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Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow F_i \rightarrow F_{i+1}\to \dots)$ be a bounded complex of sheaves of vector spaces over a given field on $X$. It is well known that there exists a spectral sequence $E_{r}^{p,q}(\mathcal{F})$ converging to the cohomology sheaves of the push-forward $R^\bullet f_*({\cal F})$ (in the derived category) such that the first term of it is equal to $$E_1^{p,q}({\cal F})=R^qf_*(F^p).$$

Now let us have one more complex of sheaves of vector spaces ${\cal G}$ on $X$. The claim I am interested in is the following one: There exists a canonical morphism of the tensor product of spectral sequences of ${\cal F}$ and ${\cal G}$ to the spectral sequence of ${\cal F}\otimes {\cal G}$.

After I talked to few experts in homological algebra and algebraic geometry and made some google search, I got an impression that this fact is a folklore and basically well known to experts. I would need a reference to this fact.

There is some discussion of multiplicative structure in various spectral sequences in the book "A user’s guide to spectral sequences" by J. McCleary, but I did not find the one I need.

Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow F_i \rightarrow F_{i+1}\to \dots)$ be a bounded complex of sheaves of vector spaces over a given field on $X$. It is well known that there exists a spectral sequence $E_{r}^{p,q}(\mathcal{F})$ converging to the cohomology sheaves of the push-forward $R^\bullet f_*({\cal F})$ (in the derived category) such that the first term of it is equal to $$E_1^{p,q}({\cal F})=R^qf_*(F^p).$$

Now let us have one more complex of sheaves of vector spaces ${\cal G}$ on $X$. The claim I am interested in is the following one: There exists a canonical morphism of the tensor product of spectral sequences of ${\cal F}$ and ${\cal G}$ to the spectral sequence of ${\cal F}\otimes {\cal G}$.

After I talked to few experts in homological algebra and algebraic geometry and made some google search, I got an impression that this fact is a folklore and basically well known to experts. I would need a reference to this fact.

Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow F_i \rightarrow F_{i+1}\to \dots)$ be a bounded complex of sheaves of vector spaces over a given field on $X$. It is well known that there exists a spectral sequence $E_{r}^{p,q}(\mathcal{F})$ converging to the cohomology sheaves of the push-forward $R^\bullet f_*({\cal F})$ (in the derived category) such that the first term of it is equal to $$E_1^{p,q}({\cal F})=R^qf_*(F^p).$$

Now let us have one more complex of sheaves of vector spaces ${\cal G}$ on $X$. The claim I am interested in is the following one: There exists a canonical morphism of the tensor product of spectral sequences of ${\cal F}$ and ${\cal G}$ to the spectral sequence of ${\cal F}\otimes {\cal G}$.*

After I talked to few experts in homological algebra and algebraic geometry and made some google search, I got an impression that this fact is a folklore and basically well known to experts. I would need a reference to this fact.

There is some discussion of multiplicative structure in various spectral sequences in the book "A user’s guide to spectral sequences" by J. McCleary, but I did not find the one I need.

Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow F_i \rightarrow F_{i+1}\to \dots)$ be a bounded complex of sheaves of vector spaces over a given field on $X$. It is well known that there exists a spectral sequence $E_{r}^{p,q}(\mathcal{F})$ converging to the cohomology sheaves of the push-forward $R^\bullet f_*({\cal F})$ (in the derived category) such that the first term of it is equal to $$E_1^{p,q}({\cal F})=R^qf_*(F^p).$$

Now let us have one more complex of sheaves of vector spaces ${\cal G}$ on $X$. The claim I am interested in is the following one: There exists a canonical morphism of the tensor product of spectral sequences of ${\cal F}$ and ${\cal G}$ to the spectral sequence of ${\cal F}\otimes {\cal G}$.

After I talked to few experts in homological algebra and algebraic geometry and made some google search, I got an impression that this fact is a folklore and basically well known to experts. I would need a reference to this fact.

There is some discussion of multiplicative structure in various spectral sequences in the book "A user’s guide to spectral sequences" by J. McCleary, but I did not find the one I need.

Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow F_i \rightarrow F_{i+1}\to \dots)$ be a bounded complex of sheaves of vector spaces over a given field on $X$. It is well known that there exists a spectral sequence $E^{p,q}_r({\cal F})$ converging to the cohomology sheaves of the push-forward $R^\bullet f_*({\cal F})$ (in the derived category) such that the first term of it is equal to $$E^{p,q}_1({\cal F})=R^qf_*(F^p).$$

Now let us have one more complex of sheaves of vector spaces ${\cal G}$ on $X$. The claim I am interested in is the following one: There exists a canonical morphism of the tensor product of spectral sequences of ${\cal F}$ and ${\cal G}$ to the spectral sequence of ${\cal F}\otimes {\cal G}$.

After I talked to few experts in homological algebra and algebraic geometry and made some google search, I got an impression that this fact is a folklore and basically well known to experts. I would need a reference to this fact.

There is some discussion of multiplicative structure in various spectral sequences in the book "A user’s guide to spectral sequences" by J. McCleary, but I did not find the one I need.

Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow F_i \rightarrow F_{i+1}\to \dots)$ be a bounded complex of sheaves of vector spaces over a given field on $X$. It is well known that there exists a spectral sequence $E_{r}^{p,q}(\mathcal{F})$ converging to the cohomology sheaves of the push-forward $R^\bullet f_*({\cal F})$ (in the derived category) such that the first term of it is equal to $$E_1^{p,q}({\cal F})=R^qf_*(F^p).$$

Now let us have one more complex of sheaves of vector spaces ${\cal G}$ on $X$. The claim I am interested in is the following one: There exists a canonical morphism of the tensor product of spectral sequences of ${\cal F}$ and ${\cal G}$ to the spectral sequence of ${\cal F}\otimes {\cal G}$.*

After I talked to few experts in homological algebra and algebraic geometry and made some google search, I got an impression that this fact is a folklore and basically well known to experts. I would need a reference to this fact.

There is some discussion of multiplicative structure in various spectral sequences in the book "A user’s guide to spectral sequences" by J. McCleary, but I did not find the one I need.