Nearby cycles for schemes with semi-stable reduction

Let $$R$$ be a henselian DVR with fraction field $$K$$ and residue field $$k$$ of characteristic $$p>0$$. Let $$\overline K$$ be an algebraic closure of $$K$$, $$\overline R$$ the normalization of $$R$$ in $$\overline K$$ and $$\overline k$$ the residue field of $$\overline R$$. Let $$G=\mathrm{Gal}(\overline K/K)$$ be the absolute Galois group of $$K$$. Write $$S, \eta, \overline{\eta}, s, \overline s$$ for the $$\mathrm{Spec}$$ respectively of $$R, K, \overline K, k, \overline k$$. Let $$\Lambda =\mathbb Z_{\ell}$$ with $$\ell \not = p$$ (or $$\mathbb Q_{\ell}$$ or any torsion sheaf where $$p$$ is invertible). Let $$X$$ be a proper scheme over $$S$$. We consider the nearby cycle sheaf $$\mathrm R\Psi\Lambda$$ on the geometric special fiber $$X_{\overline s}$$.

Assume now that $$X$$ has semi-stable reduction, implying that the special fiber $$X_s = \bigcup_{1\leq i \leq r} Y_i$$ is a normal crossing divisor. For $$E\subset \{1,\ldots,r\}$$ let $$Y_E := \bigcap_{i\in E} Y_i$$ and for $$1\leq m \leq r$$ define $$Y^{(m)} = \bigsqcup_{\#E=m} Y_E$$.

In his paper "Exposé 1: autour du théorème de monodromie locale", Illusie explains how to compute the nearby cycle sheaf $$\mathrm R\Psi\Lambda$$. Théorème 3.2.(c) states that there are isomorphisms of $$G-\Lambda-$$sheaves on $$X_{\overline s}$$

• $$\mathrm R^0\Psi\Lambda \simeq \Lambda,$$
• $$\mathrm R^1\Psi\Lambda \simeq \left(\bigoplus_{i} \Lambda_{Y_i}/\Lambda\right)(-1)$$ where $$\Lambda$$ injects diagonally in the sum,
• $$\mathrm R^q\Psi\Lambda \simeq \bigwedge^q \mathrm R^1\Psi\Lambda.$$

I have a doubt regarding this statement that I'd like to clear up. It is my geometric intuition that $$\mathrm R^q\Psi\Lambda$$ should be supported on $$Y^{(q)}_{\overline s}$$, but according to the statement is looks like it is a constant sheaf on the whole of $$X_{\overline s}$$. May I ask which is true?

(Edit) In regards to Will Sawin's reply, I understood the origin of my confusion. I thought that the notation $$\Lambda_{Y_i}$$ merely meant a copy of the constant sheaf $$\Lambda$$ (on the whole special fiber) for each $$Y_i$$, but it actually denotes the constant sheaf $$\Lambda$$ supported on $$Y_i$$.

According to this statement, $$\mathrm R^q\Psi\Lambda$$ is supported on points where $$\mathrm R^1\Psi\Lambda$$ has rank at least $$q$$ (by a property of wedge powers) and thus supported on points where $$\bigoplus_{i} \Lambda_{Y_i}$$ has rank at least $$q+1$$ (by a property of quotients) and thus on $$Y^{(q+1)}$$ (since each summand has rank $$1$$ at $$x$$ if $$x\in Y_i$$ and rank $$0$$ at $$x$$ otherwise).

• Ah, got it! Thank you very much, it's all clear now. Oct 11, 2022 at 0:25