About the definition of the exterior power of a vector space

Question

There are basically two different definitions of the exterior algebra and exterior powers of a vector space $V$. I here want to concentrate on the one where we define the exterior algebra as a quotient of the tensor algebra. Now for this, i have seen two different definitions: In some books the ideal generated by all tensors of the form $v \otimes v, v \in V$ is factorized out. In other books(e.g. Kostrikin,Manin: Linear Algebra and Geometry) the ideal generated by all tensors of the form $t- sgn(\sigma)\phi_\sigma(t), t \in \underbrace{V \otimes ... \otimes V}_{k}=:T^k(V), \sigma \in S_k, k=1,2,3,...$, where the symmetric group $S_k$ is acting on tensors in the natural way (denoted by $\phi_\sigma$). Or formulated in terms of the $k$-th homogeous part of the respective ideal: the former corresponds to factorizing out the subspace of $T^k(V)$, spanned by all $v_1 \otimes ... \otimes v_k$ with $v_i = v_{i+1}$ (or equivalently with $v_i = v_j$ for some $i \neq j$). The latter corresponds to factorizing out the subspace generated by $v_1 \otimes ... \otimes v_k - sgn(\sigma)v_{\sigma(1)} \otimes ... \otimes v_{\sigma(k)}$. So far, is this correct? My confusion now is the following: One could formulate this also in terms of the respective canonical maps $\lambda_k$ from $V^k$ to the respective quotient of $T^k(V)$. In the first case, the definition says that $\lambda_k$ is an alternating map, while in the second case, the definition says that it is a skew-symmetric map. Now, an alternating map is always skew-symmetric but the converse is only true in characteristic $\neq 2$. Why then do i find these two different definitions which are supposed to be used in all characteristics? Note: i do not ask about the definition one often finds in differential-geometric contexts, where one defines the $k$-th exterior power as the image of the alternation-operator (and which needs division by $k!$). Thanks.

Answer 1

The second definition is just wrong in characteristic $2$. Any author who uses it is either implicitly ignoring the characteristic $2$ case or is being imprecise. You find similar problems with the definition of Clifford algebras; there are again two variants and one of them is wrong in characteristic $2$.