5

I'm curious if it is possible to formulate cup products and prove that they exist in a general way which would subsume a lot of examples: e.g. group cohomology, sheaf cohomology for sheaves on topological spaces, quasicoherent sheaf cohomology, things like etale cohomology where we look at sheaves on more general sites, etc.

In the sources I've looked at (e.g. Cassels-Fröhlich, Lang's Topics in Cohomology of Groups), the existence of cup products is proven in terms of cochains, Čech cohomology, etc., even when more abstract definitions and uniqueness theorems are given.

What I'm looking for is something like this: Let $\mathscr{A}$ is an abelian category with a symmetric monoidal structure $\otimes$ such that $(\mathscr{A}, \otimes)$ satisfies certain conditions (e.g. enough injectives, existence of $\mathrm{Hom}$-objects, whatever other features are common in practice) and $\{H^i\}$ is a universal $\delta$ functor from $\mathscr{A}$ to another abelian symmetric monoidal category $\mathscr{B}$ (assuming whatever we want for $\mathscr{B}$, even $\mathscr{B} = \mathbf{Ab}$, the category of abelian groups). If $\phi \colon H^0(M) \otimes H^0(N) \rightarrow H^0(M \otimes N)$ is an additive bi-functor, then there is a unique sequence of additive bi-functors $\Phi^{p,q} \colon H^p(M) \otimes H^q(N) \rightarrow H^{p+q}(M \otimes N)$ such that:

  • $$\Phi^{0,0} = \phi$$
  • $\Phi$ is a "map of $\delta$-functors separately in $M$ and $N$": if \begin{equation}\tag{1} \label{s1} 0 \rightarrow{A'} \rightarrow A \rightarrow A'' \rightarrow 0 \end{equation} is an exact sequence in $\mathscr{A}$ with \begin{equation} \tag{2}\label{s2} 0 \rightarrow A' \otimes B \rightarrow A \otimes B \rightarrow A'' \otimes B \rightarrow 0 \end{equation} still exact, then $\Phi^{p +1 ,q} \circ (\delta_1 \otimes H^0(\mathrm{id}_B)) = \delta_2 \circ \Phi^{p+1, q}$. Here, $\delta_1 \colon H^p(A'') \rightarrow H^{p+1}(A')$ and $\delta_2 \colon H^p(A'' \otimes B) \rightarrow H^{p+1}(A' \otimes B)$ are the maps provided by the $\delta$-functor structure on $H$ via the sequences (\ref{s1}), (\ref{s2}). Similarly, if we swap the roles of $A$ and $B$, we require that $\Phi^{p +1 ,q} \circ (\delta_1 \otimes H^0(\mathrm{id}_B)) = (-1)^{p} (\delta_2 \circ \Phi^{p+1, q})$.

The answers to this question shed some light on this matter: Suppose we are in a setting where $H^0(M) = \mathrm{Hom}(O, M)$ for some object $O$ of $\mathscr{A}$ (e.g. group cohomology where we can take $O = \mathbf{Z}$, sheaf cohomology where we can take $O = \mathscr{O}_X$, etc.) Then $H^p(M) = \mathrm{Ext}^p(O, M)$, so we should get a pairing $H^p(O) \otimes H^p(O) \rightarrow H^{p+q}(O)$ induced by the 'composition' mapping $\mathrm{Hom}(O, O) \otimes \mathrm{Hom}(O, O) \rightarrow \mathrm{Hom}(O,O)$. I'm not sure exactly how to prove this part in general either, but I've at least seen it discussed in terms of classes of extensions of modules (I'm not sure how generally the result that $\mathrm{Ext}$ describes extension classes holds). This also doesn't allow general group objects, and I'm not sure how to do the extension.

