Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: When Are Torsionless Modules Projective?   In this paper, we study the problem when a finitely generated torsionless
module is projective. Let $\Lambda$ be an Artinian local algebra with radical
square zero. Then a finitely generated torsionless $\Lambda$-module $M$ is
projective if ${\rm Ext^1_\Lambda}(M,M)=0$. For a commutative Artinian ring
$\Lambda$, a finitely generated torsionless $\Lambda$-module $M$ is projective
if the following conditions are satisfied: (1) ${\rm
Ext}^i_{\Lambda}(M,\Lambda)=0$ for $i=1,2,3$; and (2) ${\rm
Ext}^i_{\Lambda}(M,M)=0$ for $i=1,2$. As a consequence of this result, we have
that for a commutative Artinian ring $\Lambda$, a finitely generated Gorenstein
projective $\Lambda$-module is projective if and only if it is selforthogonal.

36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
18
The categories of this paper are: econ
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc