# Natural Isomorphism of $S(V)$ and $(\bigwedge V)[n]$

Let $V:=\oplus_{j\in\mathbb{Z}}V_j$ be a graded $\mathbb{F}$-vector space over the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given by

$V \otimes W:= \oplus_{j\in \mathbb{Z}}\oplus_{p+q=j}V_p\otimes V_q$ and for the graded vector space $\mathbb{F}[j]$, which is $\mathbb{F}$ in degree $j$ and te zero vector space $\{0\}$ otherwise, the shift $V[j]$ is given by

$V[j]:=\mathbb{F}[j]\otimes V$

We then define a monoidal structure on the category of graded vector space, (more or less) given by the rule on homogeneous elements

$v\otimes w= (-1)^{deg(v)deg(w)}w\otimes v$

Then there is the decalage isomorphism

$dec: V_1\otimes \cdots \otimes V_n \to (V_1 \otimes \cdots \otimes V_n)[n]$

given by $dec(v_1\otimes \cdots \otimes v_n)= (-1)^{\sum_{j=1}^n(n-j)deg(v_j)}(v_1\otimes \cdots \otimes v_n)[n]$.

Now in work on graded (stuff), it is frequently said, that this isomorphism defines a natural isomorphism of the symmetric graded tensor-algebra of $V$ and the antisymmetric graded tensor algebra, that is

$S(V)\simeq (\bigwedge V)[n]$

*The question is: How does the decalage induces such an algebra isomorphism? Or What is the natural isomorphism? *

If $dec$ itself would be the isomorphism, then

$dec(v \vee w)= (dec(v_1)\wedge dec(w))$ should hold, but this isn't true in general.

You have a detailed proof (in a much more general context but easy to read) in Proposition I.4.3.2.1 of Illusie, Complexe Cotangent et Déformations I, Springer LNM 239.

• Maybe I should have left this as a comment instead of as an answer, but I have no choice (few points?).
– jjms
May 23, 2013 at 19:11
• Seems like you are right. Unfortunately I don't speak French. Anyway the answer is ok. May 25, 2013 at 6:17

Let $V$ be a $\mathbb Z$ graded vector space over a field $\mathbb k$ of characteristic $0$ (for simplicity). The suspension $V$ of $V$ is the graded vector space $V:= V\otimes_{\mathbb k} \mathbb k,$ with $\mathbb k$ concentrated in degree $-1$, and $\mathbb k_{-1}=\mathbb k$. With $s$ we denote the "suspension" morphism $s:V\rightarrow V$, $x\mapsto sx:=x$ of degree $-1$; in other words, $|sx|=|x|-1$, where $|\cdot|$ denotes the degree of any homogeneous element in $V$ (or any other graded vector space). In general, we write $s^n: V\rightarrow V[n]$, for any integer $n$. We introduce the graded symmetric resp. antisymmetric algebras over $V$:

$$S(V ) = T (V )/ \langle x \otimes y − (−1)^{|x||y|} y \otimes x \rangle,~~ \text{resp.}~~\Lambda (V ) = T (V )/ \langle x \otimes y + (−1)^{|x||y|} y \otimes x \rangle,$$

For any $n\geq 0$ the decalage is a canonical isomorphism of graded vector spaces (not of algebras)

$$\Phi_n: S_n(V )\rightarrow \Lambda(V)[n],$$

where $$\Phi_n(sx_1,\cdots, sx_n):= (-1)^{\sum_{i=1}^n(n-i)|sx_i|}s^{n}(x_1\wedge\cdots\wedge x_n).$$

Why that sign? The sign follows from the Koszul rule, once one removes the suspensions $s$ from the string $sx_1,\cdots, sx_n$ one by one applying $n$-times the desuspension morphism $s^{-1}$.

• So you say, that the map dec which the user Nevermind posted above is not correct? --Because he uses the wrong degree. ($|v_i|$ instead of $|v_i|$) Indeed the version you presented here works. Jan 3, 2015 at 22:18
• In Nevermind's formula for the decalage I would correct the sign: instead of $deg(v_j)$ one really has $deg(v_j)-1:=deg(s v_j)$, as the map picks up elements in $n$-copies of $V$ (and not $V$). Jan 4, 2015 at 12:21
• Yes. That is what I mean. I think that was the users problem after all. Jan 4, 2015 at 14:41
• I don’t think the sign should be \sum |sx_i|. It should be \sum |x_i|. Let’s write (s(x_1),...,s(x_n)) as (s_1(x_1),...,s_n(x_n)), where the subscript of s_j only means it is applied to x_j. The we move these s_j to obtain (s_1,s_2,...,s_n)(x_1,x_2,...,x_n), which really produces a sign \sum |x_j|. The sign \sum |s(x_j)| that you claimed should come with (s_n,...,s_2,s_1)(x_1,x_2,...,x_n), which I don't think is the case. Mar 25, 2021 at 3:26
• In my last comment I forgot to write the (n-j) before |x_j| and |s(x_j)|. These two signs only differ by n(n-1)/2 on V^n, so they both work for the decalage isomorphism. But I was trying to say \sum (n-j)|x_j| is consistent with the convention (f,g)(v,w)=(-1)^{|g||v|}f(v)g(w), while \sum (n-j)|s(x_j)| is not. Mar 25, 2021 at 3:34