On regularity of Rees algebras of edge ideals of cone graphs
For any simple graph G , let K [ G ] denotes the toric algebra of G over a field K and R ( I ( G )) denotes the Rees algebra of the edge ideal I ( G ). If G is a simple graph on n vertices such that G has no vertex of degree n - 1 , then we show that reg ( R ( I ∗ ) ) = a i 0 ( R ( I ∗ ) ) + i 0 = a...
Saved in:
| Published in | Indian journal of pure and applied mathematics Vol. 54; no. 1; pp. 28 - 37 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New Delhi
Indian National Science Academy
2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Summary: | For any simple graph
G
, let
K
[
G
] denotes the toric algebra of
G
over a field
K
and
R
(
I
(
G
)) denotes the Rees algebra of the edge ideal
I
(
G
). If
G
is a simple graph on
n
vertices such that
G
has no vertex of degree
n
-
1
, then we show that
reg
(
R
(
I
∗
)
)
=
a
i
0
(
R
(
I
∗
)
)
+
i
0
=
a
i
0
(
K
[
G
]
⊗
R
(
I
(
G
)
)
R
(
I
∗
)
)
+
i
0
=
reg
(
K
[
G
]
⊗
R
(
I
(
G
)
)
R
(
I
∗
)
)
, for some
i
0
, i.e., the regularities are equal and the regularities attain at the same stage, where
I
∗
denotes the edge ideal of the cone graph of
G
. We also prove that the bigraded regularities,
reg
(
0
,
1
)
(
K
[
G
]
⊗
R
(
I
)
R
(
I
∗
)
)
=
reg
(
0
,
1
)
(
R
(
I
∗
)
)
and they attain at the same stage. Finally, we show that
reg
(
1
,
0
)
(
K
[
G
]
⊗
R
(
I
)
R
(
I
∗
)
)
≤
reg
(
1
,
0
)
(
R
(
I
∗
)
)
+
1
. |
|---|---|
| ISSN: | 0019-5588 0975-7465 |
| DOI: | 10.1007/s13226-022-00226-9 |