retTable = actionOnDirectImage(I,M)retTable = actionOnDirectImage(J,N,phi)retTable = actionOnDirectImage(I,T)This method provides another representation of the direct image complex.
Let M represent a coherent sheaf G on Y, and let \pi:Y\to P^r be a Noether normalization of Y. Note that \pi is chosen among finite linear projections P^m\to P^r from certain coordinate planes. Each coordinate y_i of P^m gives a multiplication map G\to G(1), and its induced map \pi_{*}G\to (\pi_{*}G)(1). Note that these induced maps provide an O_Y-module structure on \pi_{*}G, in other words, we may recover the O_Y-module F from \pi_{*}G and this action.
If no map is specified, it computes the complex C on P^r and a list of induced maps between chain complexes C\to C(1) on P^r associated to the multiplication by y_0,...,y_m, where C represents the Beilinson monad of \pi_{*}G (or R\pi_{*}U(T)).
If a map is specified by a matrix phi, then it computes the complex C on P^r and a list of induced maps between chain complexes C\to C(1) on P^r associated to the multiplication by y_0,...,y_m, where C represents the Beilinson monad of R(\pi \cdot phi)_{*}F.
When n is quite big compared to r, it is not very efficient to deal with Beilinson bundles on P^n since they have huge rank and presentation matrices. In particular, the method directImageComplex becomes slow down.
The following is an example of direct images of the structure sheaf on a rational normal curve of degree $d$.
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RM looks complicated since it is consisted of universal bundles on $P^4$, which are of high rank.
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We see that 0 is the only key, in other words, there is no other R^i vd_{*} except i=0. To see whether it gives an action on S/J, we can use the test function isAction.
Note that list retTable#i is consisted of maps of chain complexes R^i(\pi \cdot phi)_{*}(y_j) : C\to C(1) where C represents the direct image R^i(\pi \cdot phi)_{*}F. In general, it does not give a right action on C itself. The induced maps on cohomology groups provide a right action.
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The following is a little more complicated example with nontrivial higher direct images. Let X be the product of two quartic curves C, and f : X \to C be the second projection. Let P, Q be two distinct points of C, and let L = O(P\times C + Q\times C + D) be a line bundle on X where D is the diagonal. We want to compute the higher direct images R^i f_{*}L. We choose C as the Fermat quartic, and choose P, Q as points on the intersection of C and the line V(x_2).
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To compute its Tate resolution on the ambient space P^2 \times P^2, we first consider it as a sheaf on P^2 \times P^2, and then take a linear presentation matrix via a truncation.
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We read off (a finite subquotient of) the Tate resolution of Rf_{*}L as follows.
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One can check that W has two strands (corresponding to R^0f_{*}L and R^1f_{*}L, respectively). By taking the Beilinson functor, one can check that R^0f_{*}L is the structure sheaf on C, and R^1f_{*}L is a torsion sheaf supported on two points lying on the intersection of C and the line V(x_2) other than P, Q.
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These module also can be seen as in the following way via a finite linear projection. We take a further projection \pi:C\to P^1, and check whether these modules induce an action on the direct image under \pi, in other words, provide {O_C}-module structures. As results, these actions make (the sheafification of) M0 and M1 into {O_C}-modules which are identical to R^0f_{*}L and R^1f_{*}L.
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We see that 0, 1 appear as keys, in other words, both R^0f_{*}L and R^1f_{*}L survives.
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Note that the sheafification of M0 (=R^0(\pi \cdot f)_{*}L) is a rank 4 vector bundle O \oplus O(-1) \oplus O(-2) \oplus O(-3) on P^1, and the sheafification of M1 (= R^1(\pi \cdot f)_{*} L) is a torsion sheaf on P^1 supported on the double point at [1:0]. Together with the induced action on S', they have an O_C-module structure as desired.
Note that the resulting complex is a chain complex instead of a cochain complex, so that for example HH^i RpiM = HH_{-i} RpiM. Also note that this requires a pseudo-inverse computation of a split exact sequence, which might fail over finite fields (see SVDComplexes.m2 and its documentations).
The object actionOnDirectImage is a method function.