Tensor complexes: multilinear free resolutions constructed from higher tensors

Abstract

The most fundamental complexes of free modules over a commutative ring are the Koszul complex, which is constructed from a vector (i.e., a 1-tensor), and the Eagon-Northcott and Buchsbaum-Rim complexes, which are constructed from a matrix (i.e., a 2-tensor). The subject of this paper is a multilinear analogue of these complexes, which we construct from an arbitrary higher tensor.Our construction provides detailed new examples of minimal free resolutions, as well as a unifying view on a wide variety of complexes including: the Eagon-Northcott, Buchsbaum-Rim and similar complexes, the Eisenbud-Schreyer pure resolutions, and the complexes used by Gelfand-Kapranov-Zelevinsky and Weyman to compute hyperdeterminants. In addition, we provide applications to the study of pure resolutions and Boij-Söderberg theory, including the construction of infinitely many new families of pure resolutions, and the first explicit description of the differentials of the Eisenbud-Schreyer pure resolutions.

On some quiver determinantal varieties

“…For another thing, when γ = (1, α 2 − 1) our quiver determinantal variety of K m coincides with the variety constructed from certain 3-tensor in [1]. Motivated by some ideas in [1], we consider the construction in [12] for some line bundle on Gr ( α γ ). In Proposition 4.1 we compute each term of the KLW-complex for any line bundle.…”

Zipping Tate resolutions and exterior coalgebras

“…By its part a. and Fact 2. immediately above, the t'th term of the linear strand is (25) Assume now the statements within Proposition 5.18 above hold. Nonzero injective resolutions over the exterior algebra are infinite.…”

“…Therefore a'. together with (25) for t ≫ 0, imply that the p'th linear strand is nonzero iff H −p (E • ) is nonzero. And so this latter holds iff [h p , h p+1 − 2] is nonempty.…”

“…When u 1 ≥ −1 it becomes a pure resolution, and when n ≥ u r + w r − 1 it is a pure resolution of a Cohen-Macaulay module. This gives the tensor complexes of [25], with pinching weights (0; u 1 , . .…”

“…Here we get rid of the above restrictions in [5] and show that the result holds in arbitrary characteristic, for complexes of coherent sheaves, and most importantly, without the bound on dim k V . As a consequence we can show how various old and recent resolutions in the literature, the Eagon-Northcott complex, Buchsbaum-Rim and Buchsbaum-Eisenbud complexes [6, A2.6], pure resolutions of modules supported on determinantal varieties [9], and tensor complexes [25], may be obtained by the simple procedure of zipping the Tate resolutions of various vector bundles with supernatural cohomology on P(W ), or more generally locally Cohen-Macaulay sheaves, with an exterior coalgebra on V .…”