https://en.formulasearchengine.com/w/index.php?action=history&feed=atom&title=Loss_networkLoss network - Revision history2026-07-22T10:45:46ZRevision history for this page on the wikiMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Loss_network&diff=28555&oldid=preven>Gareth Jones at 09:33, 12 September 20132013-09-12T09:33:56Z<p></p>
<p><b>New page</b></p><div>In [[algebraic geometry]], the '''projection formula''' states that,<ref>{{harvnb|Hartshorne|1977|loc=Ch III, Exercise 8.3}}</ref><ref>http://math.stanford.edu/~vakil/0708-216/216class38.pdf</ref> for a quasi-compact separated morphism of schemes <math>f:X \to Y</math>, a quasi-coherent sheaf <math>\mathcal{E}</math> on ''X'', a locally free sheaf <math>\mathcal{F}</math> on ''Y'', the natural maps of sheaves<br />
:<math>R^i f_* \mathcal{E} \otimes \mathcal{F} \to R^i f_* (\mathcal{E} \otimes f^* \mathcal{F})</math><br />
are isomorphisms.<br />
<br />
There is also a version of the formula in the intersection theory. For example,<ref>{{harvnb|Kollár|1996|loc=Ch VI. Proposition 2.11}}</ref> let <math>f:X \to Y</math> be a morphism of ''S''-schemes, <math>L_i, 1 \le i \le m</math> line bundles on ''X'' and ''F'' a sheaf on ''Y'' with support that is proper over a zero-dimensional subscheme of ''S'' and <math>m \ge \dim \operatorname{supp}F</math>. Then<br />
:<math>f^*L_1 \cdots f^* L_m \cdot F = L_1 \cdots L_m \cdot f_* F</math>.<br />
<br />
There is yet another projection formula in the setting of étale cohomology.<br />
<br />
== References ==<br />
{{reflist}}<br />
*{{Hartshorne AG}}<br />
*{{cite book |first1=János |last1=Kollár |title=Rational curves on algebraic varieties |year=1996}}<br />
<br />
[[Category:Algebraic geometry]]<br />
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{{geometry-stub}}</div>en>Gareth Jones