persh7

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             "\\ker": "\\text{ker }",
             "\\Q": "\\mathbb Q",
             "\\Set": "\\text{Set}",
-            "\\H": "\\mathscr{H}"
+            "\\H": "\\mathscr{H}",
+            "\\dim": "\\text{dim }",
+            "\\P": "\\mathbb P",
+            "\\A": "\\mathbb A",
+            "\\codim": "\\text{codim}"
         };
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+---
+layout: post
+date: 2024-1-21
+title: "Post 14: Perverse Sheaves Part 7"
+permalink: /post14.html
+katex: true
+order: 14
+---
+
+In this post, I'll define dimension and constructibility, so we can finally have a full definition of perverse sheaves with field coefficients. In the formalities section, I will also have a crash course on varieties.
+
+---
+
+# Intuition
+
+## Dimension
+
+If you've encountered manifolds before, then you know of a notion of dimension there: an $n$-dimensional real manifold has the property that each point has an open neighborhood that is homeomorphic to $\R^n$. There is a more general notion of topological dimension that I will recount here.
+
+- A nonempty topological space $X$ is *irreducible* if it cannot be expressed as a union of two proper and closed subsets.
+
+Using this, we define dimension:
+
+- The *dimension* of a topological space $X$ is the supremum of all integers $n$ for which there are distinct irreducible closed subsets $Z_0,\dots, Z_n$ of $X$ with $Z_0\subset Z_1\subset\cdots\subset Z_n$. The empty set has dimension $-\infty$, where $-\infty$ is a symbol defined by $-\infty\leq a$ for all reall numbers $a$.
+
+Note the use of the word *supremum* instead of maximum here. It may happen that a topological space has infinite dimension, where we take $\infty$ to be a unique symbol satisfying $a\leq \infty$ for all real numbers $a$. However, we will assume that all non-empty spaces we encounter are finite dimensional.
+
+This definition definitely lacks a lot of geometric intuition. Here are some basic facts about dimension that might help you get a feel for it.
+
+1. If $Y\subset X$, then $\dim Y\leq \dim X$.
+
+2. If $X$ has an open cover by subsets $U_i$, then $\dim X = \sup_i\dim U_i$.
+
+3. If $Y$ is a closed subset of an irreducible and finite dimensional space $X$, and if $\dim Y = \dim X$, then $Y=X$. Said differently, if $Y$ is a proper and closed subset of an irreducible and finite dimensional space $X$, then $\dim Y\< \dim X$.
+
+Property 2 is actually very useful in algebraic geometry.
+
+## Constructibility
+
+We want a notion of a sheaf being "locally well-behaved". This will take us through the following chain of definitions:
+
+- For an $R$-module $M$, let $\ul M\_{pre}$ be the presheaf defined by $\ul M\_{pre}(U)=M$ for all opens $U$, and $res_{V,U}=1_M$ for all $U\subset V$. This is called the *constant presheaf*.
+
+- The sheafification of $\ul M\_{pre}$ is denoted by $\ul M$ and called the *constant sheaf*.
+
+Something potentially misleading about the constant sheaf is that it is not as "constant" as the constant presheaf is. In particular, there can be open sets $U$ where $\ul M(U)$ is not just $M$. Alas, this is the terminology that has been used for decades.
+
+- A sheaf $\mc L$ on $X$ is a *local system* if there is an open covering $\\{U_i\\}$ of $X$ such that each sheaf $\mc L\|\_{U_i}$ on $U_i$ is a constant sheaf. A local system is of *finite type* if all of its stalks are finitely generated modules.
+
+Local systems are sometimes also called *locally constant*, but this is also a little misleading in my opinion.
+
+Before we can define constructibility, we need a bit more topology terminology:
+
+- A subset $Y$ of a space $X$ is *locally closed* if it is the intersection of an open subset and a closed subset.
