On regularity of Rees algebras of edge ideals of cone graphs

Rimpa Nandi and Ramakrishna Nanduri ()
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Rimpa Nandi: Indian Institute of Technology
Ramakrishna Nanduri: Indian Institute of Technology

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 1, 28-37

Abstract: Abstract 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 $$n-1$$ n - 1 , then we show that $$\displaystyle \mathrm{reg}(R(I^*))=a_{i_0}(R(I^*))+i_0=a_{i_0}(K[G] \otimes _{R(I(G))}R(I^*))+i_0= \mathrm{reg}(K[G] \otimes _{R(I(G))}R(I^*))$$ 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 0 , i.e., the regularities are equal and the regularities attain at the same stage, where $$I^*$$ I ∗ denotes the edge ideal of the cone graph of G. We also prove that the bigraded regularities, $$\displaystyle \mathrm{reg}_{(0,1)}(K[G]\otimes _{R(I)}R(I^*))= \mathrm{reg}_{(0,1)}(R(I^*))$$ 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 $$\displaystyle \mathrm{reg}_{(1,0)}(K[G]\otimes _{R(I)}R(I^*))\le \mathrm{reg}_{(1,0)}(R(I^*))+1$$ reg ( 1 , 0 ) ( K [ G ] ⊗ R ( I ) R ( I ∗ ) ) ≤ reg ( 1 , 0 ) ( R ( I ∗ ) ) + 1 .

Keywords: 13D02; 13D45; 13A30; 05E40; 05C38 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00226-9

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