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JackYo
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Let $X$ be a smooth connected proper scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $H^*(X, \mathbb{Z})$ via "push and pull", namely we have a cycle class map

$$ \text{cl}: \text{CH}^*(X) \to H^*(X, \mathbb{Z} ) $$

which is compatible with pull-backs & push-forwards. If $p_1, p_2: X \times X \to X $ are the projections and $\alpha $ a correspondence in $X \times X$, then it acts as

$$ \alpha^*: H^*(X \mathbb{Z}) \to H^*(X \mathbb{Z}), \ \ \ \alpha^*(s) := p_{2*}(p^*_1(s) \cap \text{cl}(\alpha)) $$

Clearly this action extends linearly to action by rational Chow groups $\text{CH}_r^*(-)= \text{CH}^* \otimes \mathbb{Q} $ on rational cohomology.

Let now specialize: Assume $X$ is a smooth proper curve and $D \subset X$ an effective divisor of degree $d$, ie a "multisection". By dividing the degree $\beta:= \frac{1}{d} \cdot D $ becomes a $\mathbb{Q}$-divisor of degree $1$, so a "virtual" section.

So far I understand it correctly in the discussion here Dan Petersen uses that the induced action by such $\mathbb{Q}$-divisor of degree $1$ in terms from above as associated Chow correspondence induces a splitting of the cohomology. (Dan Petersen used this more generally in relative setting & associated action on derived object $Rf_* \mathbb{Q}$ but for sake of simplicity I would like to understand it in absolute case for a single curve as baby version of this mechanism.)

Question: I not see why this action gives a splitting of the cohomology? In other words does such $\mathbb{Q}$-divisor "normed" to degree $1$ induce an idempotent endomorphism corresponding to the associated action described above? It seems that the degree $1$ assumption is crucial, but I not see the connection. Then it would mean that every effective divisor turned into degree $1$ $\mathbb{Q}$-divisor by dividing it's degree would acting idempotently on the cohomology, ie give a splitting of the cohomology? That seems strange. Maybe I misunderstood somewhere Dan Petersen's argument.

Let $X$ be a smooth connected scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $H^*(X, \mathbb{Z})$ via "push and pull", namely we have a cycle class map

$$ \text{cl}: \text{CH}^*(X) \to H^*(X, \mathbb{Z} ) $$

which is compatible with pull-backs & push-forwards. If $p_1, p_2: X \times X \to X $ are the projections and $\alpha $ a correspondence in $X \times X$, then it acts as

$$ \alpha^*: H^*(X \mathbb{Z}) \to H^*(X \mathbb{Z}), \ \ \ \alpha^*(s) := p_{2*}(p^*_1(s) \cap \text{cl}(\alpha)) $$

Clearly this action extends linearly to action by rational Chow groups $\text{CH}_r^*(-)= \text{CH}^* \otimes \mathbb{Q} $ on rational cohomology.

Let now specialize: Assume $X$ is a curve and $D \subset X$ an effective divisor of degree $d$, ie a "multisection". By dividing the degree $\beta:= \frac{1}{d} \cdot D $ becomes a $\mathbb{Q}$-divisor of degree $1$, so a "virtual" section.

So far I understand it correctly in the discussion here Dan Petersen uses that the induced action by such $\mathbb{Q}$-divisor of degree $1$ in terms from above as associated Chow correspondence induces a splitting of the cohomology. (Dan Petersen used this more generally in relative setting & associated action on derived object $Rf_* \mathbb{Q}$ but for sake of simplicity I would like to understand it in absolute case for a single curve as baby version of this mechanism.)

Question: I not see why this action gives a splitting of the cohomology? In other words does such $\mathbb{Q}$-divisor "normed" to degree $1$ induce an idempotent endomorphism corresponding to the associated action described above? It seems that the degree $1$ assumption is crucial, but I not see the connection. Then it would mean that every effective divisor turned into degree $1$ $\mathbb{Q}$-divisor by dividing it's degree would acting idempotently on the cohomology, ie give a splitting of the cohomology? That seems strange. Maybe I misunderstood somewhere Dan Petersen's argument.

Let $X$ be a smooth connected proper scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $H^*(X, \mathbb{Z})$ via "push and pull", namely we have a cycle class map

$$ \text{cl}: \text{CH}^*(X) \to H^*(X, \mathbb{Z} ) $$

which is compatible with pull-backs & push-forwards. If $p_1, p_2: X \times X \to X $ are the projections and $\alpha $ a correspondence in $X \times X$, then it acts as

$$ \alpha^*: H^*(X \mathbb{Z}) \to H^*(X \mathbb{Z}), \ \ \ \alpha^*(s) := p_{2*}(p^*_1(s) \cap \text{cl}(\alpha)) $$

Clearly this action extends linearly to action by rational Chow groups $\text{CH}_r^*(-)= \text{CH}^* \otimes \mathbb{Q} $ on rational cohomology.

Let now specialize: Assume $X$ is a smooth proper curve and $D \subset X$ an effective divisor of degree $d$, ie a "multisection". By dividing the degree $\beta:= \frac{1}{d} \cdot D $ becomes a $\mathbb{Q}$-divisor of degree $1$, so a "virtual" section.

So far I understand it correctly in the discussion here Dan Petersen uses that the induced action by such $\mathbb{Q}$-divisor of degree $1$ in terms from above as associated Chow correspondence induces a splitting of the cohomology. (Dan Petersen used this more generally in relative setting & associated action on derived object $Rf_* \mathbb{Q}$ but for sake of simplicity I would like to understand it in absolute case for a single curve as baby version of this mechanism.)

Question: I not see why this action gives a splitting of the cohomology? In other words does such $\mathbb{Q}$-divisor "normed" to degree $1$ induce an idempotent endomorphism corresponding to the associated action described above? It seems that the degree $1$ assumption is crucial, but I not see the connection. Then it would mean that every effective divisor turned into degree $1$ $\mathbb{Q}$-divisor by dividing it's degree would acting idempotently on the cohomology, ie give a splitting of the cohomology? That seems strange. Maybe I misunderstood somewhere Dan Petersen's argument.

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JackYo
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Let $X$ be a smooth connected scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $H^*(X, \mathbb{Z})$ via "push and pull", namely we have a cycle class map

$$ \text{cl}: \text{CH}^*(X) \to H^*(X, \mathbb{Z} ) $$

which is compatible with pull-backs & push-forwards. If $p_1, p_2: X \times X \to X $ are the projections and $\alpha $ a correspondence in $X \times X$, then it acts as

$$ \alpha^*: H^*(X \mathbb{Z}) \to H^*(X \mathbb{Z}), \ \ \ \alpha^*(s) := p_{2*}(p^*_1(s) \cap \text{cl}(\alpha)) $$

Clearly this action extends linearly to action by rational Chow groups $\text{CH}_r^*(-)= \text{CH}^* \otimes \mathbb{Q} $ on rational cohomology.

Let now specialize: Assume $X$ is a curve and $D \subset X$ an effective divisor of degree $d$, ie a "multisection". By dividing the degree $\beta:= \frac{1}{d} \cdot D $ becomes a $\mathbb{Q}$-divisor of degree $1$, so a "virtual" section.

So far I understand it correctly in the discussion here Dan Petersen uses that the induced action by such $\mathbb{Q}$-divisor of degree $1$ in terms from above as associated Chow correspondence induces a splitting of the cohomology. (Dan Petersen used this more generally in relative setting & associated action on derived object $Rf_* \mathbb{Q}$ but for sake of simplicity I would like to understand it in absolute case for a single curve as baby version of this mechanism.)

Question: I not see why this action gives a splitting of the cohomology? In other words does such $\mathbb{Q}$-divisor "normed" to degree $1$ induce an idempotent endomorphism corresponding to the associated action described above? It seems that the degree $1$ assumption is crucial, but I not see the connection. Then it would mean that every effective divisor turned into degree $1$ $\mathbb{Q}$-divisor by dividing it's degree would acting idempotently on the cohomology, ie give a splitting of the cohomology? That seems strange. Maybe I misunderstood somewhere Dan Petersen's argument.

Let $X$ be a smooth connected scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $H^*(X, \mathbb{Z})$ via "push and pull", namely we have a cycle class map

$$ \text{cl}: \text{CH}^*(X) \to H^*(X, \mathbb{Z} ) $$

which is compatible with pull-backs & push-forwards. If $p_1, p_2: X \times X \to X $ are the projections and $\alpha $ a correspondence in $X \times X$, then it acts as

$$ \alpha^*: H^*(X \mathbb{Z}) \to H^*(X \mathbb{Z}), \ \ \ \alpha^*(s) := p_{2*}(p^*_1(s) \cap \text{cl}(\alpha)) $$

Clearly this action extends linearly to action by rational Chow groups $\text{CH}_r^*(-)= \text{CH}^* \otimes \mathbb{Q} $ on rational cohomology.

Let now specialize: Assume $X$ is a curve and $D \subset X$ an effective divisor of degree $d$, ie a "multisection". By dividing the degree $\beta:= \frac{1}{d} \cdot D $ becomes a $\mathbb{Q}$-divisor of degree $1$, so a "virtual" section.

So far I understand it correctly in the discussion here Dan Petersen uses that the induced action by such $\mathbb{Q}$-divisor of degree $1$ in terms from above as associated Chow correspondence induces a splitting of the cohomology. (Dan Petersen used this more generally in relative setting & associated action on derived object $Rf_* \mathbb{Q}$ but for sake of simplicity I would like to understand it in absolute case for a single curve.)

Question: I not see why this action gives a splitting of the cohomology? In other words does such $\mathbb{Q}$-divisor "normed" to degree $1$ induce an idempotent endomorphism corresponding to the associated action described above? It seems that the degree $1$ assumption is crucial, but I not see the connection. Then it would mean that every effective divisor turned into degree $1$ $\mathbb{Q}$-divisor by dividing it's degree would acting idempotently on the cohomology, ie give a splitting of the cohomology? That seems strange. Maybe I misunderstood somewhere Dan Petersen's argument.

Let $X$ be a smooth connected scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $H^*(X, \mathbb{Z})$ via "push and pull", namely we have a cycle class map

$$ \text{cl}: \text{CH}^*(X) \to H^*(X, \mathbb{Z} ) $$

which is compatible with pull-backs & push-forwards. If $p_1, p_2: X \times X \to X $ are the projections and $\alpha $ a correspondence in $X \times X$, then it acts as

$$ \alpha^*: H^*(X \mathbb{Z}) \to H^*(X \mathbb{Z}), \ \ \ \alpha^*(s) := p_{2*}(p^*_1(s) \cap \text{cl}(\alpha)) $$

Clearly this action extends linearly to action by rational Chow groups $\text{CH}_r^*(-)= \text{CH}^* \otimes \mathbb{Q} $ on rational cohomology.

Let now specialize: Assume $X$ is a curve and $D \subset X$ an effective divisor of degree $d$, ie a "multisection". By dividing the degree $\beta:= \frac{1}{d} \cdot D $ becomes a $\mathbb{Q}$-divisor of degree $1$, so a "virtual" section.

So far I understand it correctly in the discussion here Dan Petersen uses that the induced action by such $\mathbb{Q}$-divisor of degree $1$ in terms from above as associated Chow correspondence induces a splitting of the cohomology. (Dan Petersen used this more generally in relative setting & associated action on derived object $Rf_* \mathbb{Q}$ but for sake of simplicity I would like to understand it in absolute case for a single curve as baby version of this mechanism.)

Question: I not see why this action gives a splitting of the cohomology? In other words does such $\mathbb{Q}$-divisor "normed" to degree $1$ induce an idempotent endomorphism corresponding to the associated action described above? It seems that the degree $1$ assumption is crucial, but I not see the connection. Then it would mean that every effective divisor turned into degree $1$ $\mathbb{Q}$-divisor by dividing it's degree would acting idempotently on the cohomology, ie give a splitting of the cohomology? That seems strange. Maybe I misunderstood somewhere Dan Petersen's argument.

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JackYo
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Let $X$ be a smooth connected scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $H^*(X, \mathbb{Z})$ via "push and pull", namely we have a cycle class map

$$ \text{cl}: \text{CH}^*(X) \to H^*(X, \mathbb{Z} ) $$

which is compatible with pull-backs & push-forwards. If $p_1, p_2: X \times X \to X $ are the projections and $\alpha $ a correspondence in $X \times X$, then it acts as

$$ \alpha^*: H^*(X \mathbb{Z}) \to H^*(X \mathbb{Z}), \ \ \ \alpha^*(s) := p_{2*}(p^*_1(s) \cap \text{cl}(\alpha)) $$

Clearly this action extends linearly to action by rational Chow groups $\text{CH}_r^*(-)= \text{CH}^* \otimes \mathbb{Q} $ on rational cohomology.

Let now specialize: Assume $X$ is a curve and $D \subset X$ an effective divisor of degree $d$, ie a "multisection". By dividing the degree $\beta:= \frac{1}{d} \cdot D $ becomes a $\mathbb{Q}$-divisor of degree $1$, so a "virtual" section.

So far I understand it correctly in the discussion here Dan Petersen uses that the induced action by such $\mathbb{Q}$-divisor of degree $1$ in terms from above as associated Chow correspondence induces a splitting of the cohomology. (Dan Petersen used this more generally in relative setting & associated action on derived object $Rf_* \mathbb{Q}$ but for sake of simplicity I would like to understand it in absolute case for a single curve.)

