Looking for reference talking about torsion theory on coherent sheaves on projective space

Question

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 $Coh(P^{1})$.

Thanks in advance

Answer 1

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 Kawamata's survey. 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).

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 original paper is by Bridgeland and in the case of $\mathbb{P}^1$ the stability manifold has been computed by Okada (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).

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 $\mathbb{P}^1$.

An example of something similar but that is not quite what you asked for is the application of cotorsion theories to relative homological algebra.

Definition: 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 cotorsion theory if
$\mathcal{F} = \{F \in \mathcal{A} \; \vert \; \mathrm{Ext}^1(F,\mathcal{C}) = 0\}$ and $\mathcal{C} = \{C \in \mathcal{A} \; \vert \; \mathrm{Ext}^1(\mathcal{F},C) = 0\}$
where the subcategories appearing in the Ext's just signifies that it is true for every object of that subcategory.

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.

The application to algebraic geometry/commutative algebra is using this formalism to build Gorenstein injective/projective/flat covers and envelopes.