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Andrea B.
Andrea B.
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In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\mathrm{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $X$ is a complex quasiprojective variety with analytic topology. This is also known to be true if $X$ is a one-point space and $R$ is commutative, Noetherian, and of finite global dimension (these conditions are necessary to define Perv).

Does anyone know of examples where this fails if $R$ is not a field? Ideally, I'd like to still assume $R$ is commutative, Noetherian, and of finite global dimension.

In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\mathrm{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $X$ is a complex quasiprojective variety with analytic topology. This is also known to be true if $X$ is a one-point space and $R$ is commutative, Noetherian, and of finite global dimension.

Does anyone know of examples where this fails if $R$ is not a field? Ideally, I'd like to still assume $R$ is commutative, Noetherian, and of finite global dimension.

In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\mathrm{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $X$ is a complex quasiprojective variety with analytic topology. This is also known to be true if $X$ is a one-point space and $R$ is commutative, Noetherian, and of finite global dimension (these conditions are necessary to define Perv).

Does anyone know of examples where this fails if $R$ is not a field?

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YCor
YCor
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What is an example of Beilinson's theorem on $D^bPerv$$D^b\mathrm{Perv}$ failing for non-field coefficients?

In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\text{Perv}(X,R)\to D^b_c(X, R)$$D^b\mathrm{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $X$ is a complex quasiprojective variety with analytic topology. This is also known to be true if $X$ is a one-point space and $R$ is commutative, Noetherian, and of finite global dimension.

Does anyone know of examples where this fails if $R$ is not a field? Ideally, I'd like to still assume $R$ is commutative, Noetherian, and of finite global dimension.

What is an example of Beilinson's theorem on $D^bPerv$ failing for non-field coefficients?

In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\text{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $X$ is a complex quasiprojective variety with analytic topology. This is also known to be true if $X$ is a one-point space and $R$ is commutative, Noetherian, and of finite global dimension.

Does anyone know of examples where this fails if $R$ is not a field? Ideally, I'd like to still assume $R$ is commutative, Noetherian, and of finite global dimension.

What is an example of Beilinson's theorem on $D^b\mathrm{Perv}$ failing for non-field coefficients?

In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\mathrm{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $X$ is a complex quasiprojective variety with analytic topology. This is also known to be true if $X$ is a one-point space and $R$ is commutative, Noetherian, and of finite global dimension.

Does anyone know of examples where this fails if $R$ is not a field? Ideally, I'd like to still assume $R$ is commutative, Noetherian, and of finite global dimension.

Source Link
Andrea B.
Andrea B.
  • 495
  • 2
  • 11

What is an example of Beilinson's theorem on $D^bPerv$ failing for non-field coefficients?

In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\text{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $X$ is a complex quasiprojective variety with analytic topology. This is also known to be true if $X$ is a one-point space and $R$ is commutative, Noetherian, and of finite global dimension.

Does anyone know of examples where this fails if $R$ is not a field? Ideally, I'd like to still assume $R$ is commutative, Noetherian, and of finite global dimension.