Update: (9/25) I just found a nice paper by Andrew Ranicki explaining the algebraic interpretation of the finiteness obstruction.

This is the second piece of a two-part post trying to understand some of the ideas in Wall’s “Finiteness conditions for CW complexes.”

In the previous post, we considered a space ${X}$ which was a homotopy retract of an ${\leq N}$-dimensional finite CW complex (where ${N \geq 3}$), and tried to express ${X}$ itself as homotopy equivalent to one such. We built a sequence of approximations

$\displaystyle K_1 \subset K_2 \subset \dots ,$

of complexes over ${X}$, such that each ${K_i}$ was an ${i}$-dimensional finite complex and such that ${\pi_j(X, K_i) = 0}$ for ${0 \leq j \leq i}$: the maps ${K_i \rightarrow X}$ increase in connectivity at each stage. In general, we cannot make this sequence stop. However, we saw that if ${K_{N-1}}$ was chosen such that the ${\mathbb{Z}[\pi_1 X]}$-module

$\displaystyle \pi_N(X, K_{N-1}) \simeq H_N(\widetilde{X}, \widetilde{K_{N-1}})$

was free (where the tilde denotes the universal cover), then we could build ${K_N}$ from ${K_{N-1}}$ (by attaching ${N}$-cells) in such a way that ${K_N \rightarrow X}$ was a homotopy equivalence: that is, ${\pi_1 K_N \simeq \pi_1 X}$ and ${H_*( \widetilde{X}, \widetilde{K_N}) = 0}$.

The goal now is to use this requirement of freeness to build a finiteness obstruction in analogy with the algebraic situation considered in the previous post. Namely, let ${X}$ be any connected space. Then the universal cover ${\widetilde{X}}$ is a ${\pi_1 X}$-space, and the singular chain complex ${C_*(\widetilde{X})}$ is a complex of ${\mathbb{Z}[\pi_1 X]}$-modules: that is, it lives in the derived category of ${\mathbb{Z}[\pi_1 X]}$-modules. We will see below that if ${X}$ is a finite complex, then it lives in the “finitely presented” derived category introduced in the previous post—so that if ${X}$ is finitely dominated, then ${C_*(\widetilde{X})}$ is in the perfect derived category of ${\mathbb{Z}[\pi_1 X]}$.

Definition 1 The Wall finiteness obstruction of ${X}$ is the class in ${\widetilde{K}^0(\mathbb{Z}[\pi_1 X])}$ represented by the complex ${C_*(\widetilde{X})}$: that is, choose a finite complex ${P_\bullet}$ of finitely generated projective modules representing ${C_*(\widetilde{X})}$, and take ${\sum (-1)^i [P_i]}$. (more…)

Let ${\mathcal{A}}$ be an abelian category with enough projectives. In the previous post, we described the definition of the derived ${\infty}$-category ${D^-(\mathcal{A})}$ of ${\mathcal{A}}$. As a simplicial category, this consisted of bounded-below complexes of projectives, and the space of morphisms between two complexes ${A_\bullet, B_\bullet}$ was obtained by taking the chain complex of maps ${\underline{Hom}(A_\bullet, B_\bullet)}$ between ${A_\bullet, B_\bullet}$ and turning that into a space (by truncation ${\tau_{\geq 0}}$ and the Dold-Kan correspondence).

Last time, we proved most of the following result:

Theorem 5 ${D^-(\mathcal{A})}$ is a stable ${\infty}$-category whose suspension functor is given by shifting by ${1}$. ${D^-(\mathcal{A})}$ has a ${t}$-structure whose heart is ${\mathcal{A}}$, and the homotopy category of ${D^-(\mathcal{A})}$ is the usual derived category.

Note for instance that this means that ${\mathcal{A}}$ sits as a full subcategory inside ${D^-(\mathcal{A})}$: that is, there is a full subcategory ${{D}^-(\mathcal{A})^{\heartsuit}}$ (the “heart”) of ${D^-(\mathcal{A})}$ (spanned by those complexes homologically concentrated in degree zero).

This heart has the property that the mapping spaces in ${D^-(\mathcal{A})^{\heartsuit}}$ are discrete, and the functor

$\displaystyle \pi_0: D^-(\mathcal{A}) \rightarrow \mathcal{A}$

restricts to an equivalence ${D^-(\mathcal{A})^{\heartsuit} \rightarrow \mathcal{A}}$; one can prove this by examining the chain complex of maps between two complexes homologically concentrated in degree zero. The inverse to this equivalence runs ${\mathcal{A} \rightarrow D^-(\mathcal{A})^{\heartsuit}}$, and it sends an element of ${\mathcal{A}}$ to a projective resolution. This is functorial in the ${\infty}$-categorical sense.

Most of the above theorem is exactly the same as the description of the ordinary derived category of ${\mathcal{A}}$ (i.e., the homotopy category of ${D^-(\mathcal{A})}$), The goal of this post is to describe what’s special to the ${\infty}$-categorical setting: that there is a universal property. I will start with the universal property for the subcategory ${D_{\geq 0}(\mathcal{A})}$.

Theorem 6 ${D_{\geq 0}(\mathcal{A})}$ is the ${\infty}$-category obtained from ${\mathcal{P} \subset \mathcal{A}}$ (the projective objects) by freely adding geometric realizations.

The purpose of this post is to sketch a proof of the above theorem, and to explain what it means. (more…)

The next thing I’d like to do on this blog is to understand the derived ${\infty}$-category of an abelian category.

Given an abelian category ${\mathcal{A}}$ with enough projectives, this is a stable ${\infty}$-category ${D^-(\mathcal{A})}$ with a special universal property. This universal property is specific to the ${\infty}$-categorical case: in the ordinary derived category of an abelian category (which is the homotopy category of ${D^-(\mathcal{A})}$), forming cofibers is not quite the natural process it is in ${D^-(\mathcal{A})}$ (in which it is a type of colimit), and one cannot expect the same results.

For instance, ${\mathcal{A}}$, and given a triangulated category ${\mathcal{T}}$ and a functor ${\mathcal{A} \rightarrow \mathcal{T}}$ taking exact sequences in ${\mathcal{A}}$ to triangles in ${\mathcal{T}}$, we might want there to be an extended functor

$\displaystyle D_{ord}^b(\mathcal{A}) \rightarrow \mathcal{T},$

where ${D_{ord}^b(\mathcal{A})}$ is the ordinary (1-categorical) bounded derived category of ${\mathcal{A}}$. We might expect this by the following rough intuition: given an object ${X}$ of ${D^b(\mathcal{A})}$ we can represent it as obtained from objects ${A_1, \dots, A_n}$ in ${\mathcal{A}}$ by taking a finite number of cofibers and shifts. As such, we should take the image of ${X}$ to be the appropriate combination of cofibers and shifts in ${\mathcal{T}}$ of the images of ${A_1, \dots, A_n}$. Unfortunately, this does not determine a functor because cofibers are not functorial or unique up to unique isomorphism at the level of a trinagulated category.

The derived ${\infty}$-category, though, has a universal property which, among other things, makes very apparent the existence of derived functors, and which makes it very easy to map out of it. One formulation of it is specific to the nonnegative case: ${D_{\geq 0}(\mathcal{A})}$ is obtained from the category of projective objects in ${\mathcal{A}}$ by freely adjoining geometric realizations. In other words:

Theorem 1 (Lurie) Let ${\mathcal{A}}$ be an abelian category with enough projectives, which form a subcategory ${\mathcal{P}}$. Then ${D_{\geq 0}(\mathcal{A})}$ has the following property. Let ${\mathcal{C}}$ be any ${\infty}$-category with geometric realizations; then there is an equivalence

$\displaystyle \mathrm{Fun}(\mathcal{P}, \mathcal{C}) \simeq \mathrm{Fun}'( D_{\geq 0}(\mathcal{A}), \mathcal{C})$

between the ${\infty}$-categories of functors ${\mathcal{P} \rightarrow \mathcal{C}}$ and geometric realization-preserving functors ${D_{\geq 0}(\mathcal{A}) \rightarrow \mathcal{C}}$.

