%compile using AMS-LaTeX
\documentclass{amsart}

\newcommand{\ie}{{\em i.e.}}



\begin{document}
\title[Shifted simplicial complexes and algebraic
shifting]{Bibliography to accompany ``Shifted simplicial complexes and
algebraic shifting''}


\author{Art M. Duval}
\email{artduval@math.utep.edu}
\address{Department of Mathematical Sciences\\
         University of Texas at El Paso\\
         El Paso, TX 79968-0514}

\begin{abstract}
This is a bibliography to accompany the slides from my talk at the AMS
Regional Meeting at Binghamton University in October, 2003.  
\end{abstract}

\maketitle

In general, a reference in the slides with a name and date has an
obvious unique corresponding entry in the bibliography.  Exceptions,
and a few further explanations, are given below, organized by slide
number.

{\sf Slide 6}: One place to find an exposition of the idea that shifted
complexes are ``iterated near-cones''is \cite{DuvalRose}.

{\sf Slide 7}: The basics of non-pure shellability are in
\cite{BjornerWachs1}, but also see \cite{BjornerWachs2}, for instance
for the canonical shelling of a shifted complex.

{\sf Slide 8}: Exterior algebraic shifting goes back to \cite{Kalai84},
though the best expositions may be \cite{BjornerKalai}, where I first
read of it, or \cite{Kalai02}, which contains (or at least seems to
contain) everything that Gil Kalai knows or suspects to be true about
algebraic shifting, including some brief historical notes.  An earlier
version of \cite{Kalai02} is \cite{Kalai93}.

{\sf Slide 9}: Herzog's survey \cite{Herzog} is probably a good place to
start with symmetric shifting, including the cases where the ideal $I$
is not squarefree monomial, \ie, does not come from a simplicial
complex.  The three papers \cite{AramovaHerzog, AramovaHerzogHibi1,
AramovaHerzogHibi2} represent just the (combinatorial) tip of the
iceberg of work on generic initial ideals with the revlex order.

{\sf Slide 10}: Kalai \cite{Kalai02} lists more ``axioms''
\cite[Theorems 2.1 and 2.2]{Kalai02}, and a more extensive example
\cite[Example 2.4]{Kalai02} of how to use them to figure out the
algebraic shift of a complex.  That ``axiom'' 5 follows from the four
previous ``axioms'' I learned from a talk by Isabella Novik.

{\sf Slide 11}: The ``earlier version'' due to Kalai is from
\cite{Kalai93}, and may be found in \cite[Section 4.3]{Kalai02}.

{\sf Slide 21}: Kook and Reiner made their conjecture based on one example
in a coffeehouse.  I learned of it through personal communication.
The theorem at the bottom of this slide, as well as the theorem at the
bottom of slide 22, is from \cite{Duval}.

{\sf Slides 23 and 24}: Theorems about integrality are from
\cite{DuvalReiner}, and theorems about ``spectrality'' are from
\cite{Duval}.

\begin{thebibliography}{AHH00b}

\bibitem[AH00]{AramovaHerzog}
A. Aramova, J. Herzog, 
``Almost regular sequences and Betti numbers'',
{\it Amer.\ J. Math.}\ {\bf 122} (2000), 689--719.

\bibitem[AHH00a]{AramovaHerzogHibi1}
A. Aramova, J. Herzog, T. Hibi, 
``Ideals with stable Betti numbers'',
{\it Adv. Math.}\  {\bf 152} (2000), 72--77.

\bibitem[AHH00b]{AramovaHerzogHibi2}
A. Aramova, J. Herzog, T. Hibi, 
``Shifting operations and graded Betti numbers'',
{\it J. Alg.\ Combin.}\ {\bf 12} (2000), 207--222.

\bibitem[BNT02]{BabsonNovikThomas}
E. Babson, I. Novik, R. Thomas,
``Symmetric iterated Betti numbers'', preprint, 2002.  
{\tt arXiv:math.CO/0206063}


\bibitem[BCP99]{BayerCharalambousPopescu}
D. Bayer, H. Charalambous, and S. Popescu,
``Extremal Betti numbers and applications to monomial ideals'',
{\it J. Algebra} {\bf 221} (1999), 497--512.

\bibitem[BK88]{BjornerKalai}
A. Bj\"orner and G. Kalai,
``An extended Euler-Poincar\'e theorem'',
{\it Acta Math.}\ {\bf 161} (1988), 279--303.

\bibitem[BW96]{BjornerWachs1}
A. Bj\"orner and M. L. Wachs,
``Shellable nonpure complexes and posets. I'',
{\it Trans.\ Amer.\ Math.\ Soc.}\ {\bf 348} (1996),
	1299--1327.

\bibitem[BW97]{BjornerWachs2}
A. Bj\"orner and M. L. Wachs,
``Shellable nonpure complexes and posets. II'',
{\it Trans.\ Amer.\ Math.\ Soc.}\ {\bf 349} (1997),
	3945--3975.

\bibitem[DW02]{DongWachs}
X. Dong and M. L. Wachs,
``Combinatorial Laplacian of the matching complex'',
{\it Electron.\ J. Combin.}\ {\bf 9} (2002), \#R17, 11 pp.

\bibitem[Du03]{Duval}
A. M. Duval,
``A common recursion for Laplacians of matroids and shifted simplicial
complexes'', preprint, 2003.
{\tt arXiv:math.CO/0310327}

\bibitem[DRe02]{DuvalReiner}
A. M. Duval and V. Reiner,
``Shifted simplicial complexes are Laplacian integral'', 
{\it Trans.\ Amer.\ Math.\ Soc.}, {\bf 354} (2002),
	4313--4344.

\bibitem[DRo00]{DuvalRose}
A. M. Duval and L. L. Rose,
``Iterated homology of simplicial complexes'',
{\it J. Alg. Comb.}\ {\bf 12} (2000), 279--294. 

\bibitem[DZ01]{DuvalZhang}
A. M. Duval and P. Zhang,
``Iterated homology and decompositions of simplicial complexes'',
{\it Israel J. Math.}\ {\bf 121} (2001), 313--331. 

\bibitem[FH98]{FriedmanHanlon}
J. Friedman and P. Hanlon,  
``On the Betti numbers of chessboard complexes'',
{\it J. Alg.\ Comb.}\ {\bf 8} (1998), 193--203. 

\bibitem[Ga79]{Garsia}
A. M. Garsia,
``Combinatorial methods in the theory of Cohen-Macaulay rings'',
{\it Adv.\ Math.}\ {\bf 38} (1980), 229--266. 

\bibitem[He02]{Herzog}
J. Herzog,
``Generic initial ideals and graded Betti numbers'', 
in {\it Computational commutative algebra and combinatorics} (Osaka,
1999), pp.\ 75--120,
Adv.\ Stud.\ Pure Math., vol.\ 33, 
Math.\ Soc.\ Japan, Tokyo, 2002.

\bibitem[Ka84]{Kalai84}
G. Kalai,
``Characterization of $f$-vectors of families of convex sets in
$R^d$. I. Necessity of Eckhoff's conditions'',  
{\it Israel J. Math.}\ {\bf 48} (1984), 175--195.

\bibitem[Ka91]{Kalai91}
G. Kalai, 
``The diameter of graphs of convex polytopes and $f$-vector theory'',
in {\it Applied geometry and discrete mathematics}, pp.\ 387--411, 
DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol.\ 4, 
Amer.\ Math.\ Soc., Providence, RI, 1991.

\bibitem[Ka93]{Kalai93}
G. Kalai, 
``Algebraic Shifting'', 
unpublished manuscript (July 1993 version); 
 updated and polished in \cite{Kalai02}.

\bibitem[Ka02]{Kalai02}
G. Kalai, 
``Algebraic shifting'', 
in {\it Computational commutative algebra and combinatorics} (Osaka,
1999), pp.\ 121--163, 
Adv.\ Stud.\ Pure Math., vol.\ 33, 
Math.\ Soc.\ Japan, Tokyo, 2002.

\bibitem[Ko99]{Kook}
W. Kook, ``Recursions in spectrum polynomial of matroids'', preprint,
1999.
{\tt http://hypatia.math.uri.edu/\verb+~+andrewk/abstracts/}

\bibitem[KRS00]{KookReinerStanton}
W. Kook, V. Reiner, and D. Stanton,
``Combinatorial Laplacians of matroid complexes'',
{\it J. Amer.\ Math.\ Soc.}\ {\bf 13} (2000), 129--148.

\bibitem[St79]{Stanley}
R. P. Stanley,
``Balanced Cohen-Macaulay complexes,''
{\it Trans.\ Amer.\ Math.\ Soc.}\ {\bf 249} (1979), 139--157.

\end{thebibliography}

\end{document}

