Invariants of linkage of modules


  • Tony J. Puthenpurakal



Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$, $N$ be two Cohen-Macaulay $A$-modules with $M$ linked to $N$ via a Gorenstein ideal $\mathfrak{q}$. Let $L$ be another finitely generated $A$-module. We show that $\mathrm{Ext}^i_A(L,M) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Tor}^A_i(L,N) = 0$ for all $i \gg 0$. If $D$ is a Cohen-Macaulay module then we show that $\mathrm{Ext}^i_A(M, D) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Ext}^i_A(D^\dagger , N) = 0$ for all $i \gg 0$, where $D^\dagger = \mathrm{Ext}^r_A(D,A)$ and $r = \mathrm{codim}(D)$. As a consequence we get that $\mathrm{Ext}^i_A(M, M) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Ext}^i_A(N, N) = 0$ for all $i \gg 0$. We also show that $\mathrm{End}_A(M)/\mathrm{rad}\,\mathrm{End}_A(M) \cong (\mathrm{End}_A(N)/\mathrm{rad}\,\mathrm{End}_A(N))^{\mathrm{op}}$. We also give a negative answer to a question of Martsinkovsky and Strooker.


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How to Cite

Puthenpurakal, T. J. . (2021). Invariants of linkage of modules: Array. MATHEMATICA SCANDINAVICA, 127(2), 223–242.