Is there an easy proof of the fact that the intermediate image functor respects weights?

It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially complicated in the case when $j$ is not affine. Does an easier proof (or a plan of it:)) exist? I would like to have a proof that (mostly) relies on the properties of $j_{!*}$ (and on the 'formal' properties of weights).

• What's BBD? Feb 6 '11 at 14:03
• Beilinson, Bernstein, Deligne... Feb 6 '11 at 14:07
• Perhaps this should be added to the question (for the non-experts). Feb 6 '11 at 18:19

The proof in BBD is not that complicated, and it doesn't matter much whether $j$ is affine or not. It uses the three following facts :

• If $f$ is a morphism of schemes, then $f_*$ sends a complex of weight $\geq a$ to a complex of weight $\geq a$, and $f_!$ sends a complex of weight $\leq a$ to a complex of weight $\leq a$ (a very natural property of weights; of course that's not so easy to prove for weights of $\ell$-adic complexes, and it is the main result of Deligne's Weil II). Cf BBD 5.1.14.

• If $K$ is an $\ell$-adic complex, then $K$ is of weight $\leq a$ (resp. $\geq a$) if and only if, for every $k\in\mathbb{Z}$, the $k$th perverse cohomology sheaf of $K$ (call it ${}^pH^k K$) is of weight $\leq a+k$ (resp. $\geq a+k$). Cf BBD 5.4.1. Again, hard to prove, but a natural enough property of weights, and a reason in my opinion why perverse sheaves are so much more natural than constructible sheaves (one out of many).

• If $j$ is a locally closed immersion (more generally, a quasi-finite map), then $j_{!*}$ is the image of ${}^pH^0j_!$ in ${}^pH^0j_*$. This is the definition of the intermediate extension.

Now the result you want is obvious : Take $j:X\rightarrow Y$ a quasi-finite morphism. If the perverse sheaf $K$ on $X$ is of weight $\leq a$, then $j_!K$ is of weight $\leq a$ (as a complex), so the perverse sheaf ${}^pH^0j_!K$ is of weight $\leq a$, and so is its quotient $j_{!*}K$. Likewise for weights $\geq a$, using this time $j_*$.

Note that you could also define $j_{!*}K$ (for $K$ pure of weight $a$) as the weight $\leq a$ part of $j_*K$, or as the weight $\geq a$ part of $j_!K$. I think it's not too hard to recover the usual properties of $j_{!*}K$ from that definition, but I would have to think more to see how to make it work for mixed (but not pure) perverse sheaves.

Edited to add two remarks :

(1) I don't think that it is so hard to go from the affine case to the general case. Consider an open embedding $j:U\rightarrow X$, let $i:Y\rightarrow X$ be the complement. Let $\pi:X'\rightarrow X$ be the blowup of $Y$ in $X$, and $j':U\rightarrow X'$ be the inclusion. Then $j'$ is affine, and, for every perverse sheaf $K$ on $U$, $j_{!*}K$ is a direct factor of ${}^pH^0\pi_*j'_{!*}K$, so the result for $j_{!*}K$ follows if you know it for $j'_{!*}K$, without any need of BBD 5.3.1. (You don't need the decomposition theorem to prove my claim. It is an exercise in perverse sheaves to prove that the map ${}^pH^0\pi_*j'_!K={}^p H^0j_!K\rightarrow j_{!*}K$ factors through a map ${}^pH^0\pi_*j'_{!*}K\rightarrow j_{!*}K$. Likewise, or by duality, there is a natural map $j_{!*}K\rightarrow{}^pH^0\pi_*j_{!*}K$. The composition $j_{!*}K\rightarrow{}^pH^0\pi_*j'_{!*}K\rightarrow j_{!*}K$ is the identity when restricted to $U$, so it is the identity.)

(2) If $K$ is pure, there is a slightly different way to prove what you want (you might be able to do something if $K$ is mixed too, but I didn't try to work it out). Notation : $j$ is an open immersion from $U$ to $X$. First, the problem is local in $X$, so you can assume that $X$ is affine. Then $Y:=X-U$ is defined by a finite number of functions on $X$. By induction over the number of functions necessary to define $Y$, you can reduce to the case where there exists a function $f:X\rightarrow\mathbb{A}^1$ such that $Y=f^{-1}(0)$. Now you can use the result of Beilinson-Bernstein (cf "A proof of Jantzen conjectures") that the Jantzen filtration on $j_!K$ coincides with (a shift of) the weight filtration if $K$ is pure. The Jantzen filtration on $j_!K$ is induced by the monodromy filtration on the maximal extension $\Xi_f K$, and it is an exercise to identify the quotient $j_{!*}K$ of $j_!K$ with one of the graded pieces of this filtration and to conclude that it has the expected weight. This proof avoids BBD 5.2, but it relies on the article of Beilinson-Bernstein instead; as fat as I can tell, the methods Beilinson-Bernstein use to prove the result that you need are natural extensions of the methods of Weil II, and you have to assume Weil II anyway, so maybe this is slightly more natural.

• Your reasoning is certainly very logical; yet it does not seem to follow BBD. In BBD 5.3.2 is not deduced from 5.4.1. Conversely, 5.4.1 relies on 5.3.7, which uses 5.3.5, which uses 5.3.4, and the latter is a corollary from 5.3.2.:) And 5.3.2 is deduced from 5.3.1, which relies on 5.2.1, whereas the proof of the latter is pretty complicated.:) Feb 6 '11 at 19:58
• You did not ask for a proof that follows BBD, you asked for a proof that uses the natural properties of $j_{!*}$ and the formal properties of weights. But none of the properties of weights are easy to prove for $\ell$-adic complexes, so there are no formal properties of weights. I think the best you can hope for is a proof that uses the natural properties of weights. The precise proof of those natural properties is going to depend on your particular context (and on your definitions). For example some of those properties are very hard to prove in the $\ell$-adic world...
– Alex
Feb 6 '11 at 20:16
• ...but very easy to prove for motives. And some of these properties are still conjectural for motives. Note also that your statement is corollary 5.4.3 of BBD, which is deduced in BBD from theorem 5.4.1. So I was actually following BBD afer all. :) As for your original question, if what you want is a proof that is very simple and uses only the definition of weights of $\ell$-adic complexes but not, say, Weil II (which is very hard to prove), then I know of no such proof. Sorry.
– Alex
Feb 6 '11 at 20:18
• Well, I do not know for sure which sort of a reasoning could help me.:) I would prefer one that relies on 5.1.14 of BBD. Could you say more about what is easy to prove for motives? Feb 6 '11 at 20:47
• Okay, the problem is that 5.3.1 (if a perverse sheaf is of weights between $a$ and $b$, then so is any subobject) is crucial, and I do not see a more direct way to get at it than the one in BBD. The definition of weights by looking at stalks is not well adapted to perverse sheaves... About motives, I should not have written "easy", because very few things are easy in maths, and I realized that I called part of your work easy although I do not think it is, so I did not mean any offense. I meant something like "the proof seems more natural to me" (more natural than the proof of Weil II).
– Alex
Feb 6 '11 at 22:49