Let $R$ be a $k$-algebra (not necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested in the class of objects $V$ of $\mathbf{D}(R^{op})$ having the following property (P):

For any (infinite) product $\prod_{i\in I} M_{i}\in \mathbf{D}(R)$ the natural map $$ [\prod_{i\in I} M_{i}]\otimes^{L}_{R} V\longrightarrow \prod_{i\in I}[ M_{i}\otimes^{L}_{R} V]$$ is an isomorphism in $\mathbf{D}(k)$.

I have two questions.

1. Does every perfect complex $V$ satisfy the property (P) ?

2. If the answer to the first question is positive, what is the biggest class of objects of $\mathbf{D}(R)$ having the property (P)? Does it have a name ?

PS: $R^{op}$ is the ring $R$ but with opposite multiplication. Thank you.

Ed.

Let $R$ be a $k$-algebra (not necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested in the class of objects $V$ of $\mathbf{D}(R^{op})$ having the following property (P):

For any (infinite) product $\prod_{i\in I} M_{i}\in \mathbf{D}(R)$ the natural map $$ [\prod_{i\in I} M_{i}]\otimes^{L}_{R} V\longrightarrow \prod_{i\in I}[ M_{i}\otimes^{L}_{R} V]$$ is an isomorphism in $\mathbf{D}(k)$.

I have two questions.

1. Does every perfect complex $V$ satisfy the property (P) ?

2. If the answer to the first question is positive, what is the biggest class of objects of $\mathbf{D}(R)$ having the property (P)? Does it have a name ?

PS: $R^{op}$ is the ring $R$ but with opposite multiplication. Thank you.

Ed.

Let $R$ be a $k$-algebra (not necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested on the class of objects $V$ of $\mathbf{D}(R^{op})$ having the following property (P):

For any (infinite) product $\prod_{i\in I} M_{i}\in \mathbf{D}(R)$ the natural map $$ [\prod_{i\in I} M_{i}]\otimes^{L}_{R} V\longrightarrow \prod_{i\in I}[ M_{i}\otimes^{L}_{R} V]$$ is an isomorphism in $\mathbf{D}(k)$.

I have two questions.

1. Does any perfect complex $V$ verifies the property (P) ?

2. If the answer to the first question is positive, what is the biggest class of objects of $\mathbf{D}(R)$ having the property (P)? Does it have a name ?

PS: $R^{op}$ is the ring $R$ but with opposite multiplication. Thank you.

Ed.

Let $R$ be a $k$-algebra (not necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested in the class of objects $V$ of $\mathbf{D}(R^{op})$ having the following property (P):

For any (infinite) product $\prod_{i\in I} M_{i}\in \mathbf{D}(R)$ the natural map $$ [\prod_{i\in I} M_{i}]\otimes^{L}_{R} V\longrightarrow \prod_{i\in I}[ M_{i}\otimes^{L}_{R} V]$$ is an isomorphism in $\mathbf{D}(k)$.

I have two questions.

1. Does every perfect complex $V$ satisfy the property (P) ?

2. If the answer to the first question is positive, what is the biggest class of objects of $\mathbf{D}(R)$ having the property (P)? Does it have a name ?

PS: $R^{op}$ is the ring $R$ but with opposite multiplication. Thank you.

Ed.

Let $R$ be a $k$-algebra (non necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested on the class of objects $V$ of $\mathbf{D}(R^{op})$ having the following property (P):

For any (infinite) product $\prod_{i\in I} M_{i}\in \mathbf{D}(R)$ the natural map $$ [\prod_{i\in I} M_{i}]\otimes^{L}_{R} V\longrightarrow \prod_{i\in I}[ M_{i}\otimes^{L}_{R} V]$$ is an isomorphism in $\mathbf{D}(k)$. I have two questions.

1. Does any perfect complex $V$ verifies the property (P) ?

2. If the answer to the first question is positive, what is the biggest class of objects of $\mathbf{D}(R)$ having the property (P)? Does it have a name ?

PS: $R^{op}$ is the ring $R$ but with opposite multiplication. Thank you.

Ed.

Let $R$ be a $k$-algebra (not necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested on the class of objects $V$ of $\mathbf{D}(R^{op})$ having the following property (P):

For any (infinite) product $\prod_{i\in I} M_{i}\in \mathbf{D}(R)$ the natural map $$ [\prod_{i\in I} M_{i}]\otimes^{L}_{R} V\longrightarrow \prod_{i\in I}[ M_{i}\otimes^{L}_{R} V]$$ is an isomorphism in $\mathbf{D}(k)$. 

I have two questions.

1. Does any perfect complex $V$ verifies the property (P) ?

2. If the answer to the first question is positive, what is the biggest class of objects of $\mathbf{D}(R)$ having the property (P)? Does it have a name ?

PS: $R^{op}$ is the ring $R$ but with opposite multiplication. Thank you.

Ed.