The Stacks project

24.8 Sheaves of graded bimodules and tensor-hom adjunction

Please skip this section.

Definition 24.8.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. A graded $(\mathcal{A}, \mathcal{B})$-bimodule is given by a family $\mathcal{M}^ n$ indexed by $n \in \mathbf{Z}$ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps

\[ \mathcal{M}^ n \times \mathcal{B}^ m \to \mathcal{M}^{n + m},\quad (x, b) \longmapsto xb \]

and

\[ \mathcal{A}^ n \times \mathcal{M}^ m \to \mathcal{M}^{n + m},\quad (a, x) \longmapsto ax \]

called the multiplication maps with the following properties

  1. multiplication satisfies $a(a'x) = (aa')x$ and $(xb)b' = x(bb')$,

  2. $(ax)b = a(xb)$,

  3. the identity section $1$ of $\mathcal{A}^0$ acts as the identity by multiplication, and

  4. the identity section $1$ of $\mathcal{B}^0$ acts as the identity by multiplication.

We often denote such a structure $\mathcal{M}$. A homomorphism of graded $(\mathcal{A}, \mathcal{B})$-bimodules $f : \mathcal{M} \to \mathcal{N}$ is a family of maps $f^ n : \mathcal{M}^ n \to \mathcal{N}^ n$ of $\mathcal{O}$-modules compatible with the multiplication maps.

Given a graded $(\mathcal{A}, \mathcal{B})$-bimodule $\mathcal{M}$ and an object $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we use the notation

\[ \mathcal{M}(U) = \Gamma (U, \mathcal{M}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{M}^ n(U) \]

This is a graded $(\mathcal{A}(U), \mathcal{B}(U))$-bimodule.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a right graded $\mathcal{A}$-module and let $\mathcal{N}$ be a graded $(\mathcal{A}, \mathcal{B})$-bimodule. In this case the graded tensor product defined in Section 24.6

\[ \mathcal{M} \otimes _\mathcal {A} \mathcal{N} \]

is a right graded $\mathcal{B}$-module with obvious multiplication maps. This construction defines a functor and a functor of graded categories

\[ \otimes _\mathcal {A} \mathcal{N} : \textit{Mod}(\mathcal{A}) \longrightarrow \textit{Mod}(\mathcal{B}) \quad \text{and}\quad \otimes _\mathcal {A} \mathcal{N} : \textit{Mod}^{gr}(\mathcal{A}) \longrightarrow \textit{Mod}^{gr}(\mathcal{B}) \]

by sending homomorphisms of degree $n$ from $\mathcal{M} \to \mathcal{M}'$ to the induced map of degree $n$ from $\mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ to $\mathcal{M}' \otimes _\mathcal {A} \mathcal{N}$.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{N}$ be a graded $(\mathcal{A}, \mathcal{B})$-bimodule. Let $\mathcal{L}$ be a right graded $\mathcal{B}$-module. In this case the graded internal hom defined in Section 24.7

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, \mathcal{L}) \]

is a right graded $\mathcal{A}$-module with multiplication maps1

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n_\mathcal {B}(\mathcal{N}, \mathcal{L}) \times \mathcal{A}^ m \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{n + m}_\mathcal {B}(\mathcal{N}, \mathcal{L}) \]

sending a section $f = (f_{p,q})$ of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n_\mathcal {B}(\mathcal{N}, \mathcal{L})$ over $U$ and a section $a$ of $\mathcal{A}^ m$ over $U$ to the section $f a$ if $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{n + m}_\mathcal {B}(\mathcal{N}, \mathcal{L})$ over $U$ defined as the family of maps

\[ \mathcal{N}^{-q - m}|_ U \xrightarrow {a \cdot -} \mathcal{N}^{-q}|_ U \xrightarrow {f_{p, q}} \mathcal{M}^ p|_ U \]

We omit the verification that this is well defined. This construction defines a functor and a functor of graded categories

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, -) : \textit{Mod}(\mathcal{B}) \longrightarrow \textit{Mod}(\mathcal{A}) \quad \text{and}\quad \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, -) : \textit{Mod}^{gr}(\mathcal{B}) \longrightarrow \textit{Mod}^{gr}(\mathcal{A}) \]

by sending homomorphisms of degree $n$ from $\mathcal{L} \to \mathcal{L}'$ to the induced map of degree $n$ from $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, \mathcal{L})$ to $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, \mathcal{L}')$.

Lemma 24.8.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a right graded $\mathcal{A}$-module. Let $\mathcal{N}$ be a graded $(\mathcal{A}, \mathcal{B})$-bimodule. Let $\mathcal{L}$ be a right graded $\mathcal{B}$-module. With conventions as above we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{B})}( \mathcal{M} \otimes _\mathcal {A} \mathcal{N}, \mathcal{L}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}( \mathcal{M}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, \mathcal{L})) \]

and

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}( \mathcal{M} \otimes _\mathcal {A} \mathcal{N}, \mathcal{L}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {A}^{gr}( \mathcal{M}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}(\mathcal{N}, \mathcal{L})) \]

functorially in $\mathcal{M}$, $\mathcal{N}$, $\mathcal{L}$.

Proof. Omitted. Hint: This follows by interpreting both sides as $\mathcal{A}$-bilinear graded maps $\psi : \mathcal{M} \times \mathcal{N} \to \mathcal{L}$ which are $\mathcal{B}$-linear on the right. $\square$

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. As a special case of the above, suppose we are given a homomorphism $\varphi : \mathcal{A} \to \mathcal{B}$ of graded $\mathcal{O}$-algebras. Then we obtain a functor and a functor of graded categories

\[ \otimes _{\mathcal{A}, \varphi } \mathcal{B} : \textit{Mod}(\mathcal{A}) \longrightarrow \textit{Mod}(\mathcal{B}) \quad \text{and}\quad \otimes _{\mathcal{A}, \varphi } \mathcal{B} : \textit{Mod}^{gr}(\mathcal{A}) \longrightarrow \textit{Mod}^{gr}(\mathcal{B}) \]

On the other hand, we have the restriction functors

\[ res_\varphi : \textit{Mod}(\mathcal{B}) \longrightarrow \textit{Mod}(\mathcal{A}) \quad \text{and}\quad res_\varphi : \textit{Mod}^{gr}(\mathcal{B}) \longrightarrow \textit{Mod}^{gr}(\mathcal{A}) \]

We can use the lemma above to show these functors are adjoint to each other (as usual with restriction and base change). Namely, let us write ${}_\mathcal {A}\mathcal{B}_\mathcal {B}$ for $\mathcal{B}$ viewed as a graded $(\mathcal{A}, \mathcal{B})$-bimodule. Then for any right graded $\mathcal{B}$-module $\mathcal{L}$ we have

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{gr}({}_\mathcal {A}\mathcal{B}_\mathcal {B}, \mathcal{L}) = res_\varphi (\mathcal{L}) \]

as right graded $\mathcal{A}$-modules. Thus Lemma 24.8.2 tells us that we have a functorial isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{B})}( \mathcal{M} \otimes _{\mathcal{A}, \varphi } \mathcal{B}, \mathcal{L}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}( \mathcal{M}, res_\varphi (\mathcal{L})) \]

We usually drop the dependence on $\varphi $ in this formula if it is clear from context. In the same manner we obtain the equality

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{gr}_\mathcal {B}( \mathcal{M} \otimes _\mathcal {A} \mathcal{B}, \mathcal{L}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {A}^{gr}(\mathcal{M}, \mathcal{L}) \]

of graded $\mathcal{O}$-modules.

[1] Our conventions are here that this does not involve any signs.

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FR4. Beware of the difference between the letter 'O' and the digit '0'.