a strange product of an inverse limit - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:20:46Z http://mathoverflow.net/feeds/question/77938 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77938/a-strange-product-of-an-inverse-limit a strange product of an inverse limit uuuuuu 2011-10-12T17:24:47Z 2011-10-12T17:24:47Z <p>Hi,</p> <p>consider the polytope in $\mathbb{R}$ given by the segment $[1,2]$. Consider the paving given by $s\in {1},[1,2],{2}$ ("the faces"). For every of these faces $s$ consider the cone $Cone(1,s)\subset \mathbb{R}^2$ and let $N_s$ the integrals points of these cones. The $N_s$ are monoids. Since the faces give an ordered set (inclusion) one can consider $$N=\varinjlim_s N_s^{gp}$$</p> <p>where $N_s^{gp}$ is the group associated to the monoids. Let $P$ be the set of integral points of the cone $Cone(1,[1,2])$. We have maps $a_s:N_s\rightarrow N$ inducing $a:P\rightarrow N$. For $p,q\in P$ define $p*q=a(p)+a(q)-a(p+q)$.</p> <p>Can you tell me which is a couple of points for which $p*q\neq 0$? If in this case the products are all zero can you give to me an example where instead of $[1,2]$ we have an integral polytope in $\mathbb{R}^n$ ($n\geq 2$) for which this product is non trivial?</p> <p>Thanks</p>