Spectrum of the Grothendieck ring of varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:59:36Z http://mathoverflow.net/feeds/question/319 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/319/spectrum-of-the-grothendieck-ring-of-varieties Spectrum of the Grothendieck ring of varieties Ilya Nikokoshev 2009-10-11T22:10:05Z 2009-12-02T17:29:41Z <p>Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my memory it appears this was posed by Grothendieck as part of the big program of motives.</p> <p>Consider <strong>classes of complex algebraic varieties X modulo relations</strong> </p> <pre><code> [X] - [Y] = [X\Y], [X x Y] = [X] x [Y], </code></pre> <p>Also, if you're familiar with taking inverse of an affine line, let's do that too: $$\exists \mathbb A^{-1}\quad \text{such that}\quad [\mathbb A] \cdot [\mathbb A^{-1}] = [\mathbb A^0].$$ </p> <p>(+ if you want, you can also take idempotent completion and formal completion by A^-1).</p> <p>It's not hard to see that you can add (formally) and multiply (geometric product as above) those things, so they <strong>form a ring</strong>. Let's denote this ring&nbsp; <strong><code>Mot</code></strong> (It's actually very close to what Grothendieck called <em>baby motives</em>.) </p> <p>And for things that form a ring you can <strong>study their <code>Spec</code></strong>. For example, you can talk about <strong>points of the ring</strong> &mdash; each point is by definition a homomorphism to complex numbers.</p> <blockquote> <p><strong>Question:</strong> what are the properties of <code>Spec Mot</code>? How to describe its points?</p> </blockquote> <p>For example, one point is <strong>Euler characteristics</strong> <code>$\chi \in \text{Spec}\,\mathbf{Mot}$</code>, since it's additive and multiplicative (it's even integral!) Any other homomorphism to complex numbers is thus sometimes called <em>generalized Euler characteristics</em>.</p> <p>There's also a plane there given by mixed Hodge polynomials (that is, polynomials whose coefficients are weighted Hodge numbers $h^{p,q}_k$), since Hodge polynomial at a given point satisfies those relations too (see the references below). </p> <p>As Ben says below, things would become even more interesting if we considered this ring for schemes over $\mathbb Z$, because then each $q$ would give a generalized Euler characteristic $\chi_q$ that counts points of $X(\mathbb F_q).$</p> <p>Are there any other points? Any more information? </p> http://mathoverflow.net/questions/319/spectrum-of-the-grothendieck-ring-of-varieties/371#371 Answer by Ben Webster for Spectrum of the Grothendieck ring of varieties Ben Webster 2009-10-12T19:14:50Z 2009-10-12T19:14:50Z <p>If you considered varieties over Z instead of over C, you would have homomorphisms given by counting points over all the different finite fields.</p> http://mathoverflow.net/questions/319/spectrum-of-the-grothendieck-ring-of-varieties/692#692 Answer by JSE for Spectrum of the Grothendieck ring of varieties JSE 2009-10-16T01:22:56Z 2009-10-16T15:18:29Z <p>One interesting fact about Spec M is that it isn't integral; i.e., the ring M has zero divisors. This was proved by Poonen in 2002:</p> <p><a href="http://arxiv1.library.cornell.edu/abs/math/0204306v1" rel="nofollow">"The Grothendieck ring of varieties is not a domain"</a></p> <p>Re points of Spec M: I suppose if you considered varieties over R instead of C, you would in addition have the map sending X to the Euler characteristic of X(R), though I've never seen this used. </p> <p><strong>Update</strong>: Oh, I've never seen this used because it's totally wrong. For instance, A^0(R) and A^1(R) have Euler characteristic 1 but P^1(R) doesn't have Euler characteristic 2. I think the mod-2 Euler characteristic would probably be OK here.</p> http://mathoverflow.net/questions/319/spectrum-of-the-grothendieck-ring-of-varieties/1280#1280 Answer by Ilya Nikokoshev for Spectrum of the Grothendieck ring of varieties Ilya Nikokoshev 2009-10-19T20:36:12Z 2009-12-02T17:29:41Z <p>I think I'll be collecting references I found in this answer, rather then in the original (already large) post:</p> <ul> <li><a href="http://www-fourier.ujf-grenoble.fr/~peters/hodge.f/peters-proc.pdf" rel="nofollow">http://www-fourier.ujf-grenoble.fr/~peters/hodge.f/peters-proc.pdf</a></li> </ul> <p>It's a community wiki &mdash; feel free to add!</p> http://mathoverflow.net/questions/319/spectrum-of-the-grothendieck-ring-of-varieties/7430#7430 Answer by Mikhail Bondarko for Spectrum of the Grothendieck ring of varieties Mikhail Bondarko 2009-12-01T13:49:58Z 2009-12-01T20:42:20Z <p>This ring is very important for motivic integration; so it might be useful for you to read surveys on this subject. </p> <p>Yet I would say that this ring is too large and complicated. A reasonable factor-ring of it is K_0 of Chow motives. If you take Chow motives with rational coefficients then as a group it (conjecturally!) would be a free abelian group with generators being isomorphism classes of indecomposable numerical motives.</p> <p>You could also be interested in weight complexes: see H. Gillet, C. Soule, Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996) or my own paper <a href="http://arxiv.org/abs/math/0601713" rel="nofollow">http://arxiv.org/abs/math/0601713</a></p>