Let $R$ be a (commutative) domain and let $Q$ be its fraction field. Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$.

Let $I_k=(\det \operatorname{min}_k(A))$ be the ideal of $R$ generated by all determinants of minors of $A$ of size $k$. I wonder if \begin{equation} \operatorname{Tor}_1^R(K,Q/R) \simeq \prod_{k=1}^{\operatorname{rk} A} \frac{I_{k-1}}{I_{k}}. \end{equation} If $R$ is a PID, the above isomorphism holds by Smith Normal Form.

Does it hold for other classes of rings?

I'm particulary interested in the case of $R$ is an order in a quadratic immaginary extension of $\mathbb{Q}$.

Let $R$ be a (commutative) domain and let $Q$ be its fraction field. Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$.

Let $I_k=(\det \operatorname{min}_k(A))$ be the ideal of $R$ generated by all determinants of minors of $A$ of size $k$. I wonder if \begin{equation} \operatorname{Tor}_1^R(K,Q/R) \simeq \prod_{k=1}^{\operatorname{rk} A} \frac{I_{k-1}}{I_{k}}. \end{equation} If $R$ is a PID, the above isomorphism holds by Smith Normal Form.

Does it hold for other classes of rings?

I'm particularily interested in the case of $R$ is an order in a quadratic imaginary extension of $\mathbb{Q}$.

Let $R$ be a (commutative) domain and let $Q$ be its fraction field. Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$.

Let $I_k=(\det \operatorname{min}_k(A))$ be the ideal of $R$ generated by all determinants of minors of $A$ of size $k$. I wonder if \begin{equation} \operatorname{Tor}_1^R(K,Q/R) \simeq \prod_{k=1}^{\operatorname{rk} A} \frac{I_{k-1}}{I_{k}}. \end{equation} If $R$ is a PID, the above isomorphism holds by Smith Normal Form.

Does it hold for other classes of rings?

I'm particulary interested in the case of $R$ is a order in a quadratic immaginary extension of $\mathbb{Q}$.

Let $R$ be a (commutative) domain and let $Q$ be its fraction field. Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$.

Let $I_k=(\det \operatorname{min}_k(A))$ be the ideal of $R$ generated by all determinants of minors of $A$ of size $k$. I wonder if \begin{equation} \operatorname{Tor}_1^R(K,Q/R) \simeq \prod_{k=1}^{\operatorname{rk} A} \frac{I_{k-1}}{I_{k}}. \end{equation} If $R$ is a PID, the above isomorphism holds by Smith Normal Form.

Does it hold for other classes of rings?

I'm particulary interested in the case of $R$ is an order in a quadratic immaginary extension of $\mathbb{Q}$.