# Sheaf of $1$-forms on affine opens

First some definitions:

Let $$k$$ be algebraically closed. Let $$X$$ be a topological space. We define an algebraic variety $$X$$ to be a $$k$$-space $$(X,\mathcal O_X)$$, such that for each $$x\in X$$, we have an open $$U\subset X$$ with $$x\in U$$, such that $$(U,\mathcal O_X\vert_U)\cong (Z,\mathcal O_Z)$$, where $$Z\subset\mathbb A^n$$ is an (embedded) affine variety, and $$\mathcal O_Z$$ is its sheaf of regular functions. In short: $$X$$ is locally affine.

An (embedded) affine variety $$Z\subset\mathbb A^n$$ is a closed subset of $$\mathbb A^n$$ (with the Zariski topology), and for open $$U\subset Z$$, $$\mathcal O_Z(U)$$ is the $$k$$-algebra consisting of functions $$U\to k$$ that locally can be written as $$f/g$$ with $$f,g\in k[x_1,\dots,x_n]$$.

Given an algebraic variety $$X$$ we have the so-called sheaf of 1-forms $$\Omega^1_X$$, which is defined to be the sheafification of the presheaf $$U\mapsto \Omega^1_{\mathcal O_X(U)}$$, where $$\Omega^1_{\mathcal O_X(U)}$$ is the module of Kähler differentials of $$\mathcal O_X(U)$$ (the regular functions on $$U$$).

Now, my lecture notes state the following two results without proof:

For an affine open $$U\subset X$$, we have $$\Omega^1_X(U)\cong\Omega^1_{\mathcal O_X(U)}$$.

For affine opens $$V\subset U\subset X$$, we have $$\Omega^1_{\mathcal O_X(V)}\cong\Omega^1_{\mathcal O_X(U)}\otimes_{\mathcal O_X(U)}\mathcal O_X(V)$$, and furthermore, the restriction map $$\Omega^1_X(U)\to\Omega^1_X(V)$$ is given by $$\omega\mapsto\omega\otimes 1$$.

When they say the restriction map is given this way, do they mean that we have the following commutative diagram:

Here, the restriction mapping $$\Omega^1_{\mathcal O_X(U)}\to\Omega^1_{\mathcal O_X(V)}$$ is the one induced from $$\iota^*$$ (exactly such that this diagram commutes).

• Giving a good explanation for your first item requires knowing the definition you're using. Please add this to the post. Feb 3, 2021 at 10:21
• @KReiser If I need to add more (e.g., how we're defining an algebraic variety), let me know. Feb 3, 2021 at 10:28
• The short answer is yes, you get the commutative diagram. Feb 3, 2021 at 10:29

Your question: "For an affine open $$U \subseteq X$$, we have

$$\text{O1}. \text{ }\Omega^1_X(U) \cong \Omega^1_{\mathcal{O}_X(U)}.$$

For affine open sets $$V\subseteq U \subseteq X$$ we have

$$\text{O2}. \text{ } \Omega^1_{\mathcal{O}_X(V)} \cong \Omega^1_{\mathcal{O}_X(U)}\otimes_{\mathcal{O}_X(U)} \mathcal{O}_X(V),$$

and furthermore, the restriction map $$\Omega^1_X(U) \rightarrow \Omega^1_X(V)$$ is given by $$ω \rightarrow ω\otimes 1$$.

It seems to me the following is true:

Question O1. Since your sheaf $$\Omega^1_X$$ is defined as the sheafification $$F^{+}$$ of the pre-sheaf $$F(U):= \Omega^1_{\mathcal{O}_X(U)}$$, it seems to me $$F^{+}$$ and $$F$$ should have she same sections on affine open subsets $$U:=Spec(A) \subseteq X$$. Hence

$$F(U)=F^{+}(U) \cong \Omega^1_{A/k}:=\Omega^1_{\mathcal{O}_X(U)}.$$

Question O2. The construction of the module of differentials in functorial in the sense that for any open subscheme $$U \subseteq X$$ it follows $$i^*(\Omega^1_{X/k}) \cong \Omega^1_{U/k}$$ where $$i: U \rightarrow X$$ is the inclusion map.

Moreover for any pair of maps of commutative rings $$A \rightarrow B$$ and $$A \rightarrow S$$ let $$B_S :=S\otimes_A B$$. There is (Matsumura's book "Commutative ring theory", Exercise 25.4 page 198) a canonical isomorphism

$$B_S \otimes_B \Omega^1_{B/A} \cong \Omega^1_{B_S/S}.$$

Hence for any inclusion of basic open sets $$D(f) \subseteq D(g) \subseteq U:=Spec(A)$$ you should get the following:

$$(\Omega^1_{D(g)/k})_{D(f)}:= \Omega^1_{A_g/k} \otimes_{A_g} A_f \cong \Omega^1_{A_{fg}/k}\cong \Omega^1_{A_f/k}.$$

By $$(\Omega^1_{D(g)/k})_{D(f)}$$ I mean the restriction of $$\Omega^1_{D(g)/k}$$ to the open set $$D(f)$$. Here we have used that $$D(f)\cap D(g):=D(fg) =D(f)$$ since $$D(f) \subseteq D(g)$$. The restriction map

$$\rho: \Omega^1_{D(g)/k}\rightarrow (\Omega^1_{D(g)/k})_{D(f)} := \Omega^1_{D(g)/k}\otimes_{A_g} A_f\cong \Omega^1_{D(f)/k}$$

is the canonical map $$\rho(\omega):=\omega \otimes 1$$ where $$1\in A_f$$ is the multiplicative identity. Formula O2 should follow since $$U$$ and $$V$$ have open covers on the form $$D(f)$$.

• Thanks for the elaborate answer! I'm not able to follow every detail (which is entirely on me), but I think I got the idea. Am I correct in saying that you only explained that the diagram commutes as it does for basic opens (instead of for general affine opens)? Feb 3, 2021 at 11:39