Looking for reference talking about torsion theory on coherent sheaves on projective space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T05:10:29Z http://mathoverflow.net/feeds/question/17424 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17424/looking-for-reference-talking-about-torsion-theory-on-coherent-sheaves-on-project Looking for reference talking about torsion theory on coherent sheaves on projective space Shizhuo Zhang 2010-03-07T22:27:13Z 2010-03-08T01:14:56Z <p>I am looking for reference talking about how torsion theory play roles in algebraic geometry. I will be really happy to see some concrete examples. Say, talking about torsion theory in $Coh(P^{1})$. </p> <p>Thanks in advance</p> http://mathoverflow.net/questions/17424/looking-for-reference-talking-about-torsion-theory-on-coherent-sheaves-on-project/17438#17438 Answer by Greg Stevenson for Looking for reference talking about torsion theory on coherent sheaves on projective space Greg Stevenson 2010-03-08T01:14:56Z 2010-03-08T01:14:56Z <p>Depending upon how strict you are with your definition of torsion theory a good source of examples is the theory of semi-orthogonal decompositions. A really nice example of this is the appearance of such decompositions which parallel operations in the minimal model program. A good introduction to this is <a href="http://arxiv.org/abs/0804.3150" rel="nofollow"> Kawamata's survey</a>. Of course there are other interesting things one can do with such decompositions in algebraic geometry (e.g. the work of Bondal, Orlov, Kapranov, and many others).</p> <p>For torsion theories on abelian categories a good example is stability conditions. Here what is interesting is the interplay between torsion theory on hearts and t-structures. The <a href="http://arxiv.org/abs/math/0212237" rel="nofollow"> original paper</a> is by Bridgeland and in the case of $\mathbb{P}^1$ the stability manifold has been computed by <a href="http://arxiv.org/abs/math/0411220" rel="nofollow"> Okada</a> (I suggest looking at the journal version, I recall that there were at one point some typos in the arxiv version which were fixed in the published one).</p> <p>As far as torsion theories on $\mathrm{Coh}(\mathbb{P}^1)$ goes it is a reasonable exercise to actually classify them (I did this at one point but never wrote it up properly). The closest place to this being written down that I know of is in the paper of Gorodentsev, Kuleshov, and Rudakov "t-stabilities and t-structures on triangulated categories" where they classify the minimal t-stabilities on the derived category of coherent sheaves on $\mathbb{P}^1$. </p> <p>An example of something similar but that is not quite what you asked for is the application of cotorsion theories to relative homological algebra.</p> <p><b>Definition</b>: Suppose that $\mathcal{A}$ is an abelian category and that $(\mathcal{F},\mathcal{C})$ is a pair of full subcategories. Then we say that $(\mathcal{F},\mathcal{C})$ is a <i>cotorsion theory </i> if<br> $\mathcal{F} = \{F \in \mathcal{A} \; \vert \; \mathrm{Ext}^1(F,\mathcal{C}) = 0\}$ and $\mathcal{C} = \{C \in \mathcal{A} \; \vert \; \mathrm{Ext}^1(\mathcal{F},C) = 0\}$<br> where the subcategories appearing in the Ext's just signifies that it is true for every object of that subcategory.</p> <p>There is a notion of a cotorsion theory having enough injectives and projectives and this guarantees for sufficiently good $\mathcal{A}$, say $R$-modules, (by a theorem of Eklof and Trlifaj) that $\mathcal{F}$-covers and $\mathcal{C}$-envelopes exist. In particular this can be used to show that flat covers exist. A good reference for this is Chapter 7 of Relative Homological Algebra by Enochs and Jenda.</p> <p>The application to algebraic geometry/commutative algebra is using this formalism to build Gorenstein injective/projective/flat covers and envelopes.</p>