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Ben Webster ♦
Ben Webster ♦
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Yes; whenever you have two objects in an abelian category such that $Ext^2(M,N)$ is not equal to 0, we have a conformalnonformal object given by coning with this morphism. More down-to-earthly, the element of $Ext^2(M,N)$ is given by some complex $N \to K \to L\to M$; the non-formal complex is just $\cdots \to 0\to K \to L \to 0\to \cdots$.

EDIT: Thanks for the example below. I was too lazy to provide one, but I also think it risks camouflaging the actual point here, since there's nothing special about coherent sheaves on $\mathbb{P}^2$, this happens in any abelian category with global dimension $>1$.

Yes; whenever you have two objects in an abelian category such that $Ext^2(M,N)$ is not equal to 0, we have a conformal object given by coning with this morphism. More down-to-earthly, the element of $Ext^2(M,N)$ is given by some complex $N \to K \to L\to M$; the non-formal complex is just $\cdots \to 0\to K \to L \to 0\to \cdots$.

EDIT: Thanks for the example below. I was too lazy to provide one, but I also think it risks camouflaging the actual point here, since there's nothing special about coherent sheaves on $\mathbb{P}^2$, this happens in any abelian category with global dimension $>1$.

Yes; whenever you have two objects in an abelian category such that $Ext^2(M,N)$ is not equal to 0, we have a nonformal object given by coning with this morphism. More down-to-earthly, the element of $Ext^2(M,N)$ is given by some complex $N \to K \to L\to M$; the non-formal complex is just $\cdots \to 0\to K \to L \to 0\to \cdots$.

EDIT: Thanks for the example below. I was too lazy to provide one, but I also think it risks camouflaging the actual point here, since there's nothing special about coherent sheaves on $\mathbb{P}^2$, this happens in any abelian category with global dimension $>1$.

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Ben Webster ♦
Ben Webster ♦
  • 44.7k
  • 12
  • 126
  • 260

Yes; whenever you have two objects in an abelian category such that $Ext^2(M,N)$ is not equal to 0, we have a conformal object given by coning with this morphism. More down-to-earthly, the element of $Ext^2(M,N)$ is given by some complex $N \to K \to L\to M$; the non-formal complex is just $\cdots \to 0\to K \to L \to 0\to \cdots$.

EDIT: Thanks for the example below. I was too lazy to provide one, but I also think it risks camouflaging the actual point here, since there's nothing special about coherent sheaves on $\mathbb{P}^2$, this happens in any abelian category with global dimension $>1$.

Yes; whenever you have two objects in an abelian category such that $Ext^2(M,N)$ is not equal to 0, we have a conformal object given by coning with this morphism. More down-to-earthly, the element of $Ext^2(M,N)$ is given by some complex $N \to K \to L\to M$; the non-formal complex is just $\cdots \to 0\to K \to L \to 0\to \cdots$.

Yes; whenever you have two objects in an abelian category such that $Ext^2(M,N)$ is not equal to 0, we have a conformal object given by coning with this morphism. More down-to-earthly, the element of $Ext^2(M,N)$ is given by some complex $N \to K \to L\to M$; the non-formal complex is just $\cdots \to 0\to K \to L \to 0\to \cdots$.

EDIT: Thanks for the example below. I was too lazy to provide one, but I also think it risks camouflaging the actual point here, since there's nothing special about coherent sheaves on $\mathbb{P}^2$, this happens in any abelian category with global dimension $>1$.

Source Link
Ben Webster ♦
Ben Webster ♦
  • 44.7k
  • 12
  • 126
  • 260

Yes; whenever you have two objects in an abelian category such that $Ext^2(M,N)$ is not equal to 0, we have a conformal object given by coning with this morphism. More down-to-earthly, the element of $Ext^2(M,N)$ is given by some complex $N \to K \to L\to M$; the non-formal complex is just $\cdots \to 0\to K \to L \to 0\to \cdots$.