All varieties appearing below are assumed smooth projective over $\mathbb C$ and all vector bundles, sections etc are assumed to be algebraic/holomorphic. We use the word resolution to mean quasi-isomorphism.
Suppose $X\subset\mathbb P^n$ is a closed subvariety of codimension $r$. Suppose we can a find a locally free sheaf $E$ on $\mathbb P^n$ of rank $r$ and a section $s$ such that $s^{-1}(0)$ is cut out transversely and equals $X$ (for example, if $X$ is a complete intersection). Then, we get the Koszul 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 view this as a resolution of $\mathcal O_X$ by a differential graded commutative $\mathcal O_{\mathbb P^n}$-algebra $K(s)$ with $E^\vee$ placed in degree $-1$ (call $K(s)$ the Koszul algebra of $s$). Notice that as a graded commutative algebra $K(s)$, is simply the (graded commutative) symmetric algebra $\text{Sym}^\bullet_{\mathcal O_{\mathbb P^n}}(E^\vee[1])$. Now, suppose we had an extension $0\to E\to E'\to F\to 0$ of vector bundles, then we could consider the section $s'$ of $E'$ which is the image of $s$. Now, the Koszul algebra $K(s')$ is no more a resolution of $\mathcal O_X$, in fact its cohomology algebra is $\mathcal O_X\otimes_{\mathcal O_{\mathbb P^n}}\wedge^\bullet F^\vee$. Thus, by adding the free generator $F^\vee$ in degree $-2$ to $K(s')$, we get a new commutative differential graded algebra with the differential mapping $F^\vee$ to $E'^\vee$ by the dual of the map $E'\to F$. The evident map from this new algebra to $K(s)$ is a quasi-isomorphism and thus, we again 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 $A$ over $\mathcal O_{\mathbb P^n}$ resolving $\mathcal O_X$ and having the following additional property. Locally on $\mathbb P^n$, $A$ is given (as a graded algebra) by a free (graded commutative) polynomial ring (in possibly infinitely many variables) over $\mathcal O_{\mathbb P^n}$ with finitely many generators in each negative degree. We'll say that $A$ is locally finitely generated if, locally on $\mathbb P^n$, it is a polynomial algebra over $\mathcal O_{\mathbb P^n}$ in finitely many variables (with the variables placed in negative degrees).
We have seen above that $K(s)\to\mathcal O_X$ is a resolution with $K(s)$ finitely generated. Are there any other examples of locally finitely generated resolutions besides the Koszul algebras $K(s)$ (and quasi-isomorphic modifications coming from exact sequences of the form $0\to E\to E'\to \cdots\to F\to 0$)?
In particular, are there examples of smooth $X\subset\mathbb P^n$ which are not complete intersections, for which $\mathcal O_X$ admits a Koszul-Tate resolution by a locally finitely generated differential graded commutative $\mathcal O_{\mathbb P^n}$-algebra? It seems pretty easy to show that Koszul-Tate resolutions exist if we do not require them to be locally finitely generated.
Note that not every $X\subset\mathbb P^n$ admits such a resolution, since a Koszul-Tate resolution of $\mathcal O_X$ would give a resolution (using the theory of the cotangent complex) of $\Omega^1_X$ by vector bundles on $X$ pulled back from $\mathbb P^n$, and therefore, if the algebra resolution is finitely generated, then so is the resolution of the cotangent bundle. Thus, taking determinants, we see that the canonical bundle $K_X$ lies in the image of the restriction map $\text{Pic}(\mathbb P^n)\to\text{Pic}(X)$. For instance, this shows that the twisted cubic in $\mathbb P^3$ doesn't admit a locally finitely generated Koszul-Tate resolution. There are also intrinsic conditions on $X$, for instance: any $X$ admitting a finite resolution must necessarily have $K_X$ to be either trivial, very ample or negative of very ample (since every line bundle on $\mathbb P^n$ is such). This, for example, rules out any $X$ which is a product of a Fano variety with a Calabi-Yau variety.
