All the varieties that appear below are assumed without projective problems on $ mathbb C $ and it is assumed that all packages of vectors, sections, etc. they are algebraic / holomorphic. We use the word resolution to mean quasi isomorphism.

Suppose $ X subset mathbb P ^ n $ It is a closed subvariety of codimension. $ r $. Suppose we can find a locally free sheaf. $ E $ in $ mathbb P ^ n $ of rank $ r $ and a section $ s $ such that $ s ^ {- 1} (0) $ It is cut transversely and is equal to $ X $ (For example, yes $ X $ It is a complete intersection). Then, we get Koszul's resolution.

$ 0 to wedge ^ rE ^ vee to cdots to wedge ^ 2E ^ vee to E ^ vee to mathcal O _ { mathbb P ^ n} to mathcal O_X to $ 0

I would like to see this as a resolution of $ mathcal O_X $ by a commutative commutative commutative $ mathcal O _ { mathbb P ^ n} $-algebra $ K (s) $ with $ E ^ vee $ put in grade $ -1 $ (call $ K (s) $ the Koszul algebra of $ s $). Note that as a commutative graduated algebra $ K (s) $, is simply the symmetric algebra (qualified commutative) $ text {Sym} ^ bullet _ { mathcal O _ { mathbb P ^ n}} (E ^ vee[1]$. Now, suppose we have an extension $ 0 a E a E & # 39; a F a 0 $ of vector packages, then we could consider the section $ s & # 39; $ of $ E & # 39; $ what is the image of $ s $. Now, the Koszul algebra $ K (s & # 39;) $ it is no longer a resolution of $ mathcal O_X $In fact, his cohomology algebra is. $ mathcal O_X otimes _ { mathcal O _ { mathbb P ^ n}} wedge ^ bullet F ^ vee $. So, adding the free generator. $ F ^ vee $ in degree $ -2 $ to $ K (s & # 39;) $, we obtain a new commutative differential graduation algebra with differential mapping $ F ^ vee $ to $ E & # 39; ^ vee $ for double the map $ E & # 39; a F $. The obvious map of this new algebra to $ K (s) $ It is a quasi-isomorphism and therefore, again we have a resolution of $ mathcal O_X $.

More generally, there is the notion of Koszul-Tate resolution of $ mathcal O_X $ which consists of a commutative differential graded algebra $ A $ finished $ mathcal O _ { mathbb P ^ n} $ solving $ mathcal O_X $ and having the following additional property. Locally in $ mathbb P ^ n $, $ A $ is administered (as a graduated algebra) by a free polynomial ring (commutative graduated) (possibly in infinite variables) on $ mathcal O _ { mathbb P ^ n} $ With finely many generators in each negative degree. We'll say that $ A $ is locally generated finitely if, locally in $ mathbb P ^ n $is a polynomial algebra $ mathcal O _ { mathbb P ^ n} $ in finite many variables (with the variables placed in negative degrees).

We have seen above that $ K (s) a mathcal O_X $ it's a resolution with $ K (s) $ finely generated. Are there other examples of resolutions generated locally finitely in addition to the Koszul algebras? $ K (s) $ (and almost isomorphic modifications that come from exact sequences of the form $ 0 a E a E & # 39; a cdots a F a 0 $)?

**In particular, are there examples of soft $ X subset mathbb P ^ n $ that are not complete intersections, so $ mathcal O_X $ Supports a Koszul-Tate resolution by means of a commutative differential gradient generated locally and finite. $ mathcal O _ { mathbb P ^ n} $-algebra?** It seems easy enough to prove that Koszul-Tate resolutions exist if we do not require that they be generated locally finitely.

Keep in mind that not all $ X subset mathbb P ^ n $ admits such a resolution, since a Koszul-Tate resolution of $ mathcal O_X $ would give a resolution (using the cotangent complex theory) of $ Omega ^ 1_X $ by vector packets in $ X $ removed from $ mathbb P ^ n $and, therefore, if the resolution of algebra is generated finitely, then so is the resolution of the cotangent package. Therefore, taking determinants, we see that the canonical package $ K_X $ It is in the restriction map image. $ text {Pic} ( mathbb P ^ n) to text {Pic} (X) $. For example, this shows that the cubic twisted in $ mathbb P ^ 3 $ does not support a Koszul-Tate resolution generated locally finitely. There are also intrinsic conditions in $ X $, for example: anyone $ X $ admit a finite resolution must necessarily have $ K_X $ be trivial, very broad or negative or very broad (since each packet of lines in $ mathbb P ^ n $ It is like). This, for example, discards any $ X $ which is a product of a Fano variety with a Calabi-Yau variety.