First recall the tensor operations on a module. Let \(A\) be a ring and \(M\) an \(A\)-module.
Let \(T^n(M)\) be the tensor product of \(M\) with itself \(n\) times, for \(n \geq 1\), and \(T^0(M) = A\). Then \(T(M)=\bigoplus_{n\ge0}T^n(M)\) is an associative \(A\)-algebra, the tensor algebra of \(M\).
The symmetric algebra \(S(M)=\bigoplus_{n\ge0}S^n(M)\) is the quotient of \(T(M)\) by the two-sided ideal generated by all \(x\tensor y-y\tensor x\) for \(x,y\in M\). Then \(S(M)\) is a commutative \(A\)-algebra, and \(S^n(M)\) is the \(n\)th symmetric product. If \(M\) is free of rank \(r\), then \(S(M)\cong A[x_1,\ldots,x_r]\).
The exterior algebra \(\bigwedge(M)=\bigoplus_{n\ge0}\bigwedge^n(M)\) is the quotient of \(T(M)\) by the two-sided ideal generated by all \(x\tensor x\) for \(x\in M\). This ideal contains all \(x \tensor y + y \tensor x\), so \(\bigwedge(M)\) is skew commutative: if \(u \in \bigwedge^r(M)\) and \(v \in \bigwedge^s(M)\), then \(u \wedge v = (-1)^{rs} v \wedge u\). The \(n\)th component \(\bigwedge^n(M)\) is the \(n\)th exterior power.
Now let \((X,\OO_X)\) be a ringed space and \(\mcf\) a sheaf of \(\OO_X\)-modules. Define the tensor, symmetric, and exterior algebras of \(\mcf\) by sheafifying the presheaf which assigns to each open set \(U\) the corresponding operation applied to \(\mcf(U)\) as an \(\OO_X(U)\)-module.
Suppose \(\mcf\) is locally free of rank \(n\ge1\), and let \(r\ge0\). Then \(T^r(\mcf)\), \(S^r(\mcf)\), and \(\bigwedge^r(\mcf)\) are also locally free, of ranks \(n^r\), \(\binom{n + r - 1}{n - 1}\), and \(\binom{n}{r}\) respectively. For rank \(n=0\), all three degree-zero sheaves are \(\OO_X\) and all their positive-degree components are zero.
Again let \(\mcf\) be locally free of rank \(n\ge0\). For \(0\le r\le n\), the multiplication map \(\bigwedge^r\mcf\tensor\bigwedge^{n-r}\mcf\to\bigwedge^n\mcf\) is a perfect pairing: it induces an isomorphism of \(\bigwedge^r\mcf\) with \(\dualof{(\bigwedge^{n-r}\mcf)}\tensor\bigwedge^n\mcf\). As a special case, if \(\mcf\) has rank \(2\) then \(\mcf \cong \dualof{\mcf} \tensor \bigwedge^2 \mcf\).
Let \(0\to\mcf'\to\mcf\to\mcf''\to0\) be an exact sequence of locally free sheaves. Then for any \(r\ge0\) there is a finite filtration \[ S^r(\mcf) = F^0 \supseteq F^1 \supseteq \cdots \supseteq F^r \supseteq F^{r+1} = 0 \] with quotients \(F^p/F^{p+1} \cong S^p(\mcf') \tensor S^{r-p}(\mcf'')\) for each \(p\).
Prove the same statement as (c), with exterior powers instead of symmetric powers. In particular, for finite ranks \(n',n,n''\) with \(n=n'+n''\), prove the induced isomorphism \[ \bigwedge^n\mcf\cong\bigwedge^{n'}\mcf'\tensor\bigwedge^{n''}\mcf''. \]
Let \(f:X\to Y\) be a morphism of ringed spaces and \(\mcf\) an \(\OO_Y\)-module. Then \(f^*\) commutes with all the tensor operations: for example, \(f^*(S^n(\mcf))\cong S^n(f^*\mcf)\) naturally.
Solution
All sheaf tensor products on \(X\) are over \(\OO_X\). Use the tensor-algebra quotient constructions in the statement. Taking a stalk commutes with tensor powers and their symmetric and exterior quotients: every tensor and every defining relation uses finitely many sections, which can be represented on one common neighborhood. Thus the stalks of these sheaf operations are the corresponding operations on the stalk module over its stalk ring.
