User calc - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:38:54Z http://mathoverflow.net/feeds/user/19452 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82603/minimal-generation-for-finite-abelian-groups Minimal generation for finite abelian groups Calc 2011-12-04T08:35:39Z 2011-12-07T11:28:53Z <p>Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups:</p> <p>1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$,</p> <p>2) With orders that are powers of not necessarily distinct primes $p_1^{\alpha_1}, \ldots, p_n^{\alpha_n}$.</p> <p>Is it true, and how can one prove that the cardinality $c$ of any minimal generating set for $G$ satisfies $k \leq c \leq n$ (I am most concerned about the second inequality)? Here minimal means irredundant.</p> http://mathoverflow.net/questions/81611/picard-groups-of-fiber-products Picard groups of (fiber) products Calc 2011-11-22T14:34:10Z 2011-11-22T16:38:08Z <p>Let us work in the "nice" situation where $X,Y,Z$ are smooth complex algebraic varieties, not necessarily compact. Assume that the fiber product $W:= X \times_Z Y$ is also smooth. What can we say about the Picard group of $W$? </p> <p>More precisely, assume that $Pic(X)=Pic(Z)=0$.</p> <p>1) Can we deduce that $Pic(W)=Pic(Y)$?</p> <p>2) If 1) is not true in general, can we draw the conclusion if $Z$ is a point and thus $W=X \times Y$?</p> <p>3) If 1) is not true in general, can we draw the conclusion if $Y \to Z$ is a finite étale cover?</p> http://mathoverflow.net/questions/82603/minimal-generation-for-finite-abelian-groups/82618#82618 Comment by Calc Calc 2011-12-05T11:18:40Z 2011-12-05T11:18:40Z Thank you again! http://mathoverflow.net/questions/82603/minimal-generation-for-finite-abelian-groups/82618#82618 Comment by Calc Calc 2011-12-05T11:11:07Z 2011-12-05T11:11:07Z Is this not a restriction on the set of all minimal generating sets of maximal cardinality? Is the cardinality of all minimal generating sets of maximal cardinality all the same? http://mathoverflow.net/questions/82603/minimal-generation-for-finite-abelian-groups/82618#82618 Comment by Calc Calc 2011-12-05T10:55:42Z 2011-12-05T10:55:42Z Sorry, one more thing. What does &quot;minimize the sum of the orders of elements of $S$ subject to that&quot; mean? http://mathoverflow.net/questions/82603/minimal-generation-for-finite-abelian-groups/82618#82618 Comment by Calc Calc 2011-12-05T10:05:09Z 2011-12-05T10:05:09Z Thanks for your answer! How can you write $s=t+u$ with orders of $t$ and $u$ respectively $a&gt;1, b&gt;1$ and such that $\gcd(a,b)=1$ and $a \dot b$ is the order of $s$? http://mathoverflow.net/questions/82603/minimal-generation-for-finite-abelian-groups Comment by Calc Calc 2011-12-04T11:59:38Z 2011-12-04T11:59:38Z Ok. Sorry. How can one erase a question? ;) http://mathoverflow.net/questions/82603/minimal-generation-for-finite-abelian-groups Comment by Calc Calc 2011-12-04T11:31:18Z 2011-12-04T11:31:18Z It is part of the fundamental theorem for finitely generated abelian groups that the $d_i, p_i, \alpha_i$ are uniquely determined by $G$ itself (up to reordering in the second case). If instead you are asking about a name, I heard of people calling $k$ and $n$ respectively the minimal and maximal rank. http://mathoverflow.net/questions/81611/picard-groups-of-fiber-products Comment by Calc Calc 2011-11-22T19:41:32Z 2011-11-22T19:41:32Z Thanks! But in the example in Hartshorne IV.4.10 $Pic(X) \neq 0$. http://mathoverflow.net/questions/81611/picard-groups-of-fiber-products/81624#81624 Comment by Calc Calc 2011-11-22T19:39:43Z 2011-11-22T19:39:43Z I was aware of IV 4.10, but I was still hoping to see an example where $Pic(X)=0$. I was not aware of III.12.4 and 12.5, thanks a lot!! http://mathoverflow.net/questions/81611/picard-groups-of-fiber-products/81617#81617 Comment by Calc Calc 2011-11-22T19:36:54Z 2011-11-22T19:36:54Z Thank you very much, your examples are both very nice and simple. The first example makes me believe that the question, as I posed it, is really wrong. This because $H^2(X \times Y)$ contains $H^1(X) \otimes H^1(Y)$, and the latters could make algebraic classes on $X \times Y$. Does this make sense?