(Sheaf Theory I) Sheaf of regular functions

In this post, we will discuss the notion of sheaf via regular functions on affine varieties. The "local information" about stalk will be also mentioned in the categorical meaning. As we will see, the language of category theory is useful in this context. This post is the first part in our program to understand the Riemann-Roch theorem as an application of sheaf theory. There are some questions left, please let me know if you have any further ideas on the topic. Also, we should have much more commutative diagrams in this note, but it is extremely difficult to draw these things on blog. Please let me know if you know how to do this.

References

[1] Andreas Gathmann, "Algebraic Geometry", class note 2002/2003.

[2] Andreas Gathmann, "Algebraic Geometry", class note 2014.

[3] Dino Lorenzini, "An Invitation to Arithmetic Geometry".

[4] B. R. Tenninson, "Sheaf Theory".

1. Regular functions on affine varieties.

We now get familiar with regular functions on affine varieties first before going further and make things more abstract.

Definition 1.1. Let $X$ be an affine variety, $a$ is a point on $X$. We call $\varphi$ a regular function at $a$ if $\varphi$ is defined on an open set $U_a$ containing $a$, and $\varphi(x)=\frac{g(x)}{f(x)}$ for all $x\in U_a$, where $f$ and $g$ are polynomials in $A(X)$-the coordinate ring of $X$.

Definition 1.2. Let  $X$ be an affine variety, and $U$ be an open subset of $X$, the map $\varphi:U\rightarrow k$ is called regular function if $\varphi$ is regular at any point in $U$.

Let $U$ be an open subset of $X$, it is not difficult to check that $\mathscr{O}_X(U)$-the set of all regular functions on $U$ forms a ring, which is called the ring of regular functions on $U$. We now look at closer on this ring via localization.

Let $\varphi$ be a regular function on $A(X)$, then $\varphi$ is defined at any point in $A(X)$, that means $\varphi$ is defined at all point $a\in X$. That means, $\varphi$ is a regular functions on $A(X)$ iff $\varphi\in\cap_{M\in\max(A(X))}A_M$, hence $\mathscr{O}_X=\cap_{M\in\max(A(X))}A_M$. Applying the basic result from Chapter II of [3], which states for all domain $A$, and $\max(A)$ denote the set of all maximal ideals of $A$, then $A=\cap_{M\in\max(A)}A_M$. From this, we have
$$\mathscr{O}_X=A(X)$$
We now consider an important kind of open subset of $X$. Let $f$ be a polynomial function in $A(X)$, we denote $D(f)=\{a\in X| f(a)\ne 0\}$. Then it can be seen that $D(f)\cap D(g) = D(fg)$. Besides, if $U$ is an arbitrary open subset, then $X-U$ is a closed, which means $X-U=V(f_1,...,f_k)$, for $f_i\in A(X)$. Hence, $U = X-V(f_1,...,f_k)=D(f_1)\cup D(f_2)\cup...\cup D(f_k)$. That means, any open subset can be expressed by finite union of open subsets of the form $D(f)$. By using localization again, we can prove

Theorem 1.3. $\mathscr{O}_X(D(f))=\{\frac{g}{f^n}|g\in A(X)\}$.

Proof. Let $S=\{1,f,f^2,...\}$, it is equivalent to prove $\mathscr{O}_X(D(f))=S^{-1}A(X)$. First, let $U$ be an arbitrary open subset of $X$, $\varphi\in\mathscr{O}_X(U)$ iff it is defined at all points of $U$, and hence, by Hilbert's nullstelensatz, if we denote the maximal ideal corresponding to a point $a$ by $M_a$, we have $\varphi\in A(X)_{M_a}$ for all points $a\in I$. And hence, $\mathscr{O}_X(U)=\cap_{a\in U}A(X)_{M_a}$.

Then we can see there exists one-to-one correspondence between the set of maximal ideals of $S^{-1}A(X)$ and the set $\mathscr{M}$ of maximal ideals of $A(X)$, which disjoints with $S$. By Hilbert's nullstelensatz, we have $\mathscr{M}={M_a|a\in D(f)}$. Furthermore,

$$S^{-1}A(X)=\cap_{M\in \max(S^{-1}A(X))}A(X)_M=\cap_{a\in D(f)}A(X)_{M(a)}=\mathscr{O}_X(D(f))$$

And this concludes our theorem (Q.E.D).

Problem 1.4. Using this theorem, compute $\mathscr{O}_\mathbb{A^2}(\mathbb{A}^2-\{(0,0)\})$.

Let $\varphi$ be a regular function on an open subset $U$, then the set $V(\varphi)=\{a\in U|\varphi(a)=0\}$ is closed in $U$. It is obvious since the complement set $X-V(\varphi)=\{a\in U|\varphi(a) \ne 0\}$ is open in $U$. Using this, we can prove the following remarkable

Theorem 1.5. Let $V$ be an non-empty open subset of $U$, and $\varphi_1,\varphi_2$ are two regular functions on $V$. If $\varphi_1,\varphi_2$ agree on $V$, then they agree on $U$.

Proof. By our previous argument, $V(\varphi_1-\varphi_2)$ is closed in $U$, and contains $V$. That means, it contains the closure of $V$ in $U$. Because of the properties of Zarisky topology, $V$ is dense in $X$, and $\overline{V}=X$, which is irreducible algebraic set. Hence, $V$ is also irreducible, and the closure of $V$ in $U$ is $U$ itself. Hence, the two regular functions agree on $U$ (Q.E.D).

Besides, another important aspect of $\mathscr{O}_X$ is the gluing property, which is stated as

Theorem 1.6. Let $U$ be an open subset of $X$, and $\{U_i\}(i\in I)$ is an arbitrary open cover of $U$. Let $\varphi_i$ be a regular function of $U_i$ such that $\varphi_{i_{|U_i\cap U_j}}=\varphi_{j_{|U_i\cap U_j}}$. Then there exists one and only one regular function $\varphi$ of $U$ such that $\varphi_{|U_i}=\varphi_i$.

Proof. Let $\{U_i\}$ be any open cover of $U$, where $U_i\ne\emptyset$. If $\varphi_i,\varphi_j$ are two regular functions defined at two points $P_i, P_j$, respectively, where $P_i\in U_i, P_j\in U_j$. Then at point $P_i$, there exists an open subset $V_i$ of $U_i$, s.t. $\varphi_i(x)=\frac{f_i(x)}{g_i(x)}(\forall x\in V_i)$, and similarly for $P_j$. We then have $f_ig_j-f_jg_i=0$ in $V_i\cap V_j$. Hence, by our definition, we can choose a regular function $\varphi$ of $U$ such that $\varphi_{V_i}=\frac{f_i}{g_i}$. From this, $\varphi_{|U_i}=\varphi_i$. The uniqueness of $\varphi$ is obvious since the value of each point in $U$ via $\varphi$ is fixed by $\varphi_i$. (Q.E.D)

2. Sheaf of regular functions and stalk at a point.

We now give the definition of presheaf.

Definition 2.1. Let $X$ be a topological space, a presheaf $\mathscr{F}$ on $X$ consists of the following data:

1. For every open subset $U$ of $X$, $\mathscr{F}(U)$ is a ring (that can be considered as the ring of functions on $U$).
2. For every inclusion of open sets $U\subset V$, there exists a ring homomorphism, which is called restriction map $\rho^V_U$ from $\mathscr{F}(V)$ to $\mathscr{F}(U)$.

that satisfies some axioms

1. $\mathscr{F}(\emptyset)=0$.
2. $\rho^U_U$ is the identity map on $\mathscr{F}(U)$ for all $U$.
3. For all $U\subset V\subset W$ are open subsets of $X$, we have $\rho^W_U=\rho^V_U\circ \rho^W_V$.

In [1] and [2], a presheaf $\mathscr{F}$ is called sheaf if it satisfies the gluing properties, that means, for any open cover $\{U_i\}$ of $U$, and $\varphi_i\in\mathscr{F}(U_i)$, s.t. $\varphi_{i_{|U_i\cap U_j}}=\varphi_{j_{|U_i\cap U_j}}$, then there exists one and only one $\varphi\in\mathscr{F}(U)$ such that $\varphi_{|U_i}=\varphi_i$.

