Suppose the Fed buys 1000 dollars worth of T-Bills in the open market to try decrease interest rates and increase the money supply. It does this by printing money and electronically increasing the sellers (a private bank) reserves by 1000 in exchange for the T-Bill thus boosting the private banks reserves and in this way the newly printed money has entered into the economy. Once the banks reserves have increased, the money multiplier process begins and the ultimate increase in the money supply is equal to 1000 dollars times the money multiplier. Depending on the money multiplier (that is, how willing banks are to lend that new money), the increase to the money supply could be only $1000 (in the case of no lending) or potentially a lot more. This much I understand as most textbooks describe the process fully.
But now what does the Fed end up doing with these T-Bills? Since they are T-Bills, they must mature within a year, so when they mature surely the government has to pay the Fed the face value of the T-Bill? The money the government uses to pay the Fed will then leave the economy and the money supply will not have changed ultimately. Even if the initial Fed purchase of the T-Bill had a high money multiplier and thus caused an increase in the money supply much larger than 1000, the government must pay the Fed with real currency or reserves at maturity. Thus, through open market operations, I can only reason that the money supply can increase at best only in the short term (maximum of 1 year when the last purchased T-Bills mature) and hence it does not constitute a permanent increase in the money supply. Also, when the government pays the Fed at maturity, this decrease in the money supply should lead to some deflation counteracting the initial boost that the open market operations were intended to do. So am I missing something or can open market operations only increase the money supply in the short term (being max 1 year assuming the Fed is purchasing T-Bills)? This reasoning would lead me to believe that quantitative easing can also only increase the money supply until the maturity date of the purchased government bonds (around 20 years) at which point the money supply must decrease to where it was before the quantitative easing began?
Any help on this would be most appreciated as all my textbooks describe how the Fed buys T-Bills and how this increases the money but not what the Fed ultimately does with these T-Bills and how that may or may not decrease the money supply.