Sheaf cohomology and free resolutions over exterior algebras

We derive an explicit version of the Bernstein-Gel'fand-Gel'fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its ``Koszul dual' exterior algebra. Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the ``linear part' of a resolution over the exterior algebra.

We explore the structure of free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their ``linear parts" in the sense that erasing all terms of degree 1$"> in the complex yields a new complex which is eventually exact.

As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms. The explicitness of our version allows us to prove two conjectures about the morphisms in the monad, and we get an efficient method for machine computation of the cohomology of sheaves. We also construct all the monads for a sheaf that can be built from sums of line bundles, and show that they are often characterized by numerical data.

     

The Auslander–Reiten translate on monomial rings

For t in Nn, E. Miller has defined a category of positively t-determined modules over the polynomial ring S in n variables. We consider the Auslander–Reiten translate, Nt, on the (derived) category of such modules. A monomial ideal I is t-determined if each generator xa has a?t. We compute the multigraded cohomology and Betti spaces of for every iterate k and also the S-module structure of these cohomology modules. This comprehensively generalizes results of Hochster and Gräbe on local cohomology of Stanley–Reisner rings.     

Enriched homology and cohomology modules of simiplicial complexes

For a simplicial complex Δ on {1, 2,…, n} we define enriched homology and cohomology modules. They are graded modules over k[x ₁,…, x _n ] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein^* complexes Δ, and also orientable homology manifolds in terms of the enriched modules. We introduce the notion of girth for simplicial complexes and make a conjecture relating the girth to invariants of the simplicial complex. We also put strong vanishing conditions on the enriched homology modules and describe the simplicial complexes we then get. They are block designs and include Steiner systems S(c, d, n) and cyclic polytopes of even dimension. This paper is to a large extent a complete rewriting of a previous preprint, “Hierarchies of simplicial complexes via the BGG-correspondence”. Also Propositions 1.7 and 3.1 have been generalized to cell complexes in [11].     

Resolutions of letterplace ideals of posets

We investigate resolutions of letterplace ideals of posets. We develop topological results to compute their multigraded Betti numbers, and to give structural results on these Betti numbers. If the poset is a union of no more than c chains, we show that the Betti numbers may be computed from simplicial complexes of no more than c vertices. We also give a recursive procedure to compute the Betti diagrams when the Hasse diagram of P has tree structure.     

The colorful Helly theorem and colorful resolutions of ideals

We demonstrate that the topological Helly theorem and the algebraic Auslander-Buchsbaum theorem may be viewed as different versions of the same phenomenon. Using this correspondence we show how the colorful Helly theorem of I. Barany and its generalizations by G. Kalai and R. Meshulam translate to the algebraic side. Our main results are algebraic generalizations of these translations, which in particular give a syzygetic version of Helly’s theorem.     

Koszul duality and equivalences of categories

Let and be dual Koszul algebras. By Positselski a filtered algebra with gr is Koszul dual to a differential graded algebra . We relate the module categories of this dual pair by a Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.

     

Curves on normal surfaces

We study locally Cohen-Macaulay space curves lying on normal surfaces. We prove some theorems on the behaviour of the cohomology functions and initial ideals of such space curves, which give a basic distinction between those curves and curves lying on non-normal surfaces.