In [4], the author requires one additional axiom, where a sheaf can be considered as a mono-presheaf. That is, let $\{U_i\}$ is an open cover of $U$, and $\varphi,\varphi'$ are in $\mathscr{F}(U)$ s.t. $\forall i, \varphi_{|U_i}=\varphi'_{|U_i}$, then $\varphi=\varphi'$.

Actually, in our case, when $X$ is an affine variety, the first section ease us to check that we have the sheaf $\mathscr{O}_X$ in this case, where $\mathscr{O}_X(U)$ is the ring of regular functions on an open subset $U$ of $X$.

Follow the definition of sheaf in [4], if we begin with the presheaf $\mathscr{F}$ on a topological space $X$. Then for any open subset $U$, and its open cover $\{U_i\}$ the map $\mathscr{F}\xrightarrow[]{f} \prod_{i\in I} \mathscr{F}(U_i)  \xrightarrow[h]{g} \prod_{(i,j)\in I\times I}\mathscr{F}(U_i\cap U_j)$ is an equalizer. We recall that if $A \xrightarrow{f} B \xrightarrow[h]{g} C$ is a diagram of sets and maps, then it is call an equalizer if $f$ is injective and $\forall a\in A, g(f(a))=h(f(a))$, or it is equivalent to say that there exists a bijective map between $A$ and a subset of $B$, where $g$ and $h$ agree.

Proposition 2.2. Follow the definition of sheaf in [4], then $\mathscr{F}$ is a sheaf on $X$ iff for any  open subset $U\subset X$ and its open cover $\{U_i\}_{i\in I}$, the following diagram is an equalizer $\{U_i\}$ the map $\mathscr{F}\xrightarrow[]{f} \prod_{i\in I} \mathscr{F}(U_i)  \xrightarrow[h]{g} \prod_{(i,j)\in I\times I}\mathscr{F}(U_i\cap U_j)$.

Proof. It is nothing but reinterpreting things (Q.E.D).

We now come to the local information, by the definition of stalk at a point. The first place we look at it is in the context of sheaf of regular functions. Let $a$ be a point on our affine variety $X$, and $U,U'$ open subsets of $X$ that contains A, with regular function $\varphi, \varphi'$ in $\mathscr{O}_X(U), and \mathscr{O}_X(U')$ respectively. We said $(U,\varphi)\sim(U',\varphi')$ if there exists an open subset $V\subset U\cap U'$ such that $\varphi_V=\varphi'_V$. It is obvious to check that $\sim$ is an equivalent relation. And the stalk at $a$ is defined

$$\mathscr{O}_{X,a}=\{(U,\varphi)|U\text{ is open in } X, a\in U, \varphi\in\mathscr{O}_X(U)\}/\sim$$

Problem 2.3. Prove that $\mathscr{O}_{X,a}$ has ring structure.

An element in the stack at $a$ is called a germ at $a$. We will see that in our current context, the stalk at $a$ is actually the local ring.

Proposition 2.4. $\mathscr{O}_{X,a}\cong A(X)_{M_a}=\{\frac{g}{f}| g,f\in A(X), f(a)\ne 0\}$.

Proof. Let us consider the map from $A(X)_{M_a}$ to $\mathscr{O}_{X,a}$ that sends $\frac{g}{f}$ to the class of $(D(f), \frac{g}{f})$. This map is well-defined since for all $\frac{g'}{f'}\sim\frac{g}{f}$, that means, there exists $h(x)\in A(X)$, and $h(a)\ne 0$, such that $h(gf'-fg')=0$. On the open subset $D(f)\cap D(f')\cap D(h)$ of $X$, we have $gf' - fg' = 0$. And hence, $\frac{g}{f}$ and $\frac{g'}{f'}$ agree on this open set. Once the map is well-defined, it is not difficult to check it is a ring homomorphism.

Furthermore, let $(U,\varphi)$ is a germ at $a$, then by the definition, there exists $U_a\subset U$ such that $\phi=\frac{g}{f}$ on $U_a$, where $f,g\in A(X)$. That means, $(U,\phi)\sim(U_a,\frac{g}{f})$, where $f(x)\ne 0$. Hence, we can see our map is surjective.

Assume that $\frac{g}{f}$ and $\frac{g'}{f'}$ agree on an open subset $U_a$ of $a$, and $f(a)\ne0, f'(a)\ne 0$. Then in this subset $fg' - gf' = 0$. Because $U_a$ is dense in $X$, and $V(fg'-gf')$ is closed in $X$, we can conclude that $fg'-gf' = 0$ in $A(X)$. Hence, $\frac{g}{f}\sim\frac{g'}{f'}$ in $A(X)_{M_a}$. This implies our map is injective.

And the theorem now follows (Q.E.D).

For the general definition of stalk at a point, we begin with a sheaf $\mathscr{F}$ on a topological space $X$. Let $a$ be a point on $X$. Let $U,U'$ be two open subsets of $X$, and $\varphi,\varphi'$ in $\mathscr{F}(U), \mathscr{F}(U')$, respectively. We say $(U,\varphi)\sim(U',\varphi')$ if there exists an open subset $V\subset U\cap U'$ such that $\varphi,\varphi'$ agree on $V$. It is obvious to see that $\sim$ is an equivalent relation. The stalk at $a$ is defined

$$\mathscr{F}_a=\{(U,\varphi)|U\text{ is open in }X,a\in U,\varphi\in\mathscr{F}(U)\}/\sim$$

By a very similar method, we can prove that $\mathscr{F}_a$ has the ring structure, and elements of $\mathscr{F}_a$ are called germs at $a$.

Let us now look at the definition of stalk in the language of category theory.

3. Stalk as direct limit.

We first recall some definitions and constructions related to direct limit.

Definition 3.1. A set $I$ with the a partial order $\le$ (that satisfies the reflexive and transitive relation: for all $\alpha\in I$, $\alpha\le \alpha$, and for all $\alpha,\beta,\gamma\in I$, that satisfies $\alpha\le\beta\le\gamma$ we have $\alpha\le\gamma$) is called directed set if for any $\alpha,\beta\in I$, there exists $\gamma$ such that $\alpha\le\gamma$ and $\beta\le\gamma$. We denote $\mathscr{I}=\{(\alpha,\beta)\in I\times I|\alpha\le\beta\}$.

For example, let $X$ be a topological space, and $\mathscr{T}$ denotes the collection of all open subsets of $X$. For $U,V\in\mathscr{T}$, we denote $U\le V$ if $V\subset U$. Then it is easy to check that $(\mathscr{T},\le)$ is a directed set.

Definition 3.2. Let $(I,\le)$ be a directed set, and $\{U_i\}_{i\in I}$ is a family of sets. We call $\{U_i\}_{i\in I}$ the directed system if for each $(\alpha,\beta)\in \mathscr{I}$, there exists a map $\rho^\alpha_\beta:U_\alpha\rightarrow U_\beta$ such that:
1. $\rho^\alpha_\alpha =id_{U_\alpha}$.
2. For any $\alpha\le\beta\le\gamma$ in $I$, we have $\rho^\alpha_\gamma=\rho^\beta_\gamma\circ\rho^\alpha_\beta$.

Let $X$ be a topological space, and $\mathscr{F}$ is a sheaf on $X$. With the directed set $(\mathscr{T},\le)$ is defined as above, then it can be seen that $\{\mathscr{F}(U)\}_{U\in\mathscr{T}}$ is a directed system. In a local context, let $a$ be a point in $X$, and $\{\mathscr{T}_a,\le\}$ is the directed set of open subsets of $X$ that contain $a$. Then it can be seen that $\{\mathscr{F}(U_a)\}_{U_a\in \mathscr{T}_a}$ is a directed system.

 Definition 3.3. Let $\{U_\alpha\}$ be a directed system of sets. A target for the system is a set $V$ with a collection of maps $\{\varphi_\alpha: U_\alpha\rightarrow V\}$ that satisfies for all $(\alpha,\beta)\in \mathscr{I}$, we have $\varphi_\beta\circ\rho^\alpha_\beta=\varphi_\alpha$.

And the direct limit is actually a target that lies on the right of everything.

Definition 3.4. Let $\{U_\alpha\}$ be a directed system of sets. A target $(U,\varphi_\alpha:U_\alpha\rightarrow U)$ of this system is called direct limit if for any target $(V,\sigma_\alpha:U_\alpha\rightarrow V)$, there exists a unique map $f: U\rightarrow V$ such that for all $\alpha$, we have $\sigma_\alpha=f\circ\varphi_\alpha$. And we denote $U=\varinjlim U_\alpha$.

By our definition, it is obvious to see that the direct limit is unique up to isomorphism. The construction of direct limit as we will see is very similar to the construction of stalk at a point. Let $\{U_\alpha\}_{\alpha\in I}$ is the directed system, and $\amalg U_\alpha$ is disjoint union of all $U_\alpha$. We define the relation $\sim$ on $\amalg U_\alpha$, $x_\alpha\in U\alpha\sim x_\beta\in U_\beta$ if there exists $\gamma\le\alpha, \gamma\le\beta$ such that $\rho^\gamma_\alpha(x_\alpha)=\rho^\gamma_\beta(x_\beta)$. It can be checked that $\sim$ is an equivalent relation, and we denote $U=\amalg U_\alpha/\sim$

The map $\varphi_\alpha$ from $U_\alpha$ to $U$ is defined by $x_\alpha\mapsto [x_\alpha]_\sim$. We now check that $(U,\varphi_\alpha)$ is actually a target of our directed system. For any $\alpha\le\beta$, we have to check that $\varphi_\beta\circ \rho^\alpha_\beta=\varphi_\alpha$, which is equivalent to say that $x_\alpha\sim\rho^\alpha_\beta(x_\alpha)$ for all $x_\alpha\in U_\alpha$. It is obvious, since there exists $U_\alpha$, and $\alpha\le\beta, \alpha\le\alpha$, and $\rho^\alpha_\alpha(\rho^\alpha_\beta(x_\alpha))=id_{U_\alpha}(\rho^\alpha_\beta(x_\alpha))=\rho^\alpha_\beta(x_\alpha)$.

We now prove the universal property for the target $(U,\varphi_\alpha)$. Let $(V,\sigma_\alpha)$ be another target, and $f:U\rightarrow V$ that sends $[x_\alpha]_\sim$ to $\sigma_\alpha(x_\alpha)$. We have to check that it is well-defined, that means, for $x_\alpha\sim x_\beta$, then $\sigma_\alpha(x_\alpha)=\sigma_\beta(x_\beta)$. It follows easily, because there exists $\gamma\le\beta,\gamma\le\alpha$ such that $\rho^\gamma_\alpha(x_\alpha)=\rho^\gamma_\beta(x_\beta)$. Taking $\sigma_\gamma$ on both side, because $\sigma_\gamma\circ\rho^\gamma_\alpha=\sigma_\alpha$, and $\sigma_\gamma\circ\rho^\gamma_\beta=\sigma_\beta$, we have $\sigma_\alpha(x_\alpha)=\sigma_\beta(x_\beta)$. The uniqueness of the map $f$ is obvious.

Problem 3.5. Via the construction above, do you see the similarity of the limit of sequences or limit of functions with direct limit?

We have constructed the direct limit of directed system. Let $X$ be a topological space, $a$ a point on $X$, and $\mathscr{F}$ a sheaf on $X$. We recall the notions of the directed set $(\mathscr{T}_a,\le)$ the collections of all open subset of $X$ that contains $a$, and the directed system $\{\mathscr{F}(U_a)_{U_a\in\mathscr{T}_a}\}$, we have $\mathscr{F}_a$ the stalk of $a$ is $\varinjlim \mathscr{F}(U_a)$, by our construction above.

Problem 3.6. The direct limit is an exact functor (prove this). Is there any exact sequence related to the stalk induced from exact sequence related to open subsets containing $a$?

Problem 3.7. What is the dual notion of stalk? And how about their geometric information?

Problem 3.8. What can we see about the further properties of stalk at a point via the construction?