# Scheme $\text{Spec}(S^*\text{Hom}(V,W)^{\vee})$ from "The Geometry of Moduli Spaces of Sheaves"

I have a couple of questions about the notations & their meaning used in "The Geometry of Moduli Spaces of Sheaves" by Huybrechts & Lehn, in Example 2.2.2 (page 38):

$$V$$ is assumed to be a be a finite dimensional vector space over field $$k$$. Let $$0 \leq r \leq dim(V)$$. the Grassmann functor is defined in the text as $$\underline{Grass}(V,r):(Sch/k)^o \rightarrow (Sets)$$ that associates every $$k$$-scheme $$S$$ of finite type to the set of all sub sheaves $$\mathfrak{U} \subset \mathcal{O}_S \otimes V$$ such that the quotient $$F = (\mathcal{O}_S \otimes V)/\mathfrak{U}$$ is locally free of rank $$r$$.

For each $$r$$-dim linear subspace $$W \subset V$$ we consider the sub functor $$\mathcal{G}_W$$ of $$\underline{Grass}(V,r)$$, that maps every $$k$$-scheme S to those locally free quotients $$F$$ for which the canonical composition $$\mathcal{O}_S \otimes W \rightarrow \mathcal{O}_S \otimes V \rightarrow F$$ is an isomorphism and therefore it induces a splitting of the inclusion $$W \subset V$$ (...splitting as what and in which category?)

From this we can conclude that $$\mathcal{G}_W$$ is represented by by an affine space $$G_W \subset \text{Spec}(S^*\text{Hom}(V,W)^{\vee})$$, "corresponding to homomorphisms that split the inclusion map $$W \subset V$$" ???

Questions:

Firstly (possibly a stupid question) what is the scheme $$\text{Spec}(S^*\text{Hom}(V,W)^{\vee})$$ concretly? my guess would be that $$S^*\text{Hom}(V,W)^{\vee}$$ is nothing by the symmetric algebra $$Sym(Hom(V, W)^{\vee})$$, is this true? If my guess is correct then I might suppose that $$\text{Spec}(S^*\text{Hom}(V,W)^{\vee})$$ is the scheme that represents the functor $$T : (Sch) \to (Sets)$$ assigning $$S \mapsto H^0(S, \mathcal{O}_S \otimes_{\mathcal{O}_{\mathbb{Z}}} Sym(Hom(V,W)^{\vee}))$$ Does it make sense ? Is there a more concrete description of $$\text{Spec}(S^*\text{Hom}(V,W)^{\vee})$$ ?

And why gives $$\mathcal{O}_S\otimes V \to \mathcal{O}_S\otimes W$$ an $$S$$-point of $$\text{Spec}(S^*\text{Hom}(V,W)^{\vee})$$ as stated in the text?

Some background: The notation "$$S$$-point" of a (affine) scheme means that via Yoneda embedding we interpret this scheme as a functor $$(Sch/k) \to (Sets)$$ given by $$S \mapsto \text{Spec}(S^*\text{Hom}(V,W)^{\vee})(S)= Hom(S, \text{Spec}(S^*\text{Hom}(V,W)^{\vee})$$. Why is $$\mathcal{O}_S\otimes V \to \mathcal{O}_S\otimes W$$ an element/"point" of it?

• I don't have time to make this a detailed answer, so here are a few remarks. A) Indeed, $S^*$ is the symmetric algebra. Moreover, $\mathrm{Spec}(S^*\mathrm{Hom}(V,W)^{\vee})$ is a coordinate-free way to get the affine scheme $\mathbb A^{\dim V\dim W}$ corresponding to the vector space $\mathrm{Hom}(V,W)^{\vee})$ and the claim is that $\mathcal G_W$ is the sub-space corresponding to those homomorphisms inducing a splitting of vector spaces. And the very rough idea why is that the splitting (of vector bundles) given by $\mathcal O\otimes W\to \mathcal O\otimes V\to F$ (to be cont'd)
– Ben
Commented Sep 26, 2019 at 17:25
• (cont.) is really a family of splittings of vector spaces parametrised by $S$. Does that help?
– Ben
Commented Sep 26, 2019 at 17:25
• a bit, but not fully. what I don't understand why every homomorphism $\phi:\mathcal{O}_S\otimes V \to \mathcal{O}_S\otimes W$ correspond exactly to the set $Hom(S, Sym(Hom(V,W)^{\vee})$? first of all what is $\mathcal{O}_S\otimes W$? if we set $n:= dim(V), m:= dim(W)$ then $\mathcal{O}_S\otimes V \simeq \mathcal{O}_S^n, \mathcal{O}_S \otimes W \simeq \mathcal{O}_S^m$ and $\phi: \mathcal{O}_S^n \to \mathcal{O}_S^m$ is fully determined by the images of $e_1,e_2,...,e_n$ (= free "base" of $\mathcal{O}_S^n$) in $\mathcal{O}_S^m$ with free base $f_1,...,f_m$.
– user705174
Commented Sep 27, 2019 at 18:37
• therefore for every point $s \in S$ we obtain equatons $\phi_s(e_i)= \sum_{k=1}^n a_{i,j;s} e_k$ in $\mathcal{O}_{S,s}^m$. Since $Hom(V,W)$ is a vector space of dimension $n \cdot m$ has canonical base $e_i \otimes f_j$ every morphism $\phi$ is determined by a "vector" $\sum_{i,j} ^{nm} c_{ij} e_i \otimes f_j$ with $c_{ij} \in \mathcal{O}_S(S)$ and therefore every morphism $\phi$ is
– user705174
Commented Sep 27, 2019 at 18:38
• determined by "linear" component of $\mathcal{O}_S \otimes_Z Sym(Hom(V,W)^{\vee})$. why do we need "higher graded elements" like $(e_1 \otimes f_1) \wedge (e_2 \otimes f_2) \in \mathcal{O}_S \otimes_Z Sym(Hom(V,W)^{\vee})$ do describe a morphism $\phi: \mathcal{O}_S\otimes V \to \mathcal{O}_S\otimes W$ ?
– user705174
Commented Sep 27, 2019 at 18:38

Even though later we will be interested in the rather specific $$k$$-vector space $$\hom(V,W)$$ of linear maps, for now, it is conceptually easier to consider any finite-dimensional $$k$$-vector space $$V$$. I like to think of it as a vector bundle over $$\mathrm{Spec}(k)$$. And a vector bundle (thought of as a sheaf) is ought to have a "total space" – a scheme $$|V|$$ over $$k$$ whose sections correspond to the elements of $$V$$, universally. Meaning that for every $$k$$-scheme $$X$$, the $$k$$-morphisms $$X\to |V|$$, being the same as the sections of the pull-back $$|V|\times_kX$$, should be the global sections of the pulled back vector bundle $$V\otimes_k\mathcal{O}_X$$, i.e., $$V\otimes_k\mathcal{O}_X(X)$$. For short, we want $$\hom_k(X,|V|) = V\otimes_k\mathcal{O}_X(X)$$. I claim that this is solved by $$\mathrm{Spec}(S^\bullet V^\vee)$$. In fact, \begin{align*} \hom_k(X, \mathrm{Spec}(S^\bullet V^\vee))&=\hom_{k\text{-alg}}(S^\bullet V^\vee,\mathcal{O}_X(X))\\ &\cong\hom_{k\text{-vect}}(V^\vee,\mathcal O_X(X))\\ &\cong V\otimes_k\mathcal O_X(X), \end{align*} where the bottom isomorphism comes from the natural map $$V\otimes_k\mathcal O_X(X)\to \hom_{k\text{-vect}}(V^\vee,\mathcal O_X(X))$$, mapping a homogeneous element $$v\otimes f$$ to the homomorphism $$(\varphi\mapsto \varphi(v)\cdot f)\in \hom_{k\text{-vect}}(V^\vee,\mathcal O_X(X))$$. It's an isomorphism since $$V$$ is finite-dimensional.
Returning to $$\hom(V,W)$$ and its associated affine scheme $$|\hom(V,W)| = \mathrm{Spec}(S^\bullet \hom(V,W)^\vee)$$: Let $$U\subset\hom(V,W)$$ be the affine subspace consisting of those linear maps $$V\to W$$ which restrict to the identity on $$W$$; equivalently, "that split the inclusion map $$W\subset V$$". Moreover, for every $$k$$-algebra $$\mathcal O_X(X)$$ it makes sense to define $$U\otimes_k\mathcal O_X(X)\subset \hom(V,W)\otimes_k \mathcal O_X(X)$$ in an obvious way and there exists an affine sub-scheme $$\mathcal U\subset |\hom(V,W)|$$ such that via the above isomorphisms, $$\hom(X,\mathcal U) = U\otimes_k \mathcal O_X(X)$$. (I'll leave the details up to you.)
What the authors claim is simply that via the indicated map $$\mathcal G_W\to |\hom(V,W)|$$, $$\mathcal G_W$$ is isomorphic to $$\mathcal U$$. (Let me know in the comments if you need more clarifications or more hints towards the proof.)
• I think I understand the point. $|V|$ might be thought as be "that what it represent" and and if we find another scheme as $\mathrm{Spec}(S^\bullet V^\vee)$ in our case that represents the same functors as induced by $V$, then Yoneda lemma tells that thy coinside up to isomorphism. hope that is it. Thank you very much for your time and help!