This is a somewhat strange (and non-abelian) universal property at first sight (though, for what it’s worth, there is another more natural one to be discussed later). I’d like to spend the next couple of posts understanding why this is such a natural universal property (and, for one thing, why projective objects make an appearance); the answer is that it is an expression of the Dold-Kan correspondence. First, we’ll need to spend some time on the actual definition of this category.

I’d like to discuss today a category-theoretic characterization of Zariski open immersions of rings, which I learned from Toen-Vezzosi’s article.

Theorem 1 If ${f: A \rightarrow B}$ is a finitely presented morphism of commutative rings, then ${\mathrm{Spec} B \rightarrow \mathrm{Spec} A}$ is an open immersion if and only if the restriction functor ${D^-(B) \rightarrow D^-(A)}$ between derived categories is fully faithful.

Toen and Vezzosi use this to define a Zariski open immersion in the derived context, but I’d like to work out carefully what this means in the classical sense. If one has an open immersion ${f: A \rightarrow B}$ (for instance, a localization ${A \rightarrow A_f}$), then the pull-back on derived categories is fully faithful: in other words, the composite of push-forward and pull-back is the identity. (more…)

I’ve been trying to re-understand some of the proofs in commutative and homological algebra. I never really had a good feeling for spectral sequences, but they seemed to crop up in purely theoretical proofs quite frequently. (Of course, they crop up in computations quite frequently, too.) After learning about derived categories it became possible to re-interpret many of these proofs. That’s what I’d like to do in this post.

Here is a toy example of a result, which does not use spectral sequences in its usual proof, but which can be interpreted in terms of the derived category.

Proposition 1 Let ${(A, \mathfrak{m})}$ be a local noetherian ring with residue field ${k}$. Then a finitely generated ${A}$-module ${M}$ such that ${\mathrm{Tor}_i(M, k) = 0, i > 0}$ is free.

Let’s try to understand the usual proof in terms of the derived category. Throughout, this will mean the bounded-below derived category ${D^-(A)}$ of ${A}$-modules: in other words, this is the category of bounded-below complexes of projectives and homotopy classes of maps. Any module ${M}$ can be identified with an object of ${D^-(A)}$ by choosing a projective resolution.

So, suppose ${M}$ satisfies ${\mathrm{Tor}_i(M, k) = 0, i > 0}$. Another way of saying this is that the derived tensor product

$\displaystyle M \stackrel{\mathbb{L}}{\otimes} k$

has no homology in negative degrees (it is ${M \otimes k}$ in degree zero). Choose a free ${A}$-module ${P}$ with a map ${P \rightarrow M}$ which induces an isomorphism ${P \otimes k \simeq M \otimes k}$. Then we have that

$\displaystyle P \stackrel{\mathbb{L}}{\otimes} k \simeq M \stackrel{\mathbb{L}}{\otimes} k$

by hypothesis. In particular, if ${C}$ is the cofiber (in ${D^-(A)}$) of ${P \rightarrow M}$, then ${C \stackrel{\mathbb{L}}{\otimes} k = 0}$.

We’d like to conclude from this that ${C}$ is actually zero, or that ${P \simeq M}$: this will imply the desired freeness. Here, we have:

Lemma 2 (Derived Nakayama) Let ${C \in D^-(A)}$ have finitely generated homology. Suppose ${C \stackrel{\mathbb{L}}{\otimes} k = 0}$. Then ${C = 0}$. (more…)

I’ve been reading the Beilinson-Bernstein-Deligne (BBD) paper “Faisceaux pervers” as part of my summer project. To help myself understand it, I thought I would try to do a few blog posts about the material. The paper is ultimately about perverse sheaves, which are suitable complexes of sheaves on a (stratified) topological space satisfying certain cohomological conditions. The surprise is that perverse sheaves actually form an abelian category. The reason behind this is a bit of homological algebra, which I would like to discuss today to start.

