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1. • CommentRowNumber1.
   • CommentAuthorvarkor
   • CommentTimeMar 23rd 2023
   • PermaLink
   Author: varkor
   Format: MarkdownItexAdd a diagram. <a href="https://ncatlab.org/nlab/revision/diff/comorphism/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/comorphism/4">v4</a>, <a href="https://ncatlab.org/nlab/show/comorphism">current</a>

   Add a diagram.

   diff, v4, current

2. • CommentRowNumber2.
   • CommentAuthorjonsterling
   • CommentTimeApr 8th 2023
   • (edited Apr 8th 2023)
   • PermaLink
   Author: jonsterling
   Format: MarkdownItexI am puzzled by this page, where it discusses morphisms between ringed spaces. In particular, it requires a morphism $f^\sharp : O_Y\to f_*O_X$ to be a homomorphism of rings and then comments that the adjointness does not imply that the transpose $f_\sharp : f^*O_Y \to O_X$ is a morphism of rings. But both Hakim and Lurie **define** morphisms of ringed spaces to be given by morphisms of rings $f_\sharp : f^*O_Y\to O_X$. Furthermore, the Stacks project [actually claims](https://stacks.math.columbia.edu/tag/01D3) that morphisms of rings $f^*O_Y \to O_X$ are the same, by adjointness, as morphisms of rings $O_Y \to f_*O_X$. Who is wrong here?

   I am puzzled by this page, where it discusses morphisms between ringed spaces. In particular, it requires a morphism $f^\sharp : O_Y\to f_*O_X$ to be a homomorphism of rings and then comments that the adjointness does not imply that the transpose $f_\sharp : f^*O_Y \to O_X$ is a morphism of rings. But both Hakim and Lurie define morphisms of ringed spaces to be given by morphisms of rings $f_\sharp : f^*O_Y\to O_X$. Furthermore, the Stacks project actually claims that morphisms of rings $f^*O_Y \to O_X$ are the same, by adjointness, as morphisms of rings $O_Y \to f_*O_X$. Who is wrong here?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2023
   • (edited Apr 16th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have deleted this passage from the entry: > However, in the case of ringed sites, a morphism $G\to f_* F$ is required to be morphism of sheaves of rings, whereas the adjunction does *not* guarantee that the corresponding morphism $f^* G\to F$ of sheaves of sets over $X$ is a morphism of sheaves of rings over $X$. Because, this contradicts standard sources (as pointed out [above](#Comment_108448)) and the author of that line ([revision 1](https://ncatlab.org/nlab/revision/comorphism/1) in 2009) does not remember why he wrote it ([here](https://nforum.ncatlab.org/discussion/16317/wolfgang-rump/?Focus=108479#Comment_108479)). Of course, better than just deleting the sentence would be to replace it by an actual analysis of what's going on. <a href="https://ncatlab.org/nlab/revision/diff/comorphism/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/comorphism/5">v5</a>, <a href="https://ncatlab.org/nlab/show/comorphism">current</a>

   I have deleted this passage from the entry:

   However, in the case of ringed sites, a morphism $G\to f_* F$ is required to be morphism of sheaves of rings, whereas the adjunction does not guarantee that the corresponding morphism $f^* G\to F$ of sheaves of sets over $X$ is a morphism of sheaves of rings over $X$.

   Because, this contradicts standard sources (as pointed out above) and the author of that line (revision 1 in 2009) does not remember why he wrote it (here).

   Of course, better than just deleting the sentence would be to replace it by an actual analysis of what’s going on.

   diff, v5, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2023
   • (edited Apr 16th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Alexander Grothendieck]], [[Jean-Louis Verdier]], Def. 13.1 of *Morphisme de topos annelés*, Section 13 in Exposé iv in: [[M. Artin]] et al., *Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos*, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 ([[SGA 4]]), Lecture Notes in Mathematics __269__, Springer (1972) &lbrack;[doi:10.1007/BFb0081551](https://link.springer.com/book/10.1007/BFb0081551), [pdf](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf)&rbrack; The following remark 13.1.1 ([p. 316](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf#page=316)) explicitly says that the two adjoint forms of comorphisms are equivalent. <a href="https://ncatlab.org/nlab/revision/diff/comorphism/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/comorphism/6">v6</a>, <a href="https://ncatlab.org/nlab/show/comorphism">current</a>

   added pointer to:

   • Alexander Grothendieck, Jean-Louis Verdier, Def. 13.1 of Morphisme de topos annelés, Section 13 in Exposé iv in: M. Artin et al., Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Mathematics 269, Springer (1972) [doi:10.1007/BFb0081551, pdf]

   The following remark 13.1.1 (p. 316) explicitly says that the two adjoint forms of comorphisms are equivalent.

   diff, v6, current