Narrow Results

Exponent matrices appear in the theory of tiled orders over a discrete valuation ring. Many properties of such an order and its quiver are fully determined by its exponent matrix. We prove that an arbitrary strongly connected simply laced quiver with a loop in every vertex is realized as the quiver of a reduced exponent matrix. The relations between exponent matrices and finite posets, Markov chains, and doubly stochastic matrices are discussed.

We give a description of Khovanov's knot homology theory in the language of sheaves. To do this, we identify two cohomology theories associated to a commutative diagram of abelian groups indexed by elements of the cube {0, 1}ⁿ. The first is obtained by taking the cohomology groups of the chain complex constructed by summing along the diagonals of the cube and inserting signs to force d² = 0. The second is obtained by regarding the commutative diagram as a sheaf on the cube (in the order-filter topology) and considering sheaf cohomology with supports. Included is a general study of sheaves on finite posets, and a review of some basic properties of knot homology in the language of sheaves.

In this paper, we present a formula for the depth of powers of generalized Hibi ideals and study the asymptotic behavior of depth. We also see that the associated primes of powers of this class of ideals are stable.