I am currently learning about algebraic viewpoint on closed embedded subsupermanifolds. In particular, I am struggling with something which should be ''easy to see''. Namely, I refer to the lemma just above Proposition 3.2.9 in "Introduction to the Theory of Supermanifolds" by D. A. Leites (http://iopscience.iop.org/0036-0279/35/1/R01).

The setting: Suppose we have a map $\psi: L \rightarrow M$ of supermanifolds, such that it is transversal to a given closed embedded subsupermanifold $N \subseteq M$.

Now, to every closed embedded subsupermanifold, one may assign a unique ideal $\mathcal{J}_{N} \leq C^{\infty}(M)$ in the superalgebra of its global functions, defined by $\mathcal{J}_{N} = \{ f \in C^{\infty}(M) \; | \; j^{*}(f) = 0 \}$, where $j: N \rightarrow M$ is the embedding. It can be then shown that $j^{\ast}: C^{\infty}(M) \rightarrow C^{\infty}(N)$ is a superalgebra epimorphism and $\mathcal{J}_{N}$ is its kernel, whence $C^{\infty}(N) \cong C^{\infty}(M) / \mathcal{J}_{N}$

Next, by definition of the map of supermanifolds, we have a superalgebra morphism $\psi^{\ast}: C^{\infty}(M) \rightarrow C^{\infty}(L)$. We can thus consider a subset $\psi^{\ast}(\mathcal{J}_{N}) \subseteq C^{\infty}(L)$. As $\psi^{\ast}$ is usually not surjective, this is in general not an ideal. We can, however, consider the ideal $\mathcal{I} = \langle \psi^{\ast}(\mathcal{J}_{N}) \rangle \leq C^{\infty}(L)$ generated by this subset.

THE ACTUAL QUESTION: $\mathcal{I}$ is supposed to have the following property: Let $\{ f_{\mu} \}_{\mu \in J}$ be any collection of functions in $\mathcal{I}$, such that $\{ supp(f_{\mu}) \}_{\mu \in J}$ is locally finite. Then also their sum $\sum_{\mu \in J} f_{\mu}$ must be function in $\mathcal{I}$.

I am stuck at this very point. According to D.A. Leites, this should be easy to see.

For the sake of completeness, let me recall some definitions:

A collection $\{C_{\mu} \}_{\mu \in J}$ of subsets of any topological space is locally finite, if for every compact subset $K$, $C_{\mu} \cap K \neq \emptyset$ only for finitely many $\mu \in J$. On (ordinary) manifolds, this is equivalent to every point $m$ having a neighborhood $U_{m}$, such that $C_{\mu} \cap U_{m} \neq \emptyset$ only for finitely many $\mu \in J$.

For a function $f$ on a supermanifold $M$, its support $supp(f)$ is a set of points $m$ of the underlying manifold $|M|$, where the germ $[f]_{m}$ of $f$ does not vanish.

Some comments: (i) The ideal $\mathcal{J}_{N}$ has this very property. For each $U \subseteq M$ open, let $j^{\ast}_{U}: C^{\infty}(U) \rightarrow C^{\infty}(U \cap N)$ be the superalgebra morphism induced by the pullback by $j$. For each point $m \in M$, we can pick a precompact neighborhood $U_{m}$. If $f_{\mu} \in \mathcal{J}_{N}$ for every $\mu \in J$, we obtain

$(j^{\ast}( \sum_{\mu \in J} f_{\mu} ))|_{U_{m} \cap N} = j^{\ast}_{U_{m}}( (\sum_{\mu \in J} f_{\mu} )|_{U_{m}}) = \sum_{\mu \in J} j^{\ast}_{U_{m}}( f_{\mu}|_{U_{m}}) = \sum_{\mu \in J} (j^{\ast}(f_{\mu}))|_{U_{m} \cap N} = 0,$

where it was important that after restriction to $U_{m}$, the sum is finite. But $m$ was arbitrary and $\{ U_{m} \cap N \}_{m \in M}$ forms an open cover of $N$, which proves that $j^{\ast}( \sum_{\mu \in J} f_{\mu} ) = 0$, that is $\sum_{\mu \in J} f_{\mu} \in \mathcal{J}_{N}$.

(ii) I don't know whether the transversality of $\psi$ to $N$ is somewhat important at this point.

(iii) D. A. Leites only assumes that the indexing set $J$ is countable. This is not very important though, as for general $J$, one can always find a countable subset $J'$, such that $\sum_{\mu \in J} f_{\mu} = \sum_{\mu' \in J'} f_{\mu'}$.

I am currently learning about algebraic viewpoint on closed embedded subsupermanifolds. In particular, I am struggling with something which should be ''easy to see''. Namely, I refer to the lemma just above Proposition 3.2.9 in "Introduction to the Theory of Supermanifolds" by D. A. Leites (http://iopscience.iop.org/0036-0279/35/1/R01).

Remark: I know there is an answer below. It, however, does not work for supermanifolds (I am not able to generalize).

The setting: Suppose we have a map $\psi: L \rightarrow M$ of supermanifolds, such that it is transversal to a given closed embedded subsupermanifold $N \subseteq M$.

Now, to every closed embedded subsupermanifold, one may assign a unique ideal $\mathcal{J}_{N} \leq C^{\infty}(M)$ in the superalgebra of its global functions, defined by $\mathcal{J}_{N} = \{ f \in C^{\infty}(M) \; | \; j^{*}(f) = 0 \}$, where $j: N \rightarrow M$ is the embedding. It can be then shown that $j^{\ast}: C^{\infty}(M) \rightarrow C^{\infty}(N)$ is a superalgebra epimorphism and $\mathcal{J}_{N}$ is its kernel, whence $C^{\infty}(N) \cong C^{\infty}(M) / \mathcal{J}_{N}$

Next, by definition of the map of supermanifolds, we have a superalgebra morphism $\psi^{\ast}: C^{\infty}(M) \rightarrow C^{\infty}(L)$. We can thus consider a subset $\psi^{\ast}(\mathcal{J}_{N}) \subseteq C^{\infty}(L)$. As $\psi^{\ast}$ is usually not surjective, this is in general not an ideal. We can, however, consider the ideal $\mathcal{I} = \langle \psi^{\ast}(\mathcal{J}_{N}) \rangle \leq C^{\infty}(L)$ generated by this subset.

THE ACTUAL QUESTION: $\mathcal{I}$ is supposed to have the following property: Let $\{ f_{\mu} \}_{\mu \in J}$ be any collection of functions in $\mathcal{I}$, such that $\{ supp(f_{\mu}) \}_{\mu \in J}$ is locally finite. Then also their sum $\sum_{\mu \in J} f_{\mu}$ must be function in $\mathcal{I}$.

I am stuck at this very point. According to D.A. Leites, this should be easy to see.

For the sake of completeness, let me recall some definitions:

A collection $\{C_{\mu} \}_{\mu \in J}$ of subsets of any topological space is locally finite, if for every compact subset $K$, $C_{\mu} \cap K \neq \emptyset$ only for finitely many $\mu \in J$. On (ordinary) manifolds, this is equivalent to every point $m$ having a neighborhood $U_{m}$, such that $C_{\mu} \cap U_{m} \neq \emptyset$ only for finitely many $\mu \in J$.

For a function $f$ on a supermanifold $M$, its support $supp(f)$ is a set of points $m$ of the underlying manifold $|M|$, where the germ $[f]_{m}$ of $f$ does not vanish.

Some comments: (i) The ideal $\mathcal{J}_{N}$ has this very property. For each $U \subseteq M$ open, let $j^{\ast}_{U}: C^{\infty}(U) \rightarrow C^{\infty}(U \cap N)$ be the superalgebra morphism induced by the pullback by $j$. For each point $m \in M$, we can pick a precompact neighborhood $U_{m}$. If $f_{\mu} \in \mathcal{J}_{N}$ for every $\mu \in J$, we obtain

$(j^{\ast}( \sum_{\mu \in J} f_{\mu} ))|_{U_{m} \cap N} = j^{\ast}_{U_{m}}( (\sum_{\mu \in J} f_{\mu} )|_{U_{m}}) = \sum_{\mu \in J} j^{\ast}_{U_{m}}( f_{\mu}|_{U_{m}}) = \sum_{\mu \in J} (j^{\ast}(f_{\mu}))|_{U_{m} \cap N} = 0,$

where it was important that after restriction to $U_{m}$, the sum is finite. But $m$ was arbitrary and $\{ U_{m} \cap N \}_{m \in M}$ forms an open cover of $N$, which proves that $j^{\ast}( \sum_{\mu \in J} f_{\mu} ) = 0$, that is $\sum_{\mu \in J} f_{\mu} \in \mathcal{J}_{N}$.

(ii) I don't know whether the transversality of $\psi$ to $N$ is somewhat important at this point.

(iii) D. A. Leites only assumes that the indexing set $J$ is countable. This is not very important though, as for general $J$, one can always find a countable subset $J'$, such that $\sum_{\mu \in J} f_{\mu} = \sum_{\mu' \in J'} f_{\mu'}$.

I am currently learning about algebraic viewpoint on closed embedded sub(super)manifold. In particular, I am struggling with something which should be ''easy to see''. Namely, I refer to the lemma just above Proposition 3.2.9 in "Introduction to the Theory of Supermanifolds" by D. A. Leites (http://iopscience.iop.org/0036-0279/35/1/R01).

To simplify things, I believe the question can be formulated and proved in the same way just for ordinary smooth manifolds (the tools at hand are essentially the same). I will recall the only necessary definition.

Definition: A collection $\{ C_{\mu} \}_{\mu \in J}$ of subsets of a given smooth manifold $M$ is said to be locally finite, if each point $m \in M$ has an open neighborhood $U$, such that $C_{\mu} \cap U \neq \emptyset$ only for finitely many $\mu \in J$.

Remarks: (i) One often encounters an equivalent definition (e.g. in the reference above), which says that $\{C_{\mu} \}_{\mu \in J}$ is locally finite, if for every compact subset $K \subseteq M$, $C_{\mu} \cap K \neq \emptyset$ only for finitely many $\mu \in J$. On manifolds (second countable Hausdorff topological spaces) both definitions agree.

(ii) If $\{ f_{\mu} \}_{\mu \in J}$ is a collection of (global) smooth functions in $M$, such that $\{ supp(f_{\mu}) \}_{\mu \in J}$ is locally finite, it makes sense to define a smooth function $\sum_{\mu \in J} f_{\mu}$. Each point has a a neighborhood $U_{m}$ obtained from the local finiteness, hence we may declare $(\sum_{\mu \in J} f_{\mu})|_{U_{m}} := \sum_{\mu \in J} f_{\mu}|_{U_{m}}$ and then use the gluing property of the sheaf $C^{\infty}$ for the open cover $\{ U_{m} \}_{m \in M}$ to define the unique global function.

The setting: Suppose we have a smooth map $\psi: L \rightarrow M$, such that it is transversal to a given closed embedded submanifold $N \subseteq M$, that is for all $l \in \psi^{-1}(N)$, one has $T_{\psi(l)}M = (T_{l}\psi)(T_{l}L) + T_{\psi(l)}N$.

Now, to every closed embedded submanifold, one may assign a unique ideal $\mathcal{J}_{N} \leq C^{\infty}(M)$ in the algebra of smooth functions, defined by $\mathcal{J}_{N} = \{ f \in C^{\infty}(M) \; | \; f|_{N} = 0 \}$. If $j: N \rightarrow M$ is the embedding, then $j^{\ast}: C^{\infty}(M) \rightarrow C^{\infty}(N)$ is an algebra epimorphism and $\mathcal{J}_{N}$ is its kernel, whence $C^{\infty}(N) \cong C^{\infty}(M) / \mathcal{J}_{N}$

Next, we have an algebra morphism $\psi^{\ast}: C^{\infty}(M) \rightarrow C^{\infty}(L)$. We can thus consider a subset $\psi^{\ast}(\mathcal{J}_{N}) \subseteq C^{\infty}(L)$. As $\psi^{\ast}$ is usually not surjective, this is in general not an ideal. We can, however, consider the ideal $\mathcal{I} = \langle \psi^{\ast}(\mathcal{J}_{N}) \rangle \leq C^{\infty}(L)$ generated by this subset.

I am stuck at this very point. According to D.A. Leites, this should be easy to see.

Some comments: (i) The ideal $\mathcal{J}_{N}$ has this very property. For each $U \subseteq M$ open, let $j^{\ast}_{U}: C^{\infty}(U) \rightarrow C^{\infty}(U \cap N)$ be the algebra morphism induced by the pullback by $j$. For each point $m \in M$, there is its neighborhood $U_{m}$ obtained from the local finiteness of $\{ supp(f_{\mu}) \}_{\mu \in J}$. If $f_{\mu} \in \mathcal{J}_{N}$ for every $\mu \in J$, we obtain

(ii) I don't know whether the transversality of $\psi$ to $N$ is somewhat important at this point. The thing is that it is a well-known fact that $\psi^{-1}(N) \subseteq L$ is a closed embedded submmanifold, hence it comes with its own ideal $\mathcal{J}_{\psi^{-1}(N)} \leq C^{\infty}(L)$. It should be true that $\mathcal{J}_{\psi^{-1}(N)} = \mathcal{I}$. However, in order to prove this, one needs the aforementioned property of $\mathcal{I}$. Moreover, I actually need this property of $\mathcal{I}$ to prove that $|\varphi|^{-1}(N)$ is a submanifold (in the supermanifold setting, the definition of an inverse image is a bit complicated).

I am currently learning about algebraic viewpoint on closed embedded subsupermanifolds. In particular, I am struggling with something which should be ''easy to see''. Namely, I refer to the lemma just above Proposition 3.2.9 in "Introduction to the Theory of Supermanifolds" by D. A. Leites (http://iopscience.iop.org/0036-0279/35/1/R01).

The setting: Suppose we have a map $\psi: L \rightarrow M$ of supermanifolds, such that it is transversal to a given closed embedded subsupermanifold $N \subseteq M$.

Now, to every closed embedded subsupermanifold, one may assign a unique ideal $\mathcal{J}_{N} \leq C^{\infty}(M)$ in the superalgebra of its global functions, defined by $\mathcal{J}_{N} = \{ f \in C^{\infty}(M) \; | \; j^{*}(f) = 0 \}$, where $j: N \rightarrow M$ is the embedding. It can be then shown that $j^{\ast}: C^{\infty}(M) \rightarrow C^{\infty}(N)$ is a superalgebra epimorphism and $\mathcal{J}_{N}$ is its kernel, whence $C^{\infty}(N) \cong C^{\infty}(M) / \mathcal{J}_{N}$

Next, by definition of the map of supermanifolds, we have a superalgebra morphism $\psi^{\ast}: C^{\infty}(M) \rightarrow C^{\infty}(L)$. We can thus consider a subset $\psi^{\ast}(\mathcal{J}_{N}) \subseteq C^{\infty}(L)$. As $\psi^{\ast}$ is usually not surjective, this is in general not an ideal. We can, however, consider the ideal $\mathcal{I} = \langle \psi^{\ast}(\mathcal{J}_{N}) \rangle \leq C^{\infty}(L)$ generated by this subset.

I am stuck at this very point. According to D.A. Leites, this should be easy to see.

For the sake of completeness, let me recall some definitions:

A collection $\{C_{\mu} \}_{\mu \in J}$ of subsets of any topological space is locally finite, if for every compact subset $K$, $C_{\mu} \cap K \neq \emptyset$ only for finitely many $\mu \in J$. On (ordinary) manifolds, this is equivalent to every point $m$ having a neighborhood $U_{m}$, such that $C_{\mu} \cap U_{m} \neq \emptyset$ only for finitely many $\mu \in J$.

For a function $f$ on a supermanifold $M$, its support $supp(f)$ is a set of points $m$ of the underlying manifold $|M|$, where the germ $[f]_{m}$ of $f$ does not vanish.

Some comments: (i) The ideal $\mathcal{J}_{N}$ has this very property. For each $U \subseteq M$ open, let $j^{\ast}_{U}: C^{\infty}(U) \rightarrow C^{\infty}(U \cap N)$ be the superalgebra morphism induced by the pullback by $j$. For each point $m \in M$, we can pick a precompact neighborhood $U_{m}$. If $f_{\mu} \in \mathcal{J}_{N}$ for every $\mu \in J$, we obtain

(ii) I don't know whether the transversality of $\psi$ to $N$ is somewhat important at this point.