# Geometry ag.algebraica – Koszul-Tate resolution for subvarieties of \$ mathbb P ^ n \$

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.