I received a tenant allowance for contruction build on a leased property. The lease states the landlord owns the permanent improvement on the asset. I consider this a credit towards capital expense,

This is something for your Controller to help you with. If you received a live check, make sure you deliver it to accounting immediately.

Accounting for the elements of leases including tenant improvement incentives is not always straight forward and yes, this is something that is dealt with in fixed asset accounting. The cash will be deducted from the final asset, but also the incentives may be required to amortized over the lease term as a credit against rent expense depending on the accounting treatment.

Talk with your Accounting team.

You should also check with your tax accountant in that for federal income tax purposes, a leashold improvement must be amortized over either 15 years under GDS or 39 years under ADS regardless of the lease term.

If the tenant is responsible to pay for the leasehold improvements but received a tenant allowance to cover a portion of those improvements, wouldn't that simply credit the leasehold improvement account?

The tenant allowance is an incentive that should be recorded as a deferred rent liability and amortized as a reduction to lease expense over the lease term.

The leasehold improvements should be amortized over the shorter of their economic lives or the lease term. The tenant allowance should not be netted against leasehold improvements, which should remain on the books as the gross amount.