than with distrust or violence to achieve a better America for everyone.<br><br>PPPPP <br>arvinoor<br>arvinoor siregar<br>arvinoor siregar sh.<br>arvinoor siregar sh.mh.<br>Word Count 710
 
If <math>X = S</math> with trivial action of ''G'', then <math>[S/G]</math> is called the '''classifying stack''' of ''G'' (in analogy with the [[classifying space]] of ''G'') and is usually denoted by ''BG''.
 
Example: For a ''S''-scheme ''X'', let <math>\operatorname{Bun}_G (X) = \operatorname{Map}(X, BG)</math>, where <math>\operatorname{Map}</math> is a [[mapping stack]].<ref>This definition is given at http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept17(Bun(G)).pdf</ref> It is called the [[moduli stack of principal bundles]] on ''X''.
 
Example:<ref>Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf</ref> Let ''L'' be the [[Lazard ring]]; i.e., <math>L = \pi_* \operatorname{MU}</math>. Then the quotient stack <math>[\operatorname{Spec}L/G]</math> by
<math>G</math>,
:<math>G(R) = \{g \in R[\![t]\!] | g(t) = b_0 t + b_1t^2 \dots, b_0 \in R^\times \}</math>,
is called the [[moduli stack of formal group laws]], denoted by <math>\mathcal{M}_\text{FG}</math>.
 
It is possible to approach the construction from the point of view of [[simplicial sheaf|simplicial sheaves]]; cf. 9.2. of Jardine's "local homotopy theory".<ref>http://www.math.uwo.ca/~jardine/papers/preprints/book.pdf</ref>
 
== References ==
{{reflist}}
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Mumford | first2=David | author2-link=David Mumford | title=The irreducibility of the space of curves of given genus | url=http://www.numdam.org/item?id=PMIHES_1969__36__75_0 | id={{MathSciNet | id = 0262240}} | year=1969 | journal=[[Publications Mathématiques de l'IHÉS]] | issue=36 | pages=75–109 | doi=10.1007/BF02684599 | volume=36}}
Some other references are
*http://www.math.missouri.edu/~edidin/Papers/mfile.pdf
*http://www.math.ubc.ca/~behrend/thesis.pdf
 
 
[[Category:Algebraic geometry]]
 
 
 
{{geometry-stub}}

Latest revision as of 03:57, 13 April 2014


Booker T. Washington

As you travel this great nation, it is no accident you will see a lot of schools given the name of Booker T. Washington. That is because this great black educator and leader set the standard and carved out a new path in the years right after UGG Boots Outlet the fall of slavery to lead his people to a better way. He showed his people a way of education, accomplishment, achievement and the prosperity that naturally comes with those goals.

In 1901, the biography of Booker T. Washington was published with the fitting title Up From Slavery. Washington's struggle to rise up from the limitations of a slave's life http://www.riversidemediagroup.com/uggboots.asp to be come one of the most respected black leaders in America is one of the reasons he is revered in black history as one of the greats who really made a difference for his people.

When Booker T. Washington's family was freed from slavery in Virginia, young Booker immediately began pursuing the path where he would make his mark, in education. Achieving success at Hampton University and then at Wayland Seminary, he was soon to pioneer new achievements for African Americans in higher education, becoming one of the first leaders of the Tuskegee University in Alabama.

But it was more than just academic success that marked Washington's career. He became prominent in many areas of leadership UGG Boots becoming a spokesperson for post slavery black America to the powerful and influential in this country. Book T. Washington lived the concept that the pen was mightier than the sword and was an early voice for moderation and learning to excel within the institutions and customs of America rather than deal in violence.

One of Washington's great strengths was finding partnership and coalitions between leaders of many communities to improve the opportunities for education and excellence for the African American community of the time. One of the most influential speeches of black history was given by Washington and became known as the Atlanta Address of 1895 in which Washington, speaking to a largely white UGG Boots audience instigated a profound change in way economic opportunity and hiring was done in America at its time. In that one speech he...

* Called up on the black community to become part of the economy and industry of America thus beginning the healing process that was so necessary at the time.

* Stated without reservation that the south was the region of the country where there were the greatest opportunities for black employment. By bringing together the strong black labor force with an economy in the midst of recovery from the civil war, Washington may have been one of the chief architects for the recovery of the south from the ravages of that war.

* Introduced to the economic institutions predominantly run by http://www.riversidemediagroup.com/uggboots.asp the white citizens of the country that it made more sense to take advantage of the large resident black population for reliable workers than to look to immigrants. The outcome was a boom in employment for the black community that was a huge leap forward in the struggle to rise up out of slavery.

The Atlanta Address of 1895 propelled Booker T. Washington into national prominence becoming a healing voice and a powerful catalyst for change in this country. Using his sophisticated network of supporters from every arena of leadership including political, academic and business leaders, Washington worked tirelessly to provide hope and new opportunities for black families trying to make their way in America.

His work ethic was profound and produced change at a rate that was phenomenal by any standard. But it took a toll on Washington who died relatively young, at the age of 59 from exhaustion and overwork. But this too points out the tremendous drive and devotion this important black leader had to use all of his talents, his intellect and his contacts to better the lives of black people and speed the road to acceptance and integration across America. We all owe a Booker T. Washington a great deal of gratitude for being "the man of the hour" to lead all people forward, black and white, to find ways to work together in partnership rather than with distrust or violence to achieve a better America for everyone.

PPPPP
arvinoor
arvinoor siregar
arvinoor siregar sh.
arvinoor siregar sh.mh.
Word Count 710

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