The above question also discusses a more homotopical/$\infty$-categorical way to think about cup products, but I'm not familiar enough in that language to really get what's going on: I'd much prefer an argument working in ordinary abelian categories.

dorebell
  • 2,968
  • 1
    What you really want is to work in with some flavour of derived categories. You want to say that the left derived functor of a lax symmetric monoidal functor is lax symmetric monoidal. This is true, but I don't know how to even state the theorem without using some kind of "homotopical" language (be it dg-categories or stable ∞-categories). – Denis Nardin Dec 12 '17 at 19:11
  • (I would like to write an answer along those lines but I'm a bit hesitant given that you explicitely asked to avoid derived categories...) – Denis Nardin Dec 12 '17 at 19:14
  • 1
    Since $\delta$-functorially ${\rm{Ext}}^i(F,G)={\rm{Hom}}_{D(A)}(F,G[i])$ for $F, G$ in an abelian category $A$ with enough injectives, define pairings ${\rm{Ext}}^i(F,G)\times {\rm{Ext}}^j(G,H)\to {\rm{Ext}}^{i+j}(F,H)$ as ${\rm{Hom}}(F,G[i])\times {\rm{Hom}}(G,H[j])\to {\rm{Hom}}(F,H[i+j])$ via $(f,f')\mapsto f'[i]\circ f$. For the category $A$ of sheaves of modules over a sheaf of rings $O$ we have ${\rm{H}}^i(X,F)={\rm{Ext}}^i(O,F)$, so use ${\rm{Ext}}^j(O,G)\to {\rm{Ext}}^j(F,F\otimes^{\mathbf{L}} G)$ (via $f\mapsto 1_F\otimes^{\mathbf{L}}f$) and $F\otimes^{\mathbf{L}}G\to F\otimes G$. – nfdc23 Dec 12 '17 at 20:22
  • 1
    This construction at the level of 1960's derived category technology adapts well to variants (such as compactly supported etale cohomology cup products), and is quite robust. For instance, it is implicit in the middle of page 303 of the book by Freitag & Kiehl on etale cohomology (just above where they discuss compactifications). This is needed to state the "derived" version of the Kunneth formula in etale cohomology, a formulation that is essential for making a manageable proof with reasonable hypotheses (even for corollaries in the non-derived setting). No need for $\infty$-stuff here. – nfdc23 Dec 12 '17 at 20:28
  • @DenisNardin I'm happy working in the derived category, but there's a big jump from this to dg categories, etc., isn't there? If such language is necessary to discuss this, that's fine, but I'd like to see a result formulated similarly to the one in the question deduced as a consequence of whatever is proved in a fancier setting. – dorebell Dec 13 '17 at 00:38
  • @nfdc23 Thanks, this clears up how to get the $\mathrm{Ext}$ construction to work more generally. Are all cohomologies one should care about of the form $H^0(F) = \mathrm{Hom}(O, F)$ where $O$ is the tensor unit in the category (it seems like you used this to get the map from $\mathrm{Ext}^j(O, G)$ to $\mathrm{Ext}^j(F, F \otimes^L G)$, right?) ? – dorebell Dec 13 '17 at 00:39
  • 1
    What you really meant to ask is if deriving global sections on the category of $O$-modules coincides with deriving on the category of abelian sheaves (nothing else has content). For this it suffices to show injective $O$-modules have vanishing higher abelian sheaf cohomology. That in turn is accomplished by Cech-theoretic methods: see Prop. 2.12(b) in Ch. III of Milne's Etale Cohomology book for the flabbiness criterion that subsumes the desired acyclicity, and note that Lemma 2.4 there works in the $O$-module category by replacing the sheaves $\mathbf{Z}V$ for $j:V\to U$ with $j{!}(O)$). – nfdc23 Dec 13 '17 at 01:49
  • 1
    In Godement's book the Godement resolution $C(F)$ (exists on topological spaces, etale site of a scheme, maybe any topos with enough points) identifies with sheafified Alexander-Spanier cochains, yielding a cup product that for any $F, G$ is induced by any "bilinear pairing" of any acyclic resolutions of $F, G, F\otimes G$ extending the identity on $F\otimes G$ (so agrees with other definitions for special $F, G$). Uniqueness axioms as for group cohomology (so agrees with Ext approach) aren't there but do hold (hint: $F_x\to C^0(F)_x$ has a section); hypergenerality in IV Thm 2 of Swan's book. – nfdc23 Dec 13 '17 at 13:01
  • @nfdc23's comment addresses the question in the cases I'm looking for. I would accept this as an answer or another answer which addresses the generality I asked about in the question. – dorebell Dec 14 '17 at 01:36

1 Answers1

2

For

$\mathcal A=$ category of sheaves of abelian groups on a space X
$\mathcal B=$ category of abelian groups

see Theorems 6.2 and 7.1 in Chapter II of Glen Bredon's Sheaf theory (Spring GTM 170, 2nd edition, 1997).

Communicative Algebra
  • 351