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Dan Turetsky
Dan Turetsky
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Let $(R,m)$ be a complete intersection ring with characteristic $p$. If $M$ is a finitely generated $R$-module with finite length i.e. $l(M)<\infty$. It, it follows that $l(F(M))\geq p^nl(M)$, where $n=\dim R$. $(\text{Here $F$ is the Frobenius functor})$.

Now under the same assumption of $R$, let $M$ and $N$ be two finitely generated $R$-modules. If $l(M\otimes N)<\infty$ and $\dim M+\dim N=n$. Is, is it true that $l(\text{Tor}_i^R(F(M),N)\geq p^nl(\text{Tor}_i^R(M,N))$?

Let $(R,m)$ be a complete intersection ring with characteristic $p$. If $M$ is a finitely generated $R$-module with finite length i.e. $l(M)<\infty$. It follows that $l(F(M))\geq p^nl(M)$, where $n=\dim R$. $(\text{Here $F$ is the Frobenius functor})$.

Now under the same assumption of $R$, let $M$ and $N$ be two finitely generated $R$-modules. If $l(M\otimes N)<\infty$ and $\dim M+\dim N=n$. Is it true that $l(\text{Tor}_i^R(F(M),N)\geq p^nl(\text{Tor}_i^R(M,N))$?

Let $(R,m)$ be a complete intersection ring with characteristic $p$. If $M$ is a finitely generated $R$-module with finite length i.e. $l(M)<\infty$, it follows that $l(F(M))\geq p^nl(M)$, where $n=\dim R$. $(\text{Here $F$ is the Frobenius functor})$.

Now under the same assumption of $R$, let $M$ and $N$ be two finitely generated $R$-modules. If $l(M\otimes N)<\infty$ and $\dim M+\dim N=n$, is it true that $l(\text{Tor}_i^R(F(M),N)\geq p^nl(\text{Tor}_i^R(M,N))$?

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Length of $\text{Tor}$ modules in complete intersection rings with characteristic $p$

Let $(R,m)$ be a complete intersection ring with characteristic $p$. If $M$ is a finitely generated $R$-module with finite length i.e. $l(M)<\infty$. It follows that $l(F(M))\geq p^nl(M)$, where $n=\dim R$. $(\text{Here $F$ is the Frobenius functor})$.

Now under the same assumption of $R$, let $M$ and $N$ be two finitely generated $R$-modules. If $l(M\otimes N)<\infty$ and $\dim M+\dim N=n$. Is it true that $l(\text{Tor}_i^R(F(M),N)\geq p^nl(\text{Tor}_i^R(M,N))$?