1. The local ranks in part (a) are\[ \boxed{\operatorname{rank}T^r(\mcf)=n^r,\quad \operatorname{rank}S^r(\mcf)=\binom{n+r-1}{n-1},\quad \operatorname{rank}\bigwedge^r\mcf=\binom nr}\qquad(n\ge1). \]
On an open set with frame \(e_1,\ldots,e_n\) for \(\mcf\), the tensor power has the frame of all words \(e_{i_1}\otimes\cdots\otimes e_{i_r}\). There are \(n^r\) such words. The symmetric quotient identifies words differing by a permutation; its frame is therefore the monomials \(e_1^{a_1}\cdots e_n^{a_n}\) with \(a_i\ge0\) and \(\sum_i a_i=r\). Choosing the positions of \(n-1\) separators among \(r+n-1\) positions counts these as \(\binom{n+r-1}{n-1}\).
For exterior powers, the square relations imply \(e_i\wedge e_j=-e_j\wedge e_i\) and kill a word with a repeated index. Thus the increasing wedges span. For each increasing \(r\)-tuple, the alternating multilinear determinant of its selected coordinates defines a coefficient functional on \(\bigwedge^r\mcf\). Its values on the increasing wedges are one on that tuple and zero on the others, proving linear independence. There are \(\binom nr\) increasing tuples, and none when \(r>n\). These constructions commute with restriction, so they give local frames of the sheaves. In degree zero the empty word is the frame of \(\OO_X\). For \(n=0\), the sheaf itself is zero, so every positive tensor power and its quotients are zero.
2. The multiplication pairing in part (b) is perfect.
Put \(\mathcal E=\bigwedge^r\mcf\), \(\mathcal G=\bigwedge^{n-r}\mcf\), and \(\mathcal L=\bigwedge^n\mcf\). Exterior multiplication defines \[ \mathcal E\longrightarrow\sheafhom(\mathcal G,\mathcal L),\qquad u\longmapsto(v\longmapsto u\wedge v). \] On a framed open set, let \(I\subseteq\{1,\ldots,n\}\) have size \(r\) and let \(I^c\) be its complementary ordered subset. The product \(e_I\wedge e_J\) is zero for \(|J|=n-r\) unless \(J=I^c\). For \(J=I^c\) it is \(\varepsilon_I e_1\wedge\cdots\wedge e_n\), where \(\varepsilon_I\in\{1,-1\}\) is the sign of the permutation listing \(I\) before \(I^c\). Consequently the displayed morphism has a signed permutation matrix in these frames and is an isomorphism over every coefficient ring. By P-AGH251DUALSHEAF, \(\sheafhom(\mathcal G,\mathcal L)\cong\dualof{\mathcal G}\otimes\mathcal L\), giving the required isomorphism. It is canonical because exterior multiplication was defined before choosing the frames. Taking \(n=2\) and \(r=1\) gives the stated rank-two identity. For \(n=r=0\), the pairing is multiplication \(\OO_X\otimes\OO_X\to\OO_X\) and is again perfect.
3. For part (c), define\[ F^p=\im\bigl(S^p(\mcf')\otimes S^{r-p}(\mcf)\longrightarrow S^r(\mcf)\bigr) \quad(0\le p\le r),\qquad F^{r+1}=0. \]Then \(F^p/F^{p+1}\cong S^p(\mcf')\otimes S^{r-p}(\mcf'')\) canonically.
The morphism is induced by inclusion of \(\mcf'\) followed by multiplication. Products with at least \(p+1\) factors from \(\mcf'\) also have at least \(p\) such factors, so \(F^{p+1}\subseteq F^p\), and \(F^0=S^r(\mcf)\).
For local sections \(u_1,\ldots,u_p\) of \(\mcf'\) and \(v_1,\ldots,v_{r-p}\) of \(\mcf''\), choose local lifts \(\widetilde v_j\) in \(\mcf\) and take the class of \[ u_1\cdots u_p\widetilde v_1\cdots\widetilde v_{r-p} \quad\text{in }F^p/F^{p+1}. \] Such lifts exist after passing to a neighborhood of each point, by surjectivity of the sheaf morphism. Changing one lift adds a term with at least \(p+1\) factors from \(\mcf'\), so does not change the class. The rule is multilinear and symmetric in each group of factors and commutes with restriction. It therefore defines the claimed comparison morphism from \(S^p(\mcf')\otimes S^{r-p}(\mcf'')\).
