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## higher algebra – Is the rank of free module spectra unique?

$$pi_0: Spto Ab$$ is a direct sum preserving functor, and it sends $$E_1$$-ring spectra to rings, and modules over them to modules over them.

In particular you get a functor $$pi_0: Mod_Rto Mod_{pi_0(R)}$$. If $$R^nsimeq R^m$$ as $$R$$-modules, then $$pi_0(R)^ncong pi_0(R)^m$$ as $$pi_0(R)$$-modules. So if $$pi_0(R)$$ is commutative and nonzero (which is a much weaker hypothesis than $$R$$ having an $$E_infty$$-structure), this implies $$n=m$$.

More generally, it suffices that $$pi_0(R)$$ have the invariant basis property.

Note that conversely, if $$pi_0(R)$$ does not have the invariant basis property, we can find inverse nonsquare matrices $$M,N$$ with coefficients in $$pi_0(R)$$, and you can view them as elements of $$pi_0map_R(R^n,R^m)$$ ($$pi_0map_R(R^m,R^n)$$ respectively), and their matrix product corresponds to the composition up to homotopy, so that $$R^nsimeq R^m$$ as $$R$$-modules.

So it’s an “if and only if” situation with $$pi_0(R)$$.

A related claim is the fact that group-completion $$K$$-theory only sees $$pi_0$$, namely if $$R$$ is a ring spectrum, then the group-completion $$K$$-theory of projective $$R$$-modules (summands of $$R^n$$ for some finite $$n$$, no shifts) is the same as that of $$tau_{geq 0}R$$, which is the same as that of $$pi_0(R)$$.

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## ag.algebraic geometry – Tensors of minimal rank in Schur modules \$S_{lambda}V subset V^{otimes |lambda|}\$

It is well known that for a vector space $$V$$ with $$dim(V)=n+1$$ the $$GL(V)-$$module $$V^{otimes d}$$ splits as a sum of irreducible representations (with suitable multiplicities) $$S_{lambda}V$$, where $$lambda=(lambda_1,dots,lambda_r)$$ is a partition of $$d$$ (with suitable properties).
For example in the case $$d=2$$ we have that $$V otimes V=Sym^2(V) oplus bigwedge^2(V)$$ associated to partitions $$(2,0)$$ and $$(1,1)$$.
Let $$S_{n,d}$$ denote the Segre variety of tensors in $$V^{otimes d}$$ parametrizing tensors of rank $$1$$, i.e. tensors of the form $$T=v_1 otimes dots otimes v_d$$
We say that a tensor $$T$$ has rank $$r$$ if $$T=alpha_1T_1+dots alpha_rT_r$$ where $$T_i$$ are tensors of rank $$1$$ and $$r$$ is the minimum among such expressions. Similar we define the border rank $$bRank(T)=r$$ if $$T$$ can be written as a limit of tensors of rank $$r$$.

Now for the example where $$d=2$$ we have that $$Sym^2(V) cap S_{n,d}=V_{n,d}$$ is the Veronese variety parametrizing symmetric tensors that are power of linear forms, i.e. $$T=v^2$$. In this case the minimal rank of tensors $$T in Sym^2(V)$$ is exactly $$1$$, the best possible. However for the second module we have that $$bigwedge^2(V) cap S_{n,d}= emptyset$$
In paritcular elements $$T in bigwedge^2(V)$$ can be identified with skew matrices and so the minimal possible rank of a skew tensor $$T in bigwedge^2(V)$$ is $$rank(T)=2$$. In the case of matrices border rank and rank coincide and so there is no distinctions.

$$textbf{My question now is the following}$$: given $$d$$ and $$lambda$$ a partition with $$|lambda|=d$$, what can we say about the minimal rank and border rank of tensors $$T in S_{lambda}V$$? Is there an explicit description of the minimal ranks or even a bound as functions of $$d$$ and $$lambda$$?

I was searching for a reference about this fact/computations but I was not able to find a proper one on the internet. Maybe you can help me.

## ag.algebraic geometry – A computation of the rank of the Jacobian of a hyperelliptic curve over a number field using MAGMA

In this paper,
the authors says that, in order to show the rank of a Jacobian over $$mathbb{Q}$$ is 0, they use the L function.

In the section 3.3, the authors compute the rank of the Jacobian of $$X_1(n)$$ for $$n = 13, 16, 18$$ over the splitting field $$K$$ of $$x^3 + x^2 – 4x + 1$$ over $$mathbb{Q}$$.

But in order to compute the rank of $$J_1(18)(K)$$, MAGMA requires the generalized Riemann hypothesis.
And I can’t find the code computing the L-function of the Jacobian of a (non-elliptic) hyperelliptic curve over a number field strictly larger than $$mathbb{Q}$$.

Is there such a function on MAGMA?
Or can we compute the rank in another way?

(If the base field is of exponent $$=2$$, then we can compute the rank of $$J(K)$$ using the Mordell Weil groups over $$mathbb{Q}$$ of twists of $$J$$.)

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