F = directImageComplex M
The computation is done using the exterior algebra method described by Eisenbud and Schreyer, in Eisenbud, David; Schreyer, Frank-Olaf ``Relative Beilinson monad and direct image for families of coherent sheaves.'' Trans. Amer. Math. Soc. 360 (2008), no. 10, 5367–5396.
The computation requires knowing the Castelnuovo-Mumford regularity of M in the variables y_i. If not provided by the user, it is computed by the function. The default is Regularity => null, which means it must be computed.
The ring A must be a polynomial ring. For the moment, the module M must be homogeneous for the variables of A as well as for the variables of S (bihomogeneous).
It is proven in loc. cit. that every complex of free modules can be realized as the direct image of a vector bundle on $\PP^n_A$.
The following example can be used to study the loci in the family of extensions of a pair of vector bundles on $\PP^1$ where the extension bundle has a given splitting type: this type is calculated by the Fitting ideals of the matrices defining the direct image complexes of various twists of the bundle. See Section 5 in loc. cite. It is conjectured there that all the sums of these Fitting ideals for the universal extension of $\mathcal O_{\PP^1}^{r-1}$ by $\mathcal O_{\PP^1}(d)$ are radical, as in the example below.
First we examine the extensions of ${\mathcal O}_{\PP^1}(1)$ by ${\mathcal O}_{\PP^1}(-3)$. There is a 3-dimensional vector space $$ Ext^1({\mathcal O}_{\PP^1}(1),{\mathcal O}_{\PP^1}(-3)) $$ of extensions. The ``universal extension'' is thus a bundle on $\PP^1\times Ext$. The locus where the extension bundle splits as ${\mathcal O}_{\PP^1}(-2) \oplus {\mathcal O}_{\PP^1}$ is the locus where the map in the direct image complex drops rank, and this is the (cone over a) conic, defined by the determinant of this matrix.
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Here is a larger example, the extension of ${\mathcal O}_{\PP^1}^2$ by ${\mathcal O}_{\PP^1}(6)$
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