At a stalk \(x\), put \(B=\OO_{X,x}\), \(M'=\mcf'_x\), \(M=\mcf_x\), and \(M''=\mcf''_x\). The quotient \(M''\) is free, so choosing lifts of a basis splits the exact module sequence and identifies \(M\cong M'\oplus M''\). The monomial bases then give \[ S^r_B(M)\cong\bigoplus_{q=0}^rS^q_B(M')\otimes_B S^{r-q}_B(M''), \qquad F^p_x\cong\bigoplus_{q=p}^rS^q_B(M')\otimes_B S^{r-q}_B(M''). \] The comparison morphism on the quotient is the identity on the summand \(q=p\). Thus it is an isomorphism at every stalk, hence an isomorphism of sheaves. These splittings only verify the globally defined morphism; no splitting on \(X\) is asserted or used.
4. For part (d), the exterior filtration is\[ G^p=\im\bigl(\bigwedge^p\mcf'\otimes\bigwedge^{r-p}\mcf \longrightarrow\bigwedge^r\mcf\bigr) \quad(0\le p\le r),\qquad G^{r+1}=0, \]with \(G^p/G^{p+1}\cong\bigwedge^p\mcf'\otimes\bigwedge^{r-p}\mcf''\) canonically.
Exterior multiplication gives the indicated subsheaves and their inclusions \(G^{p+1}\subseteq G^p\). Multiplying \(p\) local sections from \(\mcf'\) with local lifts of \(r-p\) sections from \(\mcf''\) defines a class modulo \(G^{p+1}\). It is independent of the lifts by the same extra-factor argument as in step 3 and is alternating in each group, so it defines the comparison morphism on exterior powers.
3. The increasing wedge basis, ordered with the basis of \(M'\) before that of \(M''\), decomposes \(\bigwedge^r_B M\) by the number \(q\) of factors from \(M'\).The subsheaf \(G^p\) has the summands \(q\ge p\), and its quotient by \(G^{p+1}\) has precisely the summand \(q=p\).The comparison morphism identifies this summand with \(\bigwedge^p_BM'\otimes_B\bigwedge^{r-p}_BM''\).It is therefore an isomorphism on every stalk.
For \(r=n=n'+n''\), all graded quotients are zero except the one with \(p=n'\). Indeed, \(p>n'\) makes \(\bigwedge^p\mcf'\) zero, while \(p<n'\) makes \(\bigwedge^{n-p}\mcf''\) zero. Thus \(G^{n'}=\bigwedge^n\mcf\) and \(G^{n'+1}=0\), and the remaining quotient gives \[ \bigwedge^{n'}\mcf'\otimes\bigwedge^{n''}\mcf''\xrightarrow{\cong}\bigwedge^n\mcf. \] Explicitly, it wedges a top wedge from \(\mcf'\) with any lifts of a top wedge from \(\mcf''\). Lift-independence and the fixed order of the factors determine its sign and make it canonical.
5. Pullback commutes naturally with every tensor, symmetric, and exterior power, proving part (e).
The universal properties of the sheaf tensor algebra and its defining quotients give comparison morphisms from each pullback to the corresponding operation on \(f^*\mcf\). At \(x\in X\), put \(y=f(x)\), \(A=\OO_{Y,y}\), \(B=\OO_{X,x}\), and \(M=\mcf_y\). Then \((f^*\mcf)_x\cong B\otimes_A M\). For \(q>0\), the tensor comparison is \[ B\otimes_A T_A^q(M)\longrightarrow T_B^q(B\otimes_A M), \qquad b\otimes(m_1\otimes\cdots\otimes m_q) \longmapsto(b\otimes m_1)\otimes(1\otimes m_2)\otimes\cdots\otimes(1\otimes m_q). \] Its inverse sends \((b_1\otimes m_1)\otimes\cdots\otimes(b_q\otimes m_q)\) to \((b_1\cdots b_q)\otimes(m_1\otimes\cdots\otimes m_q)\). The balancing relations make both formulas well-defined, and the two composites fix elementary tensors.
Passing to the commutation relations gives the same inverse formulas for symmetric powers. Passing to the square relations gives them for exterior powers: the resulting maps are alternating in the module arguments, and multilinearity handles arbitrary sums in \(B\otimes_A M\). Equivalently, scalar extension is right exact, so these quotient presentations are preserved; no flatness hypothesis is needed. In degree zero the comparison is \(B\otimes_A A\cong B\). These formulas are natural in the ring map and in \(M\) and preserve multiplication of homogeneous elements. Hence the sheaf comparison morphisms are isomorphisms on all stalks and give the claimed graded-algebra isomorphisms as well.
6. Q.E.D.
Steps 1, 2, 3, 4, and 5 prove parts (a), (b), (c), (d), and (e), respectively.