Methods for constructing Frobenius structures - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:25:47Z http://mathoverflow.net/feeds/question/12655 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12655/methods-for-constructing-frobenius-structures Methods for constructing Frobenius structures Hanno Becker 2010-01-22T16:39:02Z 2010-01-23T14:07:24Z <p>Let <code>${\mathbb F}:({\mathcal A},{\mathcal E}_{\mathcal A})\to({\mathcal B},{\mathcal E}_{\mathcal B})$</code> be an exact functor between exact categories, and suppose <code>${\mathbb F}$</code> has both a left adjoint <code>${\mathbb F}_\lambda$</code> and a right adjoint <code>${\mathbb F}_\rho$</code>. Then the class</p> <p><code>${\mathcal E} := \{X\to Y\to Z\in{\mathcal E}_{\mathcal A}\ |\ {\mathbb F}X\to{\mathbb F}Y\to{\mathbb F}Z\in{\mathcal E}_{\mathcal B}\}$</code></p> <p>defines another exact structure on <code>${\mathcal A}$</code>.</p> <p>It is interesting to ask for criteria to decide when this exact structure is Frobenius. One such criterion is the following:</p> <p>Suppose ${\mathcal E}_{\mathcal B}$ is the split exact structure, and that for each $X\in{\mathcal A}$ the unit <code>$\eta_X: X\to{\mathbb F}_\rho{\mathbb F}X$</code> is an <code>${\mathcal E}_{\mathcal A}$</code>-monomorphism, while the counit <code>${\mathbb F}_\lambda{\mathbb F}X\to X$</code> is an <code>${\mathcal E}_{\mathcal A}$</code>-epimorphism. Then $({\mathcal A},{\mathcal E})$ has enough projectives and injectives, and the classes ${\mathcal P}$/${\mathcal I}$ of projective/injective objects in $({\mathcal A},{\mathcal E})$ are given by</p> <p><code>${\mathcal P} = \{X\in{\mathcal A}\ |\ X\text{ is a summand of some }{\mathbb F}_\lambda , Y\in{\mathcal B}\}$</code> and</p> <p><code>${\mathcal I} = \{X\in{\mathcal A}\ |\ X\text{ is a summand of some }{\mathbb F}_\rho Y, Y\in{\mathcal B}\}$</code>,</p> <p>respectively. Consequently, if <code>${\mathbb F}_\lambda$</code> and <code>${\mathbb F}_\rho$</code> have the same image, then <code>$({\mathcal A},{\mathcal E})$</code> is Frobenius.</p> <p>This seems very restrictive, but in fact there are at least two cases I know where it can be applied:</p> <p><strong>(1)</strong> If ${\mathcal A}$ is a dg-category, then the forgetful functor ${\mathbb F}: \text{dg-mod}({\mathcal A})\to\text{gr-mod}({\mathcal A})$ fulfills the requirements of the criterion above and thus can be used to construct a Frobenius structure on $\text{dg-mod}({\mathcal A})$ (for pretriangulated dg-categories ${\mathcal A}$, this structure can in turn be restricted to ${\mathcal A}$ itself).</p> <p><strong>(2)</strong> If $G$ is a finite group, $H$ is a subgroup, then the fortgetful functor $G\text{-mod}\to H\text{-mod}$ has left adjoint $\text{Ind}^G_H$ and right adjoint $\text{Coind}^G_H$, and these two functors coincide for $(G:H)&lt;\infty$. In this case, the above criterion therefore applies to provide $G\text{-mod}$ with a Frobenius structure "relative to $H$".</p> <p><strong>Question</strong></p> <p>Do you know more criteria for constructing Frobenius structures and situations where they can be applied?</p> <p><strong>For example</strong>, I would be interested in a criterion which can be applied to show that the category of maximal Cohen-Macaulay modules over a Gorenstein ring is Frobenius.</p>