https://en.formulasearchengine.com/index.php?action=history&feed=atom&title=Interband_cascade_laserInterband cascade laser - Revision history2026-05-31T12:28:11ZRevision history for this page on the wikiMediaWiki 1.43.0-wmf.28https://en.formulasearchengine.com/index.php?title=Interband_cascade_laser&diff=28299&oldid=preven>Mogism: Cleanup/Typo fixing, typo(s) fixed: thusfar → thus far, , → , using AWB2013-10-07T00:35:23Z<p>Cleanup/<a href="/index.php?title=WP:AWB/T&action=edit&redlink=1" class="new" title="WP:AWB/T (page does not exist)">Typo fixing</a>, <a href="/index.php?title=WP:AWB/T&action=edit&redlink=1" class="new" title="WP:AWB/T (page does not exist)">typo(s) fixed</a>: thusfar → thus far, , → , using <a href="/index.php?title=Testwiki:AWB&action=edit&redlink=1" class="new" title="Testwiki:AWB (page does not exist)">AWB</a></p>
<p><b>New page</b></p><div>In mathematics, the '''Koszul cohomology''' groups ''K''<sub>''p'',''q''</sub>(''X'', ''L'') are groups associated to a projective variety ''X'' with a line bundle ''L''. They were introduced by {{harvs|txt|last=Green|year=1984|year2=1984b}}, and named after [[Jean-Louis Koszul]] as they are closely related to the [[Koszul complex]].<br />
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{{harvtxt|Green|1989}} surveys early work on Koszul cohomology, {{harvtxt|Eisenbud|2005}} gives an introduction to Koszul cohomology, and {{harvtxt|Aprodu|Nagel|2010}} gives a more advanced survey.<br />
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==Definitions==<br />
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If ''M'' is a graded module over the symmetric algebra of a vector space ''V'', then the Koszul cohomology ''K''<sub>''p'',''q''</sub>(''M'',''V'') of ''M'' is given by the cohomology of the sequence <br />
:<math>\wedge^{p+1}M_{q-1}\rightarrow \wedge^{p}M_{q} \rightarrow \wedge^{p-1}M_{q+1}</math><br />
If ''L'' is a line bundle over a projective variety ''X'', then the Koszul cohomology ''K''<sub>''p'',''q''</sub>(''X'',''L'') is given by the Koszul cohomology ''K''<sub>''p'',''q''</sub>(''M'',''V'') of the graded module ''M'' = ⊕<sub>''q''</sub>''H''<sup>0</sup>(''L''<sup>''q''</sup>), as a module over the symmetric algebra of the vector space ''V''=''H''<sup>0</sup>(''L'').<br />
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==References==<br />
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*{{Citation | last1=Aprodu | first1=Marian | last2=Nagel | first2=Jan | title=Koszul cohomology and algebraic geometry | url=http://books.google.com/books?id=Pxw9_38LlnYC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=University Lecture Series | isbn=978-0-8218-4964-4 | id={{MR|2573635}} | year=2010 | volume=52}}<br />
*{{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=The geometry of syzygies | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-22215-8 | doi=10.1007/b137572 | id={{MathSciNet | id = 2103875}} | year=2005 | volume=229}}<br />
*{{Citation | last1=Green | first1=Mark L. | title=Koszul cohomology and the geometry of projective varieties | url=http://projecteuclid.org/getRecord?id=euclid.jdg/1214438426 | id={{MR|739785}} | year=1984 | journal=Journal of Differential Geometry | issn=0022-040X | volume=19 | issue=1 | pages=125–171}}<br />
*{{Citation | last1=Green | first1=Mark L. | title=Koszul cohomology and the geometry of projective varieties. II | url=http://projecteuclid.org/getRecord?id=euclid.jdg/1214439000 | id={{MR|772134}} | year=1984 | journal=Journal of Differential Geometry | issn=0022-040X | volume=20 | issue=1 | pages=279–289}}<br />
*{{Citation | last1=Green | first1=Mark L. | editor1-last=Cornalba | editor1-first=Maurizio | editor2-last=Gómez-Mont | editor2-first=X. | editor3-last=Verjovsky | editor3-first=A. | title=Lectures on Riemann surfaces | publisher=World Sci. Publ., Teaneck, NJ | series=Proceedings of the First College on Riemann Surfaces held in Trieste, November 9–December 18, 1987 | isbn=9789971509026 | id={{MR|1082354}} | year=1989 | chapter=Koszul cohomology and geometry | pages=177–200}}<br />
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[[Category:Algebraic geometry]]</div>en>Mogism