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数学代写|复几何代写Complex Geometry代考|MATH3405 The Category of Sheaves

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数学代写|复几何代写Complex Geometry代考|The Category of Sheaves

It will be convenient to define presheaves of things other than functions. For instance, one might consider sheaves of equivalence classes of functions, distributions, and so on. For this more general notion of presheaf, the restriction maps have to be included as part of the data:

Definition 3.1.1. A presheaf $\mathscr{P}$ of sets (respectively groups or rings) on a topological space $X$ consists of a set (respectively group or ring) $\mathscr{P}(U)$ for each open set $U$, and maps (respectively homomorphisms) $\rho_{U V}: \mathscr{P}(U) \rightarrow \mathscr{P}(V)$ for each inclusion $V \subseteq U$ such that:

1. $\rho_{U U}=\mathrm{id}_{\mathscr{P}(U)}$;
2. $\rho_{V W} \circ \rho_{U V}=\rho_{U W}$ if $W \subseteq V \subseteq U$.
We will usually write $\left.f\right|V=\rho{U V}(f)$. Here is a simple example of a presheaf given abstractly.

Example 3.1.2. Let $X$ be topological space. Then
$$\mathscr{P}(U)= \begin{cases}\mathbb{Z} & \text { if } U=X, \ 0 & \text { otherwise }\end{cases}$$
with all $\rho_{U V}=0$, is a presheaf.
A more natural class of examples, which arises frequently, is given by the quotient construction.

Example 3.1.3. Let $\mathscr{P}$ be a presheaf of abelian groups. Then a subpresheaf $\mathscr{P}^{\prime} \subseteq \mathscr{P}$ is a collection of subgroups $\mathscr{P}^{\prime}(U) \subseteq \mathscr{P}(U)$ stable under the restriction maps $\rho_{U V}$. The presheaf quotient is given by
$$\left(\mathscr{P} / \mathscr{P}^{\prime}\right)^P(U)=\mathscr{P}(U) / \mathscr{P}(U)^{\prime}$$
with the induced restrictions. (This somewhat clumsy notation is used to distinguish this from the quotient sheaf to be defined later on.)
The definition of a sheaf carries over verbatim.

数学代写|复几何代写Complex Geometry代考|Exact Sequences

The categories $\operatorname{PAb}(X)$ and $\operatorname{Ab}(X)$ are additive, which means among other things that $\operatorname{Hom}(A, B)$ has an abelian group structure such that composition is bilinear.

Actually, more is true. These categories are abelian $[44,82,118]$, which implies that they possess many of the basic constructions and properties of the category of abelian groups. In particular, given a morphism, we can form kernels, cokernels, and images, characterized by the appropriate universal properties. This is spelled out more fully in the exercises. Here we just define these operations. Given a morphism of presheaves $f: \mathscr{A} \rightarrow \mathscr{B}$, we define the presheaf kernel, image, and cokernel by
\begin{aligned} (\operatorname{pker} f)(U) &=\operatorname{ker} f_U:[\mathscr{A}(U) \rightarrow \mathscr{B}(U)], \ (\operatorname{pim} f)(U) &=\operatorname{im} f_U:[\mathscr{A}(U) \rightarrow \mathscr{B}(U)], \ (\operatorname{pcoker} f)(U) &=\operatorname{coker} f_U:[\mathscr{A}(U) \rightarrow \mathscr{B}(U)] . \end{aligned}
This is an isomorphism if $f_U$ is an isomorphism for every $U$, or equivalently if pker $f=\operatorname{pcoker} f=0$.
For a morphism of sheaves $f: \mathscr{A} \rightarrow \mathscr{B}$, the sheaf kernel, etc. is given by
$$\operatorname{ker} f=(\operatorname{pker} f)^{+}, \quad \operatorname{im} f=(\operatorname{pim} f)^{+}, \quad \operatorname{coker} f=(\operatorname{pcoker} f)^{+} .$$
We may get a better sense of these by looking at the stalks:
\begin{aligned} (\operatorname{ker} f)_x &=(\operatorname{pker} f)_x=\operatorname{ker} f_x:\left[\mathscr{A}_x \rightarrow \mathscr{B}_x\right] \ (\operatorname{im} f)_x &=(\operatorname{pim} f)_x=\operatorname{im} f_x:\left[\mathscr{A}_x \rightarrow \mathscr{B}_x\right] \ (\operatorname{coker} f)_x &=(\operatorname{pcoker} f)_x=\operatorname{coker} f_x:\left[\mathscr{A}_x \rightarrow \mathscr{B}_x\right] . \end{aligned}

数学代写|复几何代写Complex Geometry代考|The Category of Sheaves

$\rho_{U U}=\operatorname{id} \mathscr{M}_{(U)}$

$\rho_{V W} \circ \rho_{U V}=\rho_{U W}$ 如果 $W \subseteq V \subseteq U$.

$$\mathscr{P}(U)={\mathbb{Z} \quad \text { if } U=X, 0 \quad \text { otherwise }$$

$$\left(\mathscr{P} / \mathscr{P}^{\prime}\right)^P(U)=\mathscr{P}(U) / \mathscr{P}(U)^{\prime}$$

数学代写|复几何代写Complex Geometry代考|Exact Sequences

$($ pker $f)(U)=\operatorname{ker} f_U:[\mathscr{A}(U) \rightarrow \mathscr{B}(U)],(\operatorname{pim} f)(U) \quad=\operatorname{im} f_U:[\mathscr{A}(U) \rightarrow \mathscr{B}(U)],($ pcoker $f)(U)=\operatorname{coker} f_U:[\mathscr{A}(U) \rightarrow \mathscr{B}(U)]$.

$\operatorname{ker} f=(\operatorname{pker} f)^{+}, \quad \operatorname{im} f=(\operatorname{pim} f)^{+}, \quad \operatorname{coker} f=(\operatorname{pcoker} f)^{+}$

$(\operatorname{ker} f)_x=(\operatorname{pker} f)_x=\operatorname{ker} f_x:\left[\mathscr{A}_x \rightarrow \mathscr{B}_x\right](\operatorname{im} f)_x=(\operatorname{pim} f)_x=\operatorname{im} f_x:\left[\mathscr{A}_x \rightarrow \mathscr{B}_x\right](\operatorname{coker} f)_x=(\operatorname{pcoker} f)_x=\operatorname{coker} f_x:\left[\mathscr{A}_x \rightarrow \mathscr{B}_x\right]$

MATLAB代写

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