# Subsupermanifolds defined using ideal, transversal example

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 $$\sum_{\mu \in J} f_{\mu} = \sum_{\mu' \in J'} f_{\mu'}$$.

Update: I added a previous version (Version 2) which does not use dimension theory and tubular neighborhoods and it might be possible to generalize it to supermanifolds. However, it requires that $$N$$ can be covered by finitely many coordinate balls.

VERSION 1: FOR SMOOTH MANIFOLDS AND ANY N AND $$\psi$$ (uses that $$N$$ can be covered by $$\mathrm{dim}(N)+1$$ charts and that there is a tubular neighborhood of $$N$$)

It holds $$\mathcal{I} = \Bigl\{ \sum_{i=1}^k (g_i \circ \psi) h_i \ \Bigl|\ k\in\mathbb{N}, g_i\in C^\infty(M), h_i\in C^\infty(L): g_i(N) = 0 \Bigr\}.$$ Given a locally finite collection of smooth functions $$(f_i\in \mathcal{I} \mid i\in I)$$, we will show that $$\sum_{i\in I} f_i \in \mathcal{I}.$$

Lemma 1: We can assume that $$f_i = (g_i \circ \psi) h_i$$ for all $$i\in\mathcal{I}$$.

Proof: Pick a partition of unity $$(\eta_j \mid j\in \mathcal{J})$$ on $$L$$ such that $$\eta_j$$ has compact support for every $$j\in\mathcal{J}$$. Let $$\mathcal{A}:= \mathcal{I}\times\mathcal{J}$$, and define $$f_\alpha:= \eta_j f_i$$ for all $$\alpha=(i,j)\in \mathcal{A}$$. The system $$(f_{\alpha}\mid \alpha\in\mathcal{A})$$ is locally finite, and it holds $$\sum_{\alpha\in\mathcal{A}} f_{\alpha} = \sum_{j\in \mathcal{J}} \eta_j \sum_{i\in\mathcal{I}} f_i = \sum_{i\in\mathcal{I}} f_i.$$ For every $$i\in\mathcal{I}$$, there is an $$m_i\in \mathbb{N}$$ such that $$f_i = \sum_{l=1}^{m_i} (g_{il}\circ\psi)h_{il}$$. It follows that for every $$\alpha = (i,j)\in\mathcal{A}$$, it holds $$f_\alpha = \eta_j f_i = \sum_{l=1}^{m_i} (g_{il}\circ\psi)(\eta_j h_{il}) = \sum_{i=1}^{m_i} (g_{il}\circ\psi)h_{\alpha l} \in \mathcal{I},$$ where we defined $$h_{\alpha l} := \begin{cases} 0 & \text{if }f_\alpha=0, \\ \eta_j h_{il} & \text{otherwise.} \end{cases}$$ Let $$x\in L$$. There is an open neighborhood $$U$$ of $$x$$ and a finite subset $$\mathcal{J}_0\subset \mathcal{J}$$ such that $$\mathrm{supp}(\eta_j) \cap U = 0$$ for all $$j\in \mathcal{J}\backslash\mathcal{J}_0$$. Because $$\sum_{j\in \mathcal{J}}\eta_j = 1$$, it holds $$U\subset \bigcup_{j\in \mathcal{J}_0} \{ x\in M \mid \eta_j(x)\neq 0\}$$. Because $$\bigcup_{j\in \mathcal{J}_0} \mathrm{supp}(\eta_j)$$ is compact, there exists a finite subset $$\mathcal{I}_0\subset \mathcal{I}$$ such that $$\mathrm{supp}(f_i)\cap \bigcup_{j\in \mathcal{J}_0} \mathrm{supp}(\eta_j) = \emptyset$$ for all $$i\in \mathcal{I}\backslash\mathcal{I}_0$$. Suppose that $$\mathrm{supp}(h_{\alpha l})\cap U \neq 0$$ for some $$\alpha = (i,j)$$ and $$l\in \{1,\dotsc,m_i\}$$. Because $$\mathrm{supp}(h_{\alpha l})\subset\mathrm{supp}(\eta_j)$$, it must hold $$j\in \mathcal{J}_0$$. By definition of $$h_{\alpha l}$$, it holds either $$h_{\alpha l} = 0$$ if $$f_\alpha = 0$$, which is equivalent to $$\{x\in M \mid f_i(x)\neq 0\}\cap\{x\in M\mid \eta_j(x)\neq 0\}=\emptyset$$ which is equivalent to $$\mathrm{supp}(f_i)\cap\{x\in M\mid \eta_j(x)\neq 0\} = \emptyset$$, or $$h_{\alpha_l} = \eta_j h_{il}$$. The second option can possibly occur only for $$i\in\mathcal{I}_0$$. This shows that the collection $$((g_{il}\circ\psi)h_{\alpha l} \mid \alpha=(i,j)\in\mathcal{A}, l\in\{1,\dotsc,m_i\})$$ is locally finite. Its sum equals $$\sum_{\alpha\in\mathcal{A}} f_\alpha$$ and hence $$\sum_{i\in \mathcal{I}}f_i$$ by construction. QED

By Lemma 1, we can assume that $$f_i = (g_i\circ\psi)h_i$$ for $$g_i\in C^\infty(M)$$ with $$g_i(N)=0$$ and $$h_i\in C^\infty(L)$$ without lost of generality.

Denote $$k:=\dim(N)$$ and $$n:=\dim(M)$$. Pick a tubular neighborhood $$\mathcal{N}(N)$$ of $$N$$ in $$M$$. The $$k$$-dimensional manifold $$N$$ can be always covered by $$k+1$$ (not necessarily connected) charts $$U_1$$, $$\dotsc$$, $$U_{k+1}$$. Every chart $$U_j$$ on $$N$$ induces a submanifold chart $$V_j = \mathcal{N}(U_j)$$ on $$M$$. Let $$V_0\subset M$$ be an open subset disjoint from $$N$$ such that $$M = \cup_{j=0}^{k+1} V_j$$. Let $$\lambda_0$$, $$\dotsc$$, $$\lambda_{k+1}$$ be a subordinate partition of unity. Let $$\mu$$ be a bump function which equals $$1$$ on $$\mathrm{supp}(\lambda_0)$$ and vanishes on $$N$$.

Let $$(x_j,y_j)\in \mathbb{R}^n$$ be coordinates on $$\mathcal{N}(U_j)$$ such that $$x_j = (x_j^1,\dotsc,x_j^k)$$ gives coordinates on the base and $$y_j = (y_j^1,\dotsc,y_j^{n-k})$$ on fibers. An important feature of $$\mathcal{N}(U_j)$$ is that it contains the vertical line $$\gamma(t) = (x_j,0) + t((x_j,y_j)-(x_j,0))$$ connecting $$(x_j,0)$$ and $$(x_j,y_j)$$. The Fundamental theorem of calculus in the form $$f(\gamma(1))-f(\gamma(0)) = \int_{0}^1 (\nabla f)(\gamma(t))\cdot\gamma'(t) \mathrm{d}t$$ then asserts that the following holds for all $$i\in I$$ and $$j\in \{1,\dotsc,k+1\}$$ on the entire $$\mathcal{N}(U_j)$$: $$(\lambda_j g_i)(x_j,y_j) - \underbrace{(\lambda_j g_i)(x_j,0)}_{=0} = \sum_{a=1}^{n-k} y^a_j \underbrace{\int_{0}^1 \frac{\partial(\lambda_j g_i)}{\partial y^a_j}(x_j,ty_j) \mathrm{d}t}_{\displaystyle=:u_{i a}^j}.$$ Let $$\tilde{y}^a_j$$ and $$\tilde{u}_{ia}^j$$ be the smooth functions on $$M$$ obtained from $$y^a_j$$ and $$u_{ia}^j$$, respectively, by multiplication with a bump function which is $$1$$ on $$\mathrm{supp} \lambda_j$$ and $$0$$ on a neighborhood of the closure of the complement of $$\mathcal{N}(U_j)$$.

For all $$i\in I$$ and $$j\in \{1,\dotsc,k+1\}$$, we have the following relations on $$M$$: $$\lambda_0 g_i = \mu \lambda_0 g_i\quad\text{and}\quad\lambda_j g_i = \sum_{a=1}^{n-k} \tilde{y}^a_j \tilde{u}_{ia}^j.$$ Using this, we compute \begin{align*} \sum_{i\in I} (g_i \circ \psi) h_i &= \sum_{i\in I} \sum_{j=0}^{k+1} (\lambda_j g_i \circ \psi) h_i \\ & = \sum_{i\in I} (\lambda_0 g_i \circ \psi) h_i + \sum_{i\in I} \sum_{j=1}^{k+1} \sum_{a=1}^{n-k} (\tilde{y}^a_j \circ \psi)(\tilde{u}_{ia}^j \circ \psi)h_i \\ & = (\mu\circ\psi)\sum_{i\in I}(\lambda_0 g_i \circ \psi) h_i+ \sum_{j=1}^{k+1} \sum_{a=1}^{n-k} (\tilde{y}^a_j\circ \psi) \sum_{i\in I} (\tilde{u}_{ia}^j \circ \psi)h_i\\ & = (G_0 \circ \psi) H_0 + \sum_{j=1}^{k+1} \sum_{a=1}^{n-k} (G_{ja}\circ\psi)H_{ja}, \end{align*} where we denoted $$G_0:= \mu,\quad G_{ja}:=\tilde{y}^a_j,\quad H_0:=\sum_{i\in I}(\lambda_0 g_i \circ \psi) h_i,\quad H_{ja}:=\sum_{i\in I} (\tilde{u}_{ia}^j\circ \psi)h_i.$$ It holds $$G_0$$, $$G_{ja}\in C^\infty(M)$$, $$G_0(N)=G_{ja}(N) = 0$$, $$H_0$$, $$H_{ja}\in C^\infty(L)$$, and it follows that $$\sum_{i\in I} (g_i \circ \psi) h_i \in \mathcal{I}$$.

VERSION 2: PROOF WHEN $$N$$ CAN BE COVERED BY FINITELY MANY COMPATIBLE COORDINATE BALLS (not using dimension theory and tubular neighborhood)

Write $$\mathbb{R}^n = \mathbb{R}^k\times \mathbb{R}^{n-k}$$ with coordinates $$(x,y)$$. Let $$f: \mathbb{R}^n\rightarrow \mathbb{R}$$ be a smooth function vanishing at $$\{(x,y) \mid x = 0\}$$. Then the fundamental theorem of calculus asserts that the following holds for all $$(x,y)\in \mathbb{R}^n$$: $$f(x,y) = \sum_{j=1}^{k} x^j \int_{0}^1 \frac{\partial f}{\partial x^j}(tx,y) dt.$$ Let $$U_\alpha$$ $$(\alpha\in\mathcal{A})$$ be a cover of $$N$$ by coordinate balls, and let $$\lambda_\alpha$$ $$(\alpha\in\mathcal{A})$$ be a subordinate partition of unity. Suppose that we are given $$\sum_{i\in I} (g_i \circ \psi) h_i$$ as above and we want to show that it lies in $$\mathcal{I}$$. We can even assume that the support lies in an arbitrary small neighborhood of $$\psi^{-1}(N)$$. Using the analytical fact above, there are smooth functions $$x_{\alpha}^j$$ vanishing on $$N$$ and smooth functions $$u^{\alpha}_{ij}$$ for all $$i\in I$$, $$\alpha\in\mathcal{A}$$ and $$j\in\{1,\dotsc,k:=\mathrm{codim} N\}$$ such that $$\lambda_\alpha g_i = \sum_{j=1}^k x_\alpha^j u_{ij}^\alpha.$$ We compute \begin{align*} \sum_{i\in I} (g_i \circ \psi) h_i = \sum_{j=1}^k \sum_{\alpha\in\mathcal{A}} (\lambda_\alpha x^j_\alpha\circ\psi) \sum_{i\in I} (u_{ij}^\alpha \circ \psi) h_i. \end{align*} If $$\mathcal{A}$$ is finite, then we are done.

• Your answer related to the original formulation of this question (I have edited it significantly). In a summary - you give an answer for the case where all the manifolds are ordinary manifolds. However, in supermanifolds, one does not have a full apparautus of tubular neighborhoods, or the fact that k-dimensional manifold can be covered by k+1 charts. – Jan Vysoky Apr 21 at 13:53
• Version 2 can be significantly simplified. By the modification of Lemma 1 above, you can assume that $f_{i} = (g_{i} \circ \psi) h_{i}$, where $\{ supp(g_{i}) \}_{i \in I}$ is locally finite. It then suffices to cover $N$ by finitely many precompact sets $\{U_{\alpha} \}_{\alpha=1}^{m}$ (e.g. coordinate balls). Let $U_{0} := M - N$ and let $\{ \lambda_{\alpha} \}_{\alpha = 0}^{m}$ be the corresponding partition of unity. Note that $\lambda_{0} \in \mathcal{J}_{N}$. Now, as $U_{\alpha}$ are compact for $\alpha > 0$, there is a finite subset $I_{\alpha} \subseteq I$ such that – Jan Vysoky Apr 28 at 12:52
• such that $\lambda_{\alpha} \cdot g_{i} \neq 0$ only for $i \in I_{\alpha}$. Also note that $\lambda_{\alpha} \cdot g_{i} \in \mathcal{J}_{N}$. Then $\sum_{i \in I} (g_{i} \circ \psi) h_{i} = \sum_{i \in I} ((\sum_{\alpha=0}^{m}\lambda_{\alpha}) \cdot g_{i} \circ \psi) h_{i} = (\sum_{i \in I} (g_{i} \circ \psi) h_{i}) \cdot (\lambda_{0} \circ \psi)$ $+ \sum_{\alpha=1}^{m} \sum_{i \in I_{\alpha}} (\lambda_{\alpha} \cdot g_{i} \circ \psi) h_{i}$. The first summand is in $\mathcal{I}$ as $\lambda_{0} \in \mathcal{J}_{N}$ and the rest is a finite sum of elements in $\mathcal{I}$. – Jan Vysoky Apr 28 at 13:02