1. t-structures

The first important notion in BBD is that of a ${t}$-structure. If ${\mathcal{A}}$ is an abelian category, and ${\mathbf{D}(\mathcal{A})}$ its derived category, we know that there is a notion of an object in positive degrees and one in negative degrees. There are two subcategories ${\mathbf{D}_{\geq 0}(\mathcal{A}), \mathbf{D}_{\leq 0}(\mathcal{A}) \subset \mathbf{D}(\mathcal{A})}$. Namely, a complex ${K^\bullet}$ can be said to be in ${\mathbf{D}_{\geq 0}(\mathcal{A})}$ if it has no cohomology in negative degrees, and similarly in ${\mathbf{D}_{\leq 0}(\mathcal{A})}$ if it has no cohomology in positive degrees. This is equivalent to saying that ${K^\bullet}$ is isomorphic to a complex all of whose nonzero terms are in nonnegative (resp. nonpositive) degrees because of the truncation functors ${\tau_{\geq 0}, \tau_{\leq 0}}$.

A ${t}$-structure is an axiomatization of this observation for the derived category.

Let ${\mathcal{D}}$ be a triangulated category. A ${t}$-structure on ${\mathcal{D}}$ is the data of two full subcategories ${\mathcal{D}_{\geq 0}, \mathcal{D}_{\leq 0}}$. We can define ${\mathcal{D}_{\leq n}}$ as ${\mathcal{D}_{\leq 0}[-n]}$ and similarly ${\mathcal{D}_{\geq 0} = \mathcal{D}_{\geq 0}[-n]}$, in accordance with what happens for the derived category.

The following axioms are required:

1. ${\mathcal{D}_{\leq 0} }$ is closed under translation ${X \mapsto X[1]}$.
2. If ${X \in \mathcal{D}_{\leq 0}}$, ${Y \in \mathcal{D}_{\geq 1}}$, then ${\hom_{\mathcal{D}}(X, Y) = 0}$.
3. If ${X \in \mathcal{D}}$, there is a triangle

$\displaystyle A \rightarrow X \rightarrow B \rightarrow X[1]$

such that ${A \in \mathcal{D}_{\leq 0}}$ and ${B \in \mathcal{D}_{\geq 1}}$.

The axioms for a ${t}$-structure are simple, but they turn out to be quite powerful, because the axioms for a triangulated category are themselves quite powerful. (more…)

We have spent a while in the past few days going through the rather categorical formalism of the upper shriek functor $f^!$ obtained from a map $f: X \to Y$ between locally compact Hausdorff spaces of finite cohomological dimension. That is, we showed that the upper shriek must exist on the derived category; this was Verdier duality. However, so far we have not seen any concrete applications of this formalism. I actually feel a bit guilty about having not indicated better some of these in the introductory post and having essentially plunged into the abstract nonsense.

Now we shall apply the existence of ${f^!}$ to questions involving manifolds. Once we know that ${f^!}$ exists, we will be able to describe it using the adjoint property rather simply (for manifolds). This will lead to clean statements of theorems in algebraic topology. For instance, Poincaré duality will be a direct consequence of the fact that, on an ${n}$-dimensional oriented manifold, the dualizing sheaf (see below) is just ${k[n]}$.

1. The dualizing complex

After wading through the details of the proof of Verdier duality, let us now consider the simpler case where ${Y = \left\{\ast\right\}}$. ${X}$ is still a locally compact space of finite dimension, and ${k}$ remains a noetherian ring. Then Verdier duality gives a right adjoint ${f^!}$ to the functor ${\mathbf{R} \Gamma_c: \mathbf{D}^+(X, k) \rightarrow \mathbf{D}^+(k)}$. In other words, for each ${\mathcal{F}^\bullet \in \mathbf{D}^+(X, k)}$ and each complex ${G^\bullet}$ of ${k}$-modules, we have an isomorphism

$\displaystyle \hom_{\mathbf{D}^+(k)}(\mathbf{R} \Gamma_c (\mathcal{F}^\bullet), G^\bullet) \simeq \hom_{\mathbf{D}^+(X, k)}(\mathcal{F}^\bullet, f^!(G^\bullet)).$

Of course, the category ${\mathbf{D}^+(k)}$ is likely to be much simpler than ${\mathbf{D}^+(X, k)}$, especially if, say, ${k}$ is a field.

Definition 1 ${\mathcal{D}^\bullet = f^!(k)}$ is called the dualizing complex on the space ${X}$. ${\mathcal{D}^\bullet}$ is an element of the derived category ${\mathbf{D}^+(X, k)}$, and is well-defined there. We will always assume that ${\mathcal{D}^\bullet}$ is a bounded-below complex of injective sheaves. (more…)

This is the fifth in a series of posts on Verdier duality, started here. In this post, I will describe the proof of the duality theorem, which itself states the existence of an adjoint to the derived version of the lower shriek functor $f_!$. This might not sound too exciting at first, but we will see that in fact, the dualizing functor will be computable in the important special case of a manifold, and Poincaré duality will fall out quickly. Moreover, the flexible interpretation of sheaf cohomology will allow other duality theorems (such as Alexander duality) to be derived very efficiently from the general formalism.

I will try to explain some of this story (namely, that using sheaf cohomology and Verdier duality one can re-derive much of the classical theory of homology and cohomology) next time. First, though, it will be good to prove the result.

1. Duality

We can now enunciate the result we shall prove in full generality.

Theorem 1 (Verdier duality) Let ${f: X \rightarrow Y}$ be a continuous map of locally compact spaces of finite dimension, and let ${k}$ be a noetherian ring. Then ${\mathbf{R} f_!: \mathbf{D}^+(X, k) \rightarrow \mathbf{D}^+(Y, k)}$ admits a right adjoint ${f^!}$. In fact, we have an isomorphism in ${\mathbf{D}^+(k)}$

$\displaystyle \mathbf{R}\mathrm{Hom}( \mathbf{R} f_! \mathcal{F}^\bullet , \mathcal{G}^\bullet) \simeq \mathbf{R}\mathrm{Hom}( \mathcal{F}^\bullet, f^! \mathcal{G}^\bullet)$

when ${\mathcal{F}^\bullet \in \mathbf{D}^+(X, k), \mathcal{G}^\bullet \in \mathbf{D}^+(Y, k)}$.

Here ${\mathbf{R}\mathrm{Hom}}$ is defined as follows. Recall that given chain complexes ${A^\bullet, B^\bullet}$ of sheaves, one may define a chain complex of ${k}$-modules ${\hom^\bullet(A^\bullet, B^\bullet)}$; the elements in degree ${n}$ are given by the product ${\prod_m \hom(A^m, B^{m+n})}$, and the differential sends a collection of maps ${\left\{f_m: A^m \rightarrow B^{m+n}\right\}}$ to ${ df_m + (-1)^{n+1} f_{m+1}d: A^m \rightarrow B^{m+n+1}}$. Then ${\mathbf{R}\mathrm{Hom}}$ is the derived functor of ${\hom^\bullet}$, and lives in the derived category ${\mathbf{D}^+(k)}$ if ${A^\bullet, B^\bullet \in \mathbf{D}^+(X, k)}$. Since the cohomology in degree zero is given by ${\hom_{\mathbf{D}^+(X, k)}(A^\bullet, B^\bullet)}$, we see that the last statement of Verdier duality implies the adjointness relation. (more…)

This is the third in a series of posts started here (in particular, the notation is kept from there) intended to cover the basics of Verdier duality. Here, I will discuss the lower shriek functors needed even to state Verdier duality (in the most general form, at least); as we will see, the class of soft sheaves will be acyclic with respect to this functor. To see this, though, we shall need to prove some general facts on how push-forward behave with respect to base change, which are themselves of independent interest.