Question: I not see why this action gives a splitting of the cohomology? In other words does thissuch $\mathbb{Q}$-divisor "normed" to degree $1$ induce an idempotent endomorphism corresponding to the associated action described above? It seems that the degree $1$ assumption is crucial, but I not see the connection. Then it would mean that every effective divisor turned into degree $1$ $\mathbb{Q}$-divisor by dividing it's degree would acting idempotently on the cohomology, ie give a splitting of the cohomology? That seems strange. Maybe I misunderstood somewhere Dan Petersen's argument.

Let $X$ be a smooth connected scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $H^*(X, \mathbb{Z})$ via "push and pull", namely we have a cycle class map

$$ \text{cl}: \text{CH}^*(X) \to H^*(X, \mathbb{Z} ) $$

which is compatible with pull-backs & push-forwards. If $p_1, p_2: X \times X \to X $ are the projections and $\alpha $ a correspondence in $X \times X$, then it acts as

$$ \alpha^*: H^*(X \mathbb{Z}) \to H^*(X \mathbb{Z}), \ \ \ \alpha^*(s) := p_{2*}(p^*_1(s) \cap \text{cl}(\alpha)) $$

Clearly this action extends linearly to action by rational Chow groups $\text{CH}_r^*(-)= \text{CH}^* \otimes \mathbb{Q} $ on rational cohomology.

Let now specialize: Assume $X$ is a curve and $D \subset X$ an effective divisor of degree $d$, ie a "multisection". By dividing the degree $\beta:= \frac{1}{d} \cdot D $ becomes a $\mathbb{Q}$-divisor of degree $1$, so a "virtual" section.

So far I understand it correctly in the discussion here Dan Petersen uses that the induced action by such $\mathbb{Q}$-divisor of degree $1$ in terms from above as associated Chow correspondence induces a splitting of the cohomology. (Dan Petersen used this more generally in relative setting & associated action on derived object $Rf_* \mathbb{Q}$ but for sake of simplicity I would like to understand it in absolute case for a single curve.)

Question: I not see why this action gives a splitting of the cohomology? In other words does this $\mathbb{Q}$-divisor induce an idempotent endomorphism corresponding to the associated action described above? It seems that the degree $1$ assumption is crucial, but I not see the connection. Then it would mean that every effective divisor turned into degree $1$ $\mathbb{Q}$-divisor by dividing it's degree would acting idempotently on the cohomology, ie give a splitting of the cohomology? That seems strange. Maybe I misunderstood Dan Petersen's argument.

Let $X$ be a smooth connected scheme over field $k$. It is known that correspondences $\alpha \subset X \times X$ regarded as objects in Chow groups $\text{CH}^*(X \times X)$ act on cohomology $H^*(X, \mathbb{Z})$ via "push and pull", namely we have a cycle class map

$$ \text{cl}: \text{CH}^*(X) \to H^*(X, \mathbb{Z} ) $$

which is compatible with pull-backs & push-forwards. If $p_1, p_2: X \times X \to X $ are the projections and $\alpha $ a correspondence in $X \times X$, then it acts as

$$ \alpha^*: H^*(X \mathbb{Z}) \to H^*(X \mathbb{Z}), \ \ \ \alpha^*(s) := p_{2*}(p^*_1(s) \cap \text{cl}(\alpha)) $$

Clearly this action extends linearly to action by rational Chow groups $\text{CH}_r^*(-)= \text{CH}^* \otimes \mathbb{Q} $ on rational cohomology.

Let now specialize: Assume $X$ is a curve and $D \subset X$ an effective divisor of degree $d$, ie a "multisection". By dividing the degree $\beta:= \frac{1}{d} \cdot D $ becomes a $\mathbb{Q}$-divisor of degree $1$, so a "virtual" section.

So far I understand it correctly in the discussion here Dan Petersen uses that the induced action by such $\mathbb{Q}$-divisor of degree $1$ in terms from above as associated Chow correspondence induces a splitting of the cohomology. (Dan Petersen used this more generally in relative setting & associated action on derived object $Rf_* \mathbb{Q}$ but for sake of simplicity I would like to understand it in absolute case for a single curve.)

Question: I not see why this action gives a splitting of the cohomology? In other words does such $\mathbb{Q}$-divisor "normed" to degree $1$ induce an idempotent endomorphism corresponding to the associated action described above? It seems that the degree $1$ assumption is crucial, but I not see the connection. Then it would mean that every effective divisor turned into degree $1$ $\mathbb{Q}$-divisor by dividing it's degree would acting idempotently on the cohomology, ie give a splitting of the cohomology? That seems strange. Maybe I misunderstood somewhere Dan Petersen's argument.

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JackYo
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