Representability of Hom implies flatness

Let $X$ be a projective scheme over a noetherian base scheme $S$, and let $F$ be a coherent sheaf on $X$. For any coherent sheaf $E$ on $X$, consider the set-valued contravariant functor $Hom_{E,F}$ on $S$-schemes, defined by $Hom_{E,F}(T) = Hom(E_T,F_T)$ where $E_T$ and $F_T$ are the pull-backs of $E$ and $F$ to $X_T = X\times_S T$. A basic result of Grothendieck ([EGA] III 7.7.8, 7.7.9) says that if $F$ is flat over $S$ then $Hom_{E,F}$ is representable for all $E$. We prove the converse of the above, in fact, we show that if $L$ is a relatively ample line bundle on $X$ over $S$ such that the functor $Hom_{L^{-n},F}$ is representable for infinitely many positive integers $n$, then $F$ is flat over $S$. As a corollary, taking $X=S$, it follows that if $F$ is a coherent sheaf on $S$ then the functor $T\mapsto H^0(T, F_T)$ on the category of $S$-schemes is representable if and only if $F$ is locally free on $S$. This answers a question posed by Angelo Vistoli. The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author's earlier result (see arXiv.org/abs/math.AG/0204047) that the automorphism group functor of a coherent sheaf on $S$ is representable if and only if the sheaf is locally free.Comment: 9 pages, LaTe

Matrix Representation of Iterative Approximate Byzantine Consensus in Directed Graphs

This paper presents a proof of correctness of an iterative approximate Byzantine consensus (IABC) algorithm for directed graphs. The iterative algorithm allows fault- free nodes to reach approximate conensus despite the presence of up to f Byzantine faults. Necessary conditions on the underlying network graph for the existence of a correct IABC algorithm were shown in our recent work [15, 16]. [15] also analyzed a specific IABC algorithm and showed that it performs correctly in any network graph that satisfies the necessary condition, proving that the necessary condition is also sufficient. In this paper, we present an alternate proof of correctness of the IABC algorithm, using a familiar technique based on transition matrices [9, 3, 17, 19]. The key contribution of this paper is to exploit the following observation: for a given evolution of the state vector corresponding to the state of the fault-free nodes, many alternate state transition matrices may be chosen to model that evolution cor- rectly. For a given state evolution, we identify one approach to suitably "design" the transition matrices so that the standard tools for proving convergence can be applied to the Byzantine fault-tolerant algorithm as well. In particular, the transition matrix for each iteration is designed such that each row of the matrix contains a large enough number of elements that are bounded away from 0

Progress on Polynomial Identity Testing - II

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve