Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebraic Shifting Increases Relative Homology   \newcommand{\rhomi}[1]{\widetilde{H}_{#1}} \newcommand{\rbeti}[1]{\beta_{#1}}
\newcommand{\kk}{\mathbf k} \newcommand{\dimk}{\dim_{\kk}}
  We show that algebraically shifting a pair of simplicial complexes weakly
increases their relative homology Betti numbers in every dimension.
  More precisely, let $\Delta(K)$ denote the algebraically shifted complex of
simplicial complex $K$, and let $\rbeti{j}(K,L)=\dimk \rhomi{j}(K,L;\kk)$ be
the dimension of the $j$th reduced relative homology group over a field $\kk$
of a pair of simplicial complexes $L \subseteq K$. Then $\rbeti{j}(K,L) \leq
\rbeti{j}(\Delta(K),\Delta(L))$ for all $j$.
  The theorem is motivated by somewhat similar results about Gr\"obner bases
and generic initial ideals. Parts of the proof use Gr\"obner basis techniques.

28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
14
The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat