Commutative algebras and Gamma-modules - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T01:58:30Z http://mathoverflow.net/feeds/question/41440 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41440/commutative-algebras-and-gamma-modules Commutative algebras and Gamma-modules Semen Podkorytov 2010-10-07T18:17:05Z 2010-11-24T13:44:00Z <p>A commutative algebra (with unity) over a field gives rise to the covariant functor F: Set_f->Vect from finite sets to vector spaces: F(E) := A^{otimes E}. Is it true that, over complex numbers, a finite dimensional algebra can be reconstructed from the corresponding functor?</p> <p>(A Gamma-module is a functor from finite <em>pointed</em> sets to vector spaces; so F is not a Gamma-module. I use this term in the title just because I do not know the correct term for F: Set_f->Vect.)</p> http://mathoverflow.net/questions/41440/commutative-algebras-and-gamma-modules/41459#41459 Answer by Andreas Blass for Commutative algebras and Gamma-modules Andreas Blass 2010-10-07T21:22:32Z 2010-10-07T21:22:32Z <p>I'd guess that the intended functor uses the multiplication operation of $A$ to provide $F(s):A\otimes A\to A$ where $s$ is the surjection <code>$\{1,2\}\to\{1\}$</code> mentioned by Mattia Talpo, that it uses the unity element of $A$ to provide $F(i):k\to A$ where $i$ is the injection <code>$\emptyset\to\{1\}$</code> and $k$ is the field of scalars, and that $F$ is to be defined on arbitrary maps between finite sets by a natural generalization and combination of these two examples. The trouble with this guess is that it makes the question trivial, since the algebra structure is contained in $F(s)$ and $F(i)$, so no work is needed to reconstruct the algebra from $F$. Unfortunately, I have no alternative guess as to what $F$ the proposer might have intended.</p> http://mathoverflow.net/questions/41440/commutative-algebras-and-gamma-modules/41506#41506 Answer by Semen Podkorytov for Commutative algebras and Gamma-modules Semen Podkorytov 2010-10-08T13:04:37Z 2010-10-08T13:04:37Z <p>Let me clarify my question. For a commutative algebra $A$ we define a functor $F:\mathrm{Set}_\mathrm{f}\to\mathrm{Vect}$ by</p> <p>$F(I)=A^{\otimes I}$ for a finite set $I$ and</p> <p>$F(t):F(I)\to F(J)$, $\bigotimes_{i\in I}a_i\mapsto\bigotimes_{j\in J}\prod_{i\in t^{-1}(j)}a_i$ for a map $t:I\to J$ (exactly as Andreas Blass proposed).</p> <p>Suppose now that two finite-dimensional algebras $A$ and $B$ over the complex numbers produce isomorphic functors $F$ and $G$. Is it true that then $A$ and $B$ are isomorphic?</p> <p>The question is not trivial. Let $e:F\to G$ be an isomorphism of functors. Then <code>$e_{\{1\}}:A\to B$</code> and <code>$e_{\{1,2\}}:A\otimes A\to B\otimes B$</code> are isomorphisms of vector spaces. If we had <code>$e_{\{1,2\}}=e_{\{1\}}\otimes e_{\{1\}}$</code>, this would imply that <code>$e_{\{1\}}$</code> is an isomorphism of algebras. The problem is that we have only linear naturality relations between $e_I$.</p> http://mathoverflow.net/questions/41440/commutative-algebras-and-gamma-modules/47229#47229 Answer by Semen Podkorytov for Commutative algebras and Gamma-modules Semen Podkorytov 2010-11-24T13:44:00Z 2010-11-24T13:44:00Z <p>Now I can prove this, see <a href="http://www.pdmi.ras.ru/~ssp/te.pdf" rel="nofollow">http://www.pdmi.ras.ru/~ssp/te.pdf</a></p>