1. The ${f_!}$ functors

Let ${f: X \rightarrow Y}$ be a map of spaces. We have defined the functor

$\displaystyle f_! : \mathbf{Sh}(X) \rightarrow \mathbf{Sh}(Y)$

earlier, such that ${f_!(U) }$ consists of the sections of ${\mathcal{F}(f^{-1}(U))}$ whose support is proper over ${U}$ ; ${f_!\mathcal{F} }$ is always a subsheaf of ${f_*\mathcal{F}}$, equal to it if ${f}$ is proper. When ${Y}$ is a point, we get the functor

$\displaystyle \mathcal{F} \mapsto \Gamma_c(X, \mathcal{F}) = \left\{\text{global sections with proper support}\right\}.$

One can check that ${f_! \mathcal{F}}$ is in fact a sheaf. The observation here is that a map ${A \rightarrow B}$ of topological spaces is proper if and only if there is an open cover ${\left\{B_i\right\}}$ of ${B}$ such that ${A \times_B B_i \rightarrow B_i}$ is proper for each ${i}$. Now ${f_!}$ is a left-exact functor, as one easily sees. We now want to show that the class of soft sheaves is acyclic with respect to ${f_!}$, and in particular so that one may use soft resolutions to compute the derived functors. To do this, we shall prove a general “base change” theorem that will compute the stalk of ${f_! \mathcal{F}}$. (more…)

I’ve been away from this blog for longer than I should have. I got stuck in my series on the cotangent complex, partially because I’ve been busy doing other things–namely, trying to learn about the foundations of etale cohomology. As I learn more I might write a few posts. And someday the cotangent complex thing will get finished as a short expository note on my website.

One thing I’ve discovered as of late is that many concepts that I learned earlier in life were in fact shadows or special cases of more powerful and general ones. I’ve consequently had to un-learn many such concepts, to replace them with the newer ones.

Sheaves

An example is basic sheaf theory: like many people, I learned this from Hartshorne chapter II, working out the exercises there. But as I have more recently discovered, many of the methods there are not the appropriate ones for the general theory of sheaves on a site. As an example, Hartshorne defines sheafification (and many other things) on a topological space using stalks. However, on a site this is meaningless because there is no analogous notion in general.

The stalk of a sheaf (or presheaf) on a space $X$ at a point corresponds to the inverse image functor via the inclusion $\{\ast\} \to X$. The analogy in the theory of sites would be the inverse image via a morphism from the site with one point (or something equivalent to this). It turns out, fortunately, in etale cohomology this more general notion does make sense, if $\{\ast\}$ is taken to be the spectrum of a separably closed field. So, if $X$ is a scheme, it is not topological points $\{\ast \} \to X$ that lead to the stalk functors in etale cohomology, but the morphisms $\mathrm{Spec} K \to X$ for $K$ a separably closed field (e.g. the separable closure of the residue fields of the topological points).

It is a curious story that there is an even more general theory of points of a (Grothendieck) topos. A point is a geometric morphism (that is, an adjunction where the left adjoint is exact) between the category of sets and the given topos. The direct and inverse image functors obtained from maps $\mathrm{Spec} K \to X$ show that there are lots of “points” in the etale topos. In fact, on general so-called “coherent” topoi there is a general theorem of Deligne that there are always enough points to detect isomorphisms of sheaves. Apparently this is a topos-theoretic reformulation of the completeness theorem in first-order logic! I’m far from understanding the story here though. (more…)