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Adrien Zabat
Adrien Zabat
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Localic or topos-theoretic definition of $\operatorname{Spec}$?

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LSpice
LSpice
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Localic or topos-theoritictheoretic definition of $\mathrm$\operatorname{Spec}$?

Usually, the construction of the spectrum of a commutative ring starts with defining the points of $\mathrm{Spec}(A)$$\operatorname{Spec}(A)$, and constructing a topology with the closed sets being the "zeroes" of elements of $A$, and the sets $D(f)$ being a basis of opens. The fact that $D(I) = D(\sqrt{I})$ is then a theorem.

Is there a construction of the spectrum of a ring, where it is defined as a locale or Grothendieck site generated by the $D(f)$, and where the relation that $D(f^k) = D(f)$ is definitional ?

Localic or topos-theoritic definition of $\mathrm{Spec}$?

Usually, the construction of the spectrum of a commutative ring starts with defining the points of $\mathrm{Spec}(A)$, and constructing a topology with the closed sets being the "zeroes" of elements of $A$, and the sets $D(f)$ being a basis of opens. The fact that $D(I) = D(\sqrt{I})$ is then a theorem.

Is there a construction of the spectrum of a ring, where it is defined as a locale or Grothendieck site generated by the $D(f)$, and where the relation that $D(f^k) = D(f)$ is definitional ?

Localic or topos-theoretic definition of $\operatorname{Spec}$?

Usually, the construction of the spectrum of a commutative ring starts with defining the points of $\operatorname{Spec}(A)$, and constructing a topology with the closed sets being the "zeroes" of elements of $A$, and the sets $D(f)$ being a basis of opens. The fact that $D(I) = D(\sqrt{I})$ is then a theorem.

Is there a construction of the spectrum of a ring, where it is defined as a locale or Grothendieck site generated by the $D(f)$, and where the relation that $D(f^k) = D(f)$ is definitional ?

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Adrien Zabat
Adrien Zabat
  • 253
  • 1
  • 6

Localic or topos-theoritic definition of $\mathrm{Spec}$?

Usually, the construction of the spectrum of a commutative ring starts with defining the points of $\mathrm{Spec}(A)$, and constructing a topology with the closed sets being the "zeroes" of elements of $A$, and the sets $D(f)$ being a basis of opens. The fact that $D(I) = D(\sqrt{I})$ is then a theorem.

Is there a construction of the spectrum of a ring, where it is defined as a locale or Grothendieck site generated by the $D(f)$, and where the relation that $D(f^k) = D(f)$ is definitional ?