Is the bounded derived category of coherent sheaves of a variety a small category? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T03:02:16Z http://mathoverflow.net/feeds/question/88372 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88372/is-the-bounded-derived-category-of-coherent-sheaves-of-a-variety-a-small-category Is the bounded derived category of coherent sheaves of a variety a small category? David Steinberg 2012-02-13T20:13:26Z 2012-02-14T02:09:09Z <p>The question is in the title. </p> <p>I am trying to apply the Mitchell (Freyd-Mitchell?) embedding theorem, which states that for every small abelian category $A$, there exists a ring $R$ such that A embeds into the category $R$-mod. The derived category is not abelian, of course, but I have a particular subcategory that is abelian, and life would be easiest if the derived category was smal, so that the subcategory was small and abelian. </p> http://mathoverflow.net/questions/88372/is-the-bounded-derived-category-of-coherent-sheaves-of-a-variety-a-small-category/88394#88394 Answer by S. Carnahan for Is the bounded derived category of coherent sheaves of a variety a small category? S. Carnahan 2012-02-14T02:09:09Z 2012-02-14T02:09:09Z <p>The answer to your question really depends on what you mean by the word "the". An unhelpful answer is that the coherent sheaves over any variety form a proper class (hence "no"). A more useful answer is (as mentioned in the comments) that there exists a small category that is equivalent to any category that can be reasonably called the bounded derived category of coherent sheaves (hence "yes").</p> <p>Furthermore, the construction of such a category can be accomplished without the use of replacement. In particular, the category lives in the same ZC universe (i.e., $V_\alpha$ for $\alpha$ a not-necessarily-inaccessible limit ordinal greater than $\omega$ - see e.g., <a href="http://en.wikipedia.org/wiki/Von_Neumann_universe" rel="nofollow">Wikipedia</a> or the set theory section of the <a href="http://math.columbia.edu/algebraic_geometry/stacks-git/browse.html" rel="nofollow">Stacks project</a>) as the defining field.</p>