12,159 research outputs found
Representability of Hom implies flatness
Let be a projective scheme over a noetherian base scheme , and let
be a coherent sheaf on . For any coherent sheaf on , consider the
set-valued contravariant functor on -schemes, defined by
where and are the pull-backs of
and to . A basic result of Grothendieck ([EGA] III
7.7.8, 7.7.9) says that if is flat over then is
representable for all . We prove the converse of the above, in fact, we show
that if is a relatively ample line bundle on over such that the
functor is representable for infinitely many positive integers
, then is flat over . As a corollary, taking , it follows that
if is a coherent sheaf on then the functor on
the category of -schemes is representable if and only if is locally free
on . 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 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
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