$\DeclareMathOperator\Cl{Cl}$**Update Feb 3 '12: now with a question 2 which is much more elementary (and should be well-known!).**


Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-module, and $g:L\to k$ be a quadratic form, i. e., a map for which the map $L\times L\to k,\ \left(x,y\right)\mapsto g\left(x+y\right)-g\left(x\right)-g\left(y\right)$ is $k$-bilinear and which satisfies


$$g\left(\lambda x\right)=\lambda^2 g\left(x\right)\quad\text{for all $x\in L$ and $\lambda\in k$.}$$


We define the *Clifford algebra* $\Cl\left(L,g\right)$ as the tensor algebra $\otimes L$ (all tensors are over $k$) divided by the two-sided ideal generated by $x\otimes x-g\left(x\right)$ for all $x\in L$.


**Question 1:** Is $\Cl\left(L,g\right)$ isomorphic to ${\bigwedge}L$ as a $k$-module? Is the associated graded object of $\Cl\left(L,g\right)$ isomorphic to ${\bigwedge}L$ as a $k$-algebra?


**Remark:** The answer is [yes](http://www.cip.ifi.lmu.de/%7Egrinberg/algebra/algebra.html#chevalley) if the quadratic form $g$ comes from a bilinear (not necessarily symmetric!) form. As a consequence, the answer is yes if $L$ is a free $k$-module or, more generally, a direct sum of quotient $k$-modules of $k$ (in fact, it is easy to see that in this case, every quadratic form on $L$ comes from a bilinear form). (I think that in the case when $L$ is a finite-free $k$-module, the answer "yes" can also be proven by the diamond lemma, though I have not checked.) I am interested in the "perverse" cases when none of these holds, but I suffer from a lack of perversion: I can't name any such case. So here is the question which obviously needs to be addressed first:


**Question 2:** Find a commutative ring $k$ with $1$ and a $k$-module $L$ with a quadratic form (this is defined as above) which doesn't come from any bilinear (symmetric or not) form on $L$. (A quadratic form $g$ is said to *come from* a bilinear form $h$ if we have $g\left(v\right)=h\left(v,v\right)$ for every $v\in L$.)


**Note:** The Clifford algebra of a quadratic form is, in some sense, a "little sister" of the universal enveloping algebra of a Lie or pseudo-Lie algebra. ("Little sister" not in the historical sense, but in the sense of partly having the same properties, but them being easier to prove in the Clifford case than in the universal enveloping algebra case.) The above question asks for a kind of [Poincaré–Birkhoff–Witt (PBW) theorem](http://en.wikipedia.org/wiki/Poincar%25C3%25A9%25E2%2580%2593Birkhoff%25E2%2580%2593Witt_theorem) for the Clifford algebra. (Note that the PBW theorem itself requires some niceness conditions such as $L$ being finite-free or $k$ being a $\mathbb Q$-algebra, so it wouldn't surprise me if the Clifford case also doesn't work in full generality. But the opposite case wouldn't surprise me either, because PBW is substantially harder than PBW for Clifford algebras, even in characteristic $0$.)



Clifford PBW theorem for quadratic form
If someone could check the below I'd be very indebted.


Found the counterexample (to Question 1 and thus also to Question 2). It is inspired by the counterexample to ring-theoretical PBW in P. M. Cohn, *A remark on the Birkhoff-Witt theorem*. J. London Math. Soc. 38 1963 pp. 197-203, MR0148717 (though I still don't know whether Cohn's counterexample is true for all primes $p$ - but here I only need $p=2$).


Let $k$ be the commutative ring $\mathbb F\_2 \left[\alpha,\beta,\gamma\right] / \left(\alpha^2,\beta^2,\gamma^2\right)$.


Let $L$ be the $k$-module $\left\langle x,y,z\right\rangle / \left\langle \alpha x - \beta y - \gamma z\right\rangle$.


Define a map $q:L\to k$ by $q\left(\overline{ax+by+cz}\right) = a^2+b^2+c^2$ for all $a,b,c\in k$. This map $q$ is easily seen to be well-defined and a quadratic form; it also satisfies $q\left(\overline x\right)=q\left(\overline y\right)=q\left(\overline z\right)=1$.


In the Clifford algebra $\mathrm{Cl}\left(L,q\right)$, we have $\left(\overline{\alpha x-\beta y-\gamma z}\right)\cdot\overline x = \alpha \underbrace{\overline{x}\cdot\overline{x}}\_{=q\left(\overline x\right)=1} - \beta \overline y \overline x - \gamma \overline z \overline x = \alpha - \beta \overline y \overline x - \gamma \overline z \overline x$. Since the left hand side of this equality is $0$ (because $\overline{\alpha x-\beta y-\gamma z}=0$), we thus have $ \alpha - \beta \overline y \overline x - \gamma \overline z \overline x = 0$. Multiplied with $\beta\gamma$, this becomes $\alpha\beta\gamma = 0$ (because $\beta^2=0$ and $\gamma^2=0$ make the other two terms disappear). This must hence hold in $\mathrm{Cl}\left(L,q\right)$. But this does not hold in $k$, and thus not in $\wedge L$ either, so we cannot have an isomorphism.