+
+- A *stratification* $\mc S$ of a space $X$ is a finite collection $\\{X_1,\dots, X_n\\}$ of subsets of $X$ with the following conditions:
+
+1. For each $i$, $X_i$ is irreducible, locally closed, and "smooth".
+
+2. $X = \bigcup\_{i=1}^n X_i$
+
+3. For any two $i,j$, either $\ol X_i\cap X_j = X_j$ or $\ol X_i \cap X_j = \varnothing$.
+
+So, "smooth" is technically important here, but I don't think it really matters to understand what it means. It is a "niceness" condition that we want the $X_i$ to have for technical reasons.
+
+An element $X_i$ of a stratification $\\{X_1,\dots, X_n\\}$ is called a *stratum*. The plural of stratum is *strata*.
+
+Now:
+
+- Let $F$ be a sheaf on $X$, and let $\mc S$ be a stratification of $X$. We say $F$ is *constructible with respect to* $S$ if each sheaf $F\|\_{X_i}$ on $X_i$ is a local system of finite type.
+
+- Let $F$ be a sheaf on $X$. We say $F$ is *constructible* if there exists a stratification $\mc S$ of $X$ for which $F$ is constructible with respect to $\mc S$.
+
+- Let $\mc F\in D^b(X)$. We say $\mc F$ is *constructible* (resp. *with respect to a stratification*) if each cohomology sheaf $H^i(\mc F)$ is constructible (resp. *with respect to a stratification*).
+
+The full subcategory of $D^b(X)$ consisting of constructible objects is denoted by $D^b_c(X)$.
+
+So, I actually used something I haven't told you yet. Namely, I never defined the restriction of a sheaf onto a non-open subset, but we are (possibly) using it in the definition of constructibility! Unfortunately, the definition of this restriction for non-open subsets requires a little bit of work that I've laid out in the formalities section. If you're happy accepting the existence of the functor $^\circ f^\*:Sh(Y)\to Sh(X)$ for a continuous function $f:X\to Y$, then $F\|_X = ^\circ j^\*(F)$ for a sheaf $F$ on $Y$, with $j:X\to Y$ the inclusion map. In fact, the definitions of $f^\*$ and restriction are so similar that one may as well not bother defining restriction separately.
+
+We finally have everything we need to make sense of the definition of perverse sheaves when the ring $R$ is a field! To make the format nicer for all readers, I'm going to put the definition of perverse sheaves at the bottom of the post, after the formalities. See you there!
+
+--
+
+# Formalities
+
+The biggest thing I've left unformalized is the notion of "smoothness" used in the definition of stratification. To formalize it, I will have to give a crash course in classical algebraic geometry.
+
+## Varieties
+
+In geometric representation theory, the spaces that one deals with are spaces that come from algebraic geometry. In the setting of Pramod Achar's book, which is my main source for all of these posts, we deal with complex quasiprojective varieties. I will just call these varieties.
+
+The short story is that varieties are locally closed subsets of some complex projective space $\C\P^n$, or simply $\P^n$, with the *Zariski topology*. Our first goal will be to define this topology.
+
+For those of you who have seen varieties before: I will not be assuming that varieties are irreducible.
+
+In what follows, $k$ will denote a fixed algebraically closed field. Since we will end up using $k=\C$, you are at no loss at assuming $k=\C$ if you want.
+
+### Affine space
+
+Affine space $\A^n$ is the set of $n$ tuples of elements of $k$, with the *Zariski topology*: the closed sets are exactly the sets of the form
+
+$$V(S)=\{ a\in\A^n\mid f(a)=0,\forall f\in S \},$$
+
+where $S$ is any subset of $k[x_1,\dots,x_n]$, and $f(a)$ is the element of $k$ defined by substituting $i$th component of $a$ in place of the variable $x_i$.
+
+We will call a topological space an *affine variety* if it is (homeomorphic to) a closed subset of some $\A^n$. Similarly, a *quasi-affine variety* is a locally closed subset of some $\A^n$.
+
+A wonderful fact is that any affine variety $V(S)$ can be written in the form $V(\\{f_1,\dots, f_m\\})$ for **finitely many** polynomials $f_i\in S$. This will also be true of the projective varieties defined below.
+
+### Projective space
+
+First, let us define projective space $\P^n$ at a set level. The element $(0,\dots,0)\in\A^n$ is the zero point; all other elements are called non-zero. For $\lambda\in k$ and $a=(a_1,\dots,a_n)\in \A^n$, we define $\lambda a=(\lambda a_1,\dots,\lambda a_n)$. Two non-zero elements $a,b\in\A^n$ are equivalent if there is a non-zero $\lambda\in k$ such that $b=\lambda a$. We then define $\P^n$ to be the set of equivalence classes of non-zero elements in $\A^{n+1}$. The equivalence class of $(a_0,\dots,a_n)$ is denoted by $\[a_0:\dots:a_n\]$.
+
+We want to define a topology on $\P^n$ that is similar to the topology on $\A^n$. However, the property of a polynomial evaluating to zero is not well-defined on equivalence classes. For instance, the polynomial $f(x)=x_0-x_1^2$ is zero at the point $(1,1)$, but not the equivalent point $(2,2)$. However, evaluating to zero *is* well-defined for *homogeneous* polynomials: polynomials $f\in k[x_0,\dots,x_n]$ such that $f(\lambda a)=\lambda^d f(a)$ for some fixed $d$, called the *degree* of $f$, and all $\lambda\in k$, $a\in \A^n$. We then define the Zariski topology to have closed sets exactly those sets of the form
+
+$$V(S)=\{ a\in\P^n\mid f(a)=0,\forall f\in S \},$$
+
+where $S$ is a set of homogeneous polynomials in $k[x_0,\dots,x_n]$.
+
+A topological space that is (homeomorphic to) a closed subset of some $\P^n$ is called a projective variety; a quasi-projective variety is a locally closed subset of some $\P^n$.
+
+An important aspect of $\P^n$ is that it is covered by $n+1$ open subsets that are homeomorphic to $\A^n$. Indeed, by definition, any point $[a_0:\dots:a_n]\in \P^n$ cannot have all of its coordinates being $0$. We define the $n+1$ subsets $U_i$ to the be the set of points whose $i$th coordinate is non-zero. If it is non-zero, then we can scale the point by its inverse, to get a representative of the same point with $i$th coordinate equal to $1$. The other $n$ coordinates can then be arbitrary elements in $k$, so $U_i$ can be canonically identified with $\A^n$. I'll call the $U_i$ the "affine slices" of $\P^n$.
+
+Note that if $X$ is a (quasi-)projective variety in $\P^n$, then $X\cap U_i$ is a (quasi-)affine variety in $\A^n$. For example, if $X$ is defined by $x_1^2-x_0x_2=0$, then $X\cap U_0$ is defined by $x_1^2-x_2=0$. You can also go in the direction $\A^n$ to $\P^n$, from which you will find that all of the four types of varieties we defined are quasi-projective. Said differently, the class of quasi-projective varieties encompasses the classes of affine, quasi-affine, and projective varieties.
+
+## Smoothness
+
+Suppose we are given a quasi-projective variety $X$ in $\P^n$ that is defined by the intersection of $V(\\{f_1,\dots,f_m\\})$ and the complement of $V(\\{g_1,\dots,g_r\\})$. Given a point $p\in X$, we want to define a notion of what it means for $X$ to be *smooth* at $p$. We do this as follows:
+
+1. Define a notion of smoothness for affine varieties. We will do this later.
+
+2. Pick an affine slice $U_i$ that contains $p$, i.e. such that $p_i\neq 0$.
+
+3. Consider the affine variety $\tilde X = V(\\{\tilde f_1,\dots,\tilde f_m\\})$ in $\A^n$, where $\tilde f$ represents setting $x_i=1$.
+
+4. Let $\tilde p\in \A^n$ be the point corresponding to $p$, i.e. the point with coordinates $p_j/p_i$ for $j\neq i$. Then $\tilde p$ is a point on $\tilde X$.
+
+5. We say $X$ is *smooth* at $p$ if $\tilde X$ is smooth at $\tilde p$.
+
+So, it remainds to discuss the notion of smoothness for affine varieties. Before we do so, we need some terminology:
+
+- Given a non-empty irreducible closed subset $Y$ of a topological space $X$, we define the codimension of $Y$ in $X$, denoted $\codim_XY$, as the supremum over all $n$ such that there is a chain $Y\subset Y_0\subset\cdots\subset Y_n\subset X$ of distinct irreducible closed subsets $Y_0,\dots,Y_n$.
+
+Now, suppose our affine variety $X$ is given by $V(f_1,\dots,f_m)\subset \A^n$. Form the $m\times n$ Jacobian matrix $J$ whose $(i,j)$ component is $\dfrac{\partial f_i}{\partial x_j}$, where this partial derivative is defined formally by using linearity and the power rule. Then the variety is *smooth* at a point $p$ if evaluating the entries of $J$ at $p$ gives a matrix of rank at least $n-\codim_X\\{p\\}$.
+
+If that went over your head, here's a small example. The variety $X=V(\\{x_1x_2\\})\subset \A^2$ looks like a union of two lines; one is the set of points $(0,a_2)$, and the other is the set of points $(a_1,0)$. Each of these lines is an irreducible closed subset of $X$, but $X$ itself is not irreducible (because it is the union of proper closed subsets). The codimension of any point is $1$. Then for $X$ to be smooth at $p$, we ought to have $J(p)$ having rank at least $2-1=1$. The Jacobian will be a row vector with two entries, $x_2$ and $x_1$. Then $J(p)$ will have rank one if and only if $p_1$ and $p_2$ are not both zero; otherwise, $J(p)$ has rank zero. Therefore, $X$ is smooth at all points that aren't $(0,0)$, where the two lines intersect. Hopefully this result makes sense geometrically.
+
+## A final note on topologies
+
+One subtle point that you may want to be aware of when studying perverse sheaves is this: $\C\P^n$ has two topologies, the Zariski topology and the more intuitive "analytic" topology[^analytop] that you may have encountered before. Recall that sheaves are defined relative to a topology. In Pramod Achar's book on perverse sheaves, and perhaps this is standard in geometric representation theory, sheaves on varieties are considered with respect to the analytic topology, not the Zariski topology. However, notions like stratification that involve topological language will refer to the Zariski topology. It's definitely a bit confusing.
+
+[^analytop]: Since $\C\P^n$ is covered by subsets that are naturally identified with $\A^n$, which is essentially just another name for $\C^n$, and $\C$ has a topology induced by the "usual" topology on $\R$, there is a topology on $\C\P^n$ also induced by the topology on $\R$. By "usual" topology I mean the one generated by open intervals.
+
+---
+
+# Perverse sheaves, finally
+
+We can now define perverse sheaves in the case where the coefficient ring $R$ is a field.
+
+- A *perverse sheaf* is a complex $\mc F\in D^b_c(X)$ satisfying the following conditions:
+
+1. For all integers $i$, we have $\text{dim}\left (\text{supp}\left ( H^i(\mc F) \right )\right )\leq -i$.
+
+2. For all integers $i$, we have $\text{dim}\left (\text{supp}\left ( H^i(\bb D(\mc F)) \right )\right )\leq -i$.
+
+The category of perverse sheaves on $X$ is denoted $Perv(X)$. It is abelian.
+
+As I mentioned at the end of Part 6, defining perverse sheaves for non-field coefficient rings is a bit of a hassle. I have decided that I won't go through with it, even in the formalities section.
+
+The next and final post in this chain of posts will concern some basic facts about and examples of perverse sheaves. Since Part 1 already contains some of the applications, I won't mention them again.
+
+---
+
+Footnotes: