show/hide this revision's text 3 Fixed my least favorite typo

In EGA IV, Sec. 16, Grothendieck defines the sheaf of principal parts as follows: Let $f:X\rightarrow S$ be a morphism of schemes and $\Delta:X\rightarrow X\times_S X$ the diagonal morphism associated to $f$. $\Delta$ is an immersion, so the corresponding morphism $\Delta^{-1}\mathcal{O}_{X\times_S X}\rightarrow\mathcal{O}_X$ is surjective. Let $\mathcal{I}$ denote its kernel and define the sheaves of principle principal parts as

\[\mathcal{P}_{X/S}^n:=\Delta^{-1}( \mathcal{O}_{X\times_S X}) / \mathcal{I}^{n+1}\]

In their book on Crystalline cohomology, Berthelot and Ogus define the sheaf of principal parts $\mathcal{P}^n_{X/S}$ as

\[(\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_S}\mathcal{O}_X)/\mathbf{I}^{n+1},\] where $\mathbf{I}$ is the kernel of the multiplication map from the tensor product to $\mathcal{O}_X$.

My question is probably simple, but I don't know how to see it: Why are those definitions equivalent if $X$ and $S$ are not affine and $n>0$?

I've not seen the second definition anywhere else, although it seems somewhat nicer than the first one...
show/hide this revision's text 2 Fixed error

In EGA IV, Sec. 16, Grothendieck defines the sheaf of principal parts as follows: Let $f:X\rightarrow S$ be a morphism of schemes and $\Delta:X\rightarrow X\times_S X$ the diagonal morphism associated to $f$. $\Delta$ is an immersion, so the corresponding morphism $\Delta^{-1}\mathcal{O}_{X\times \Delta^{-1}\mathcal{O}_{X\times_S X}\rightarrow\mathcal{O}_X$ is surjective. Let $\mathcal{I}$ denote its kernel and define the sheaves of principle parts as

\[\mathcal{P}_{X/S}^n:=\Delta^{-1}( \mathcal{O}_{X\times mathcal{O}_{X\times_S X}) / \mathcal{I}^{n+1}\]

In their book on Crystalline cohomology, Berthelot and Ogus define the sheaf of principal parts $\mathcal{P}^n_{X/S}$ as

\[\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_S}\mathcal{O}_X/\mathbf{I}^{n+1},\][(\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_S}\mathcal{O}_X)/\mathbf{I}^{n+1},\] where $\mathbf{I}$ is the kernel of the multiplication map from the tensor product to $\mathcal{O}_X$.

My question is probably simple, but I don't know how to see it: Why are those definitions equivalent if $X$ and $S$ are not affine and $n>0$?

I've not seen the second definition anywhere else, although it seems somewhat nicer than the first one...
show/hide this revision's text 1

Sheaves of Principal parts

In EGA IV, Sec. 16, Grothendieck defines the sheaf of principal parts as follows: Let $f:X\rightarrow S$ be a morphism of schemes and $\Delta:X\rightarrow X\times_S X$ the diagonal morphism associated to $f$. $\Delta$ is an immersion, so the corresponding morphism $\Delta^{-1}\mathcal{O}_{X\times X}\rightarrow\mathcal{O}_X$ is surjective. Let $\mathcal{I}$ denote its kernel and define the sheaves of principle parts as

\[\mathcal{P}_{X/S}^n:=\Delta^{-1}( \mathcal{O}_{X\times X}) / \mathcal{I}^{n+1}\]

In their book on Crystalline cohomology, Berthelot and Ogus define the sheaf of principal parts $\mathcal{P}^n_{X/S}$ as

\[\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_S}\mathcal{O}_X/\mathbf{I}^{n+1},\] where $\mathbf{I}$ is the kernel of the multiplication map from the tensor product to $\mathcal{O}_X$.

My question is probably simple, but I don't know how to see it: Why are those definitions equivalent if $X$ and $S$ are not affine and $n>0$?

I've not seen the second definition anywhere else, although it seems somewhat nicer than the first one...