# About lattice of finitely generated projective module

Let $$A$$ be euclidian ring and $$K$$ be its field of a fraction.Let $$(V,B)$$ be a nonzero IPS (inner product space) over $$K$$. A finitely generated A-submodule $$L ⊆ V$$is said to be an $$A$$ -lattice in $$V$$ if $$L$$ contains a $$K$$-basis of $$V$$ . As we have already observed, $$L$$ must be $$A$$-free since it is torsion-free, and it is easy to verify that an $$A$$-basis for $$L$$ will automatically be a $$K$$-basis for $$V$$.

Attempt: I know that $$V$$ is finitely generated projective module since $$(V, B)$$ is inner product space and $$B$$ is regular bilinear form. Now question is what is the meaning of $$L$$ contains a $$K$$-basis of $$V$$ and how to verify $$A$$-basis for L will automatically be a $$K$$-basis for $$V$$ ; it certainly means we can extend from basis of $$L$$ to basis of $$V$$?

The meaning of $$L$$ containing a $$K$$-basis for $$V$$ is that there are $$\ell_1,\ldots,\ell_n\in L\subseteq V$$ forming a basis for $$V$$ as a $$K$$-vector space (where $$n=\dim_K V$$).

Now suppose that we have $$l_1,\ldots,l_r\in L$$ are a basis for $$L$$ as a free $$A$$-module. (Note the different style of $$l$$). We want to prove that $$r=n$$ and that the elements $$l_i$$ are linearly independent over $$K$$, so that they form a $$K$$-basis.

If $$\sum_{i=1}^r c_il_i=0,$$ with $$c_i\in K$$, we can multiply by a common denominator $$d\ne 0\in A$$, so that $$c_id\in A$$. Then we get $$\sum_{i=1}^r (c_id)l_i=0,$$ but the $$l_i$$ were an $$A$$-basis for $$L$$, so $$c_id=0$$ for all $$i$$, and since $$d\ne 0$$, $$c_i=0$$ for all $$i$$.

As for why $$n=r$$, observe that $$V=KL=K\otimes_A L=K\otimes_A A^r = K^r$$, so $$r=\dim K^r=\dim V =n$$.

• @mathsstudent I don't usually participate in chat, if you have another question, feel free to ask it, and drop me a comment linking to it and I'll check it out. – jgon Mar 27 at 16:26
• @mathsstudent Yes – jgon Mar 27 at 16:29
• I ask doubt below sir. – maths student Mar 27 at 16:38
• math.stackexchange.com/questions/3164762/… I have ask it? – maths student Mar 27 at 16:49
• I have got corollary 3.9 can you please explain to me how to prove note below corollary 3.9. – maths student Mar 27 at 17:06