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Generic flatness
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<h2 id="generic-freeness">Generic freeness</h2> <p>Generic flatness is a consequence of the generic freeness lemma. Generic freeness states that if <i>A</i> is a <a href="/facts/Noetherian_ring/kYbRSdnp">noetherian</a> <a href="/facts/Integral_domain/cylt6zxW">integral domain</a>, <i>B</i> is a <a href="/facts/Algebra_of_finite_type/57a93D5A">finite type</a> <i>A</i>-algebra, and <i>M</i> is a finite type <i>B</i>-module, then there exists a non-zero element <i>f</i> of <i>A</i> such that <i>M</i><i>f</i> is a free <i>A</i><i>f</i>-module.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> Generic freeness can be extended to the <a href="/facts/Graded_ring/AIUjkSaP">graded</a> situation: If <i>B</i> is graded by the natural numbers, <i>A</i> acts in degree zero, and <i>M</i> is a graded <i>B</i>-module, then <i>f</i> may be chosen such that each graded component of <i>M</i><i>f</i> is free.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>Generic freeness is proved using Grothendieck's technique of <a href="/facts/D%C3%A9vissage/XqHbEuav">dévissage</a>. Another version of generic freeness can be proved using <a href="/facts/Noether%27s_normalization_lemma/a7jpWIHv">Noether's normalization lemma</a>. </p> <h2 id="bibliography">Bibliography</h2> <ul><li><a href="/facts/David_Eisenbud/hw2yEnnx">Eisenbud, David</a> (1995), <i>Commutative algebra with a view toward algebraic geometry</i>, Graduate Texts in Mathematics, vol. 150, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94268-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1322960">1322960</a></li> <li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexandre</a>; <a href="/facts/Jean_Dieudonn%C3%A9/gWagzMdu">Dieudonné, Jean</a> (1965). <a href="http://www.numdam.org/articles/PMIHES_1965__24__5_0">"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie"</a>. <i><a href="/facts/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S/2ur5ofxL">Publications Mathématiques de l'IHÉS</a></i>. 24. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02684322">10.1007/bf02684322</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0199181">0199181</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>EGA IV2, Théorème 6.9.1 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>EGA IV2, Corollaire 6.9.3 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>EGA IV2, Lemme 6.9.2 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Eisenbud, Theorem